THE ENCHANTING ARITHMETIC

PREFACE

Mathematics is an important subject without which we cannot imagine the modern world of science and technology. In mathematics, arithmetic is the back bone of mathematical sciences. Here we deal with some basic concepts of arithmetic and their applications which you will also feel enchanting and make more interesting to read a lot. The approach could be different than normal.

This book is written in a simple language with examples, and definitions for easy understanding of the readers from various backgrounds. I sincerely hope this book will provide a better understanding of concepts in arithmetic. Any suggestions and constructive criticism towards this book can be sent to sri_chemist@yahoo.co.in and will be gratefully acknowledged.

Author

Copyright@2011

All rights reserved.

No part of this book may be reproduced in any form or by any electronic or mechanical means including information storage and retrieval systems, without permission in writing from the Publisher. The only exception is by a reviewer, who may quote short excerpts in a review.

AB Publishing House, Div: Sydney, Australia

http://www.abpublishinghouse.com

http://www.abpublishinghouse.com/store

First printing 2011

S.No.

Contents

Page no.

1.

Basic arithmetic

9

2.

Addition

21

3.

Subtraction

26

4.

Multiplication

28

5.

Powers

44

6.

Division

64

7.

Palindrome numbers

81

8.

Arithmetic of zero and infinity

86

9.

Sequences and series

92

10.

Polygonal numbers

98

11.

Magic mathematics

120

Annexure 1. Glossary

Annexure 2. Comparison of mathematical properties

134

142

Subject index

A

Hendecagonal number

Amicable number

Heptadecagonal number

Arithmetic operations

Heptagonal hexagon number

Arithmetic progression

Heptagonal number

Armstrong number

Heptagonal pentagon number

Ascending number see Climbing number

Heptagonal pyramidal number

Automorphic number

Heptagonal square number

B

Heptagonal triangular number

Binary system

Hexadecagonal number

BODMAS

Hexadecimal number system

Branches of arithmetic

Hexagonal number

C

Hexagonal pentagon number

Captive zero

Hexagonal pyramidal number

Cardinal number

Hexagonal square number

Chenshuwen’s sequence

Hexagonal triangular number

Climbing number

Highest common factor

Complementary number

I

Composite number

Icosagonal number

Consecutive number

Icosidigonal number

Coprime number

Icosihenagonal number

Cube number

Icosiheptagonal number

Cullen number

Icosihexagonal number

Cyclic number

Icosinonagonal number

D

Icosioctagonal number

Decagonal number

Icosipentagonal number

Decenary system

Icositetragonal number

Demlo number

Icositrigonal number

Descending number

Imperfect division

Digital root (or) Digit sum

Infinite imperfect division

Divisibility tests

Infinity

Dodecagonal number

K

Dozenal system

Kaprekar number

E

L

Eratosthenes sieve

Leading zero

Even number

Least common multiplier

F

M

Face value

Magic square

Fibonacci series

Magic star

Finite imperfect division

Magic triangle

Following number see Consecutive number

Magic square border puzzle

Fractions

Mersenne number

G

Mirror image number see Reflux number

Geometric progression

Mods

H

N

Harmonic progression

Natural number

Harshad number

Negative number

Non recurring infinite division

Pyramidal number

Nonadecagonal number

Pythagorean triples

Nonagonal heptagon number

R

Nonagonal hexagon number

Rational number

Nonagonal number

Reflux number

Nonagonal octagon number

Repunit number

Nonagonal pentagon number

Ratio

Nonagonal square number

Rachnisky sequence

Nonagonal triangular number

Ramanujan number

Number digit see Digital root

Remainder theorem

Numbering system

Rectangular number

O

Radix

Oblong number see Pronic number

Real number see Natural number

Octadecagonal number

Recurring infinite division

Octagonal heptagon number

S

Octagonal hexagon number

Seesaw number

Octagonal number

Square number

Octagonal pentagon number

Square triangular number

Octagonal square number

Square pyramidal triangular number

Octagonal triangular number

Square pyramidal square number

Octahedral number

Scientific notation number

Octal number system

Square pyramidal number

Odd number

Square pyramidal tetrahedral number

Ordinal number

T

P

Ternary number system

Probable number

Tetradecagonal number

Pronic number

Tetrahedral number

Proportionality

Tetrahedral square number

Place value

Tetrahedral triangular number

Prime number

Trailing zero

Palindromic prime number

Transpose number

Pascal series

Triacontagonal number

Pentadecagonal number

Triangular number

Pentagonal number

Triangular pyramidal number

Pentagonal number

Tridecagonal number

Pentagonal square number

Twin prime number

Pentagonal triangular number

W

Percentages

Whole number

Perfect division

Z

Perfect number

Zero

References

1. Ya. Perelman: Mathematics can be fun, Mir publishers, Moscow (1985).

2. J. T. Glover: Vedic mathematics for schools Book 1, 2 & 3, Motilal Bansaridass Publishers, Delhi.

3. H.K.Gupta: Vedic mathematics, bpi India private limited.

4. T. A. Ramasubban: Speed arithmetic.

5. H.C.Khare: Issues in Vedic mathematics, Rashtriya Veda vidya Pratishthan.

6. Kenneth R. Williams: Discover Vedic mathematics, Motilal Bansaridass Publishers, Delhi.

7. Jagadhguru Sri Bharti Krishna Maharaj: Vedic mathematics, Motilal Bansaridass Publishers, Delhi.

8. Jayanth V. Narlikar: Fun and fundamentals of mathematics, University press.

9. Amit Garg: Fun with numbers, Pustak Mahal, Delhi.

10. D.Jaganmohan Rao: Mathematics quizmaster, Neelkamal publications private limited, New Delhi.

11. Vandana Sighal: Vedic mathematics for all ages, Motilal Bansaridass Publishers, Delhi.

12. D.Jaganmohan Rao: Sensations in mathematics, Neelkamal publications private limited, New Delhi.

13. Dr. S.K.Kapoor: Vedic mathematics skills, Lotus press, New Delhi.

14. Dhaval Bathia: Vedic mathematics, Jaico publishing house, Mumbai.

15. G.Polya: How to solve it, Prentice hall of India private limited, New Delhi.

16. K. Venkataraman: Think without ink, TWI foundation, Bangalore.

17. Shailesh Shirali: A primer on number sequences, University press.

18. Shailesh Shirali: A primer on divisibility, University press.

19. Shailesh Shirali: Adventures in problem solving, University press.

20. T.S. Bhanu Murthy: A modern introduction to Indian mathematics, Wiley Eastern limited, New Delhi (1995).

21. Shakuntala Devi: The book of numbers, Orient paperbacks, New Delhi (1984).

22. Ravi Narula: Brain Teasers, Jaico publishing house, Mumbai (2000).

23. Shakuntala Devi: The joy of numbers, Orient paperbacks, New Delhi (2000).

24. Shakuntala Devi: Puzzles to puzzle you, Orient paperbacks, New Delhi (2000).

Features

• Important arithmetic terms and definitions

• Lucid explanation of terminologies

• Glossary of arithmetic numbers

• Comparative study of mathematical properties

• Understanding explanation of basic arithmetic

• A guide for preparing competitive examinations

• Clearly an enchanting arithmetic book for further studies

About the author

Mr.R. Sridharan M.Sc., B.Ed. is working as a Senior Instructor to the B.E. and B.Tech. students in the Department of Chemistry at Sri Chandrasekharendra Saraswathi Viswa Maha Vidyalaya University, Kanchipuram, Tamil Nadu. Though the core subject of the author is Chemistry, he is interested in the areas of Mathematics especially in Arithmetic.

1. BASIC ARITHMETICS

(1.1)The set of positive integers 1, 2, 3, 4, 5 … naturally thought of first in the process of counting, is one component of what are called as real numbers or natural numbers and denoted as ‘N’. Real numbers are positive numbers. It starts from 1 onwards. Real numbers form the alphabets of arithmetic. The notation used is

N = {1, 2, 3, 4, 5, 6…}

Note: - Counting of the objects by 1, 2, 3, 4, 5 … is called as cardinal numbers. Counting of the objects by first, second, third, fourth, fifth… is called as ordinal numbers.

Whole numbers include zero in the real number system.

To this set ‘N’ was added the set of negative integers -1, -2, -3, -4, -5…and India contributed the “zero” to this to make up

Z = {…-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5…}

as the set of positive and negative integers including zero.

Then the need arose to include the fractions, both proper and improper fractions to form the set Q of rational numbers. This is definable as a number capable of having thrown in the form of p/q where p and q are integers and q≠0.

Even after this the system was found to be inadequate when one thought of numbers like √2, √3, √7 even ∏ (used in the calculation of circumference or area of a circle) are not numbers that can be put in the form p/q where p and q are integers and q≠0, a definable classification of rational numbers. Logically that which is not rational, got designated as irrational numbers.

This completes the formulation of real numbers – either as rational or irrational.

In this treatise, mostly simplified methods of algebraical operations, among numbers are dealt with and hence often we talk about integers.

(1.2) Consecutive number or Following number: -

(a)It is the following number which present successively in the given number sequence. For example in the case of real number sequence 6 is the following number of 5 and in the case of square number sequence 36 is the following number of 25 and so on.

(b)If a, b and c are any three consecutive numbers or following numbers in real number sequence then b^2 = ac+1. For example 2, 3 and 4 then 3^2 = (2×4) +1 in real number sequence.

(1.3) Even number or Paired number: -

(a)If a real number is divisible by 2 or multiples of 2 means it is an even number. We can also call even numbers as paired numbers.

For example 14 is divisible by 2 and so it is an even number.

The even number series is 0, 2, 4, 6, 8, 10, 12…

(b)Even numbers end with the digits 0, 2, 4, 6 or 8.

(c)All even numbers are divisible numbers but all divisible numbers are not even numbers. For example 27.

(d)All the real numbers as well as negative numbers when multiplied by 2 will give even numbers.

(1.4) Odd number or unpaired number: -

(a)If a real number is indivisible by 2 means it is an odd number. We can also call odd numbers as unpaired numbers.

For example 15 is indivisible by 2 and so it is an odd number.

The odd number series is 1, 3, 5, 7, 9, 11 …

(b)Odd numbers end with the digits 1, 3, 5, 7 or 9.

(c)All prime numbers are odd numbers but all odd numbers are not prime numbers. For example 13 is an odd number as well as a prime number but 21 is an odd number but not a prime number.

(1.5) Composite number: -

(a)All the real numbers except the prime numbers are called as composite numbers. It contains both odd number and even number and so called as composite numbers.

The composite number series is 4, 6, 8, 9, 10, 12, 14, 15 …

(b) A formula for finding out the composite number is (n^4 + 4n) is a composite number provided that n>1.

(1.6) Reflux number or Mirror image number: -

(a)This is the number obtained by the mirror image of the original number. For example if the number 176 is placed before a mirror, the image obtained is 671 and is its reflux number.

(b)If the reflux number is same as that of the original number itself means it is called as palindrome number. Note that all palindrome numbers are reflux numbers but all reflux numbers are not palindrome numbers.

(1.7) Pronic number or Oblong number or Heteromecic number: -

(a)It is a number which is the product of two consecutive integers, that is, n (n + 1) where ‘n’ is a real number. This can also be represented as n^2+n. For example the pronic number series is 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462…

(b)The nth pronic number is twice the nth triangular number. For example the fifth pronic number is 20 and its corresponding fifth triangular number is 10.

(c) The nth pronic number is ‘n’ less than the nth square number. For example the fifth pronic number is 20 and its corresponding fifth square number is 25.

(1.8) Transpose number: -

(a)The numbers derived by interchanging the digits of the original number are called as transpose number. For example 21 and its transpose number is 12

(b)Obviously for a single digit number there is only one transpose number available and is the original number itself. For 213 the transposed numbers are 231, 132, 123, 312 and 321.

(1.9) Cyclic numbers: -

(a)The numbers formed by constructing new numbers from the same digits by moving them cyclically are called as cyclic numbers.

For example for 2387 the cyclic numbers are 3872, 8723 and 7238.

(b) For an ‘n’ digit number, ‘n’ cyclic numbers are possible. That is for a five digits number five cyclic numbers are possible including the original number itself.

(c) All cyclic numbers are transpose numbers but all transpose numbers will not become cyclic numbers.

(d) Obviously for a single digit number there is only one cyclic number available and is the original number itself.

(e) For a two digit number except for palindromic two digit number, two cyclic numbers are possible. For a palindromic two digit number only one cyclic number available and is the original number itself.

(f) For a three digit number except for repunit palindromic three digit number, three cyclic numbers are possible. For a repunit palindromic three digit number only one cyclic number available and is the original number itself. The same is applicable for higher digit numbers.

(g) The resultant of summation of cyclic numbers with ‘n’ digits is found by multiplying the sum of digits of the original number and ‘n’ 1s. For example the summation of cyclic numbers for 123 is (1+2+3) ×111=666. That is 123+231+312=666. Likewise for a 4 digit number we have to multiply with 1111 and so on.

(1.10) Probable number: -

(a)The number of probable numbers that can be written using the given digits is called as the probable number. For example the number of probable three digit numbers that can be formed by using ‘m’ digits is m^3. That is by using 1 and 2 the number of probable three digit numbers are 2^3 (here m=2). The numbers are 111, 222, 112, 221, 211, 122, 121 and 212.

(b) For repunit numbers there is no probable number.

(1.11) Repunit numbers: -

(a)Numbers formed with repeated digit is called as repunit numbers. For example numbers formed with only 1’s are called as repunit numbers. That is 1, 11, 111, 333, 77… are called as repunit numbers. These are denoted by Rn where ‘n’ is the number of repeated digit.

(b) Repunit numbers will not become power numbers. That is 2222, 33, 888 and so on like numbers are not power numbers.

(c)When the number 111 is multiplied by 12, 23, 34, 45, 56, 67, 78 and 89, that is multiplication by consecutive ascending number the answer differs from previous one only by 1221.

111×12=1332

111×23=2553

111×34=3774

If the repunit number 111 is multiplied by 123, 234, 345 and so on the answer differs from previous one only by 12321. Similarly when the repunit number 111 is multiplied by 1234, 2345, 3456… the answer differs from previous one only by 123321.

(d)If we multiply 11 by repeated digits of 2 such as 2, 22, 222… then the answer will contain the first and last digits as 2 and 4s between the two 2s. The number of 4s is given by 1 less than the number of 2s present in the multiplier. For example 11×222=2442.

(1.12) Number digit or Digit sum: -

(a) The unit digit derived after the sum of the digits of a number is called as the number digit. This can also be called as digital root. Obviously the number digit for a single digit number is the same number itself.

For example the number digit of 4519 is 4+5+1+9=19 and again 1+9=10 to give 1+0=1. That is 1 is the number digit of 4519.

(b) A quicker way of finding the number digit is by the method of casting out of nines. For example 2351068 then 3+6=9; 1+8=9 and so the left out digits are 5, 2 and 0 and the addition of which will give the number digit, that is 7. Another example is 991 and if we remove 9 then the left out number 1 is the digit sum.

(c) Number digit of a number will not be zero. Also the number digit of a number is always a positive number.

(d) The number digit of a negative number like -732 is -3. Since the number digit should not be a negative number add 9 to the resultant to get 6 and is the number digit of the given negative number.

(e) Every number has its unique number digit and the number digit may be associated with more than one number. For example 3254 has the number digit 5 while the number digit 5 is also possessed by numbers 32, 41, 536, 2327 and so on.

(1.13) Complementary number: - It is the number complement to the given decimal number with respect to the base 10. It is found out by the subtraction of all the digits from nine but the last from ten. For example the complementary number of 123 is 877. Another example is the complementary number of 758120 is 241880 (Note: - The last digit zero in the number is subtracted from 10, keeping zero 1 is carried over. Again subtract the second digit from 9 and the resultant is added with the carryover).

(1.14) Climbing number or Ascending number: - The digits in the number 1789 are in ascending order. So it is called as climbing number or ascending number. The next climbing number after 1789 is 2345.

(1.15) Descending number: - (a) If the digits in a number are in descending order means then it is called as descending number. For example 8731. (b) The reflux number of the descending number is ascending number and vice versa.

(1.16) Seesaw numbers: - (a) If the left hand side of the digits in a number with respect to the middle digit/s is increasing/decreasing and again the right hand side of the number is decreasing/increasing then the number is called as seesaw number. For example 786, 7553, 613... (b) Positive seesaw number: - If the left hand side of the digits in a number with respect to the middle digit/s is increasing and again the right hand side of the number is decreasing, then the number is called as positive seesaw number. For example 785, 694... (c) Negative seesaw number: - If the left hand side of the digits in a number with respect to the middle digit/s is decreasing and again the right hand side of the number is increasing then the number is called as negative seesaw number. For example 612, 902, 923...

(1.17) Pandigital number: - (a) A pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1223334444555567890 is a pandigital number in base 10. The pandigital numbers to base 10 sequence is given by 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689… (b)The following table lists the smallest pandigital numbers of a few selected bases:

Base

Smallest pandigital number

Equivalent value in base 10

2

10

2

3

102

11

4

1023

75

10

1023456789

1023456789

16

1023456789ABCDEF

1162849785405935

Roman numerals

MCDXLIV

1444

(c)It is obvious that if the base value is high more pandigital numbers without redundant digits or repeated digits possible and if the base value is small fewer pandigital numbers without redundant digits are possible. (d)Pandigital numbers are useful in fiction and in advertising. The Social Security Number 987-65-4321 is a zeroless pandigital number reserved for use in advertising. Some credit card companies use pandigital numbers with redundant digits as fictitious credit card numbers (while others use strings of zeroes). (e) Examples of pandigital numbers 123456789 is the first zeroless pandigital number. 987654321 is the largest zeroless pandigital number without redundant digits. 1023456789 is the first pandigital number. 1234567890 is the first pandigital number with the digits in order. 9876543210 is the largest pandigital number without redundant digits. 12345678987654321 is a pandigital number with all the digits except zero in both ascending and descending order. It is the square of 111111111 and is also a palindrome number.

(1.18) Numbering systems: -Based upon the whole numbers represented by the repeating terms or digits various numbering systems are followed. For example if the numbering system involves the digits from 0 to 9 it is called as decenary system and if the numbering system involves 0 and 1 means it is called as binary system and so on.

(a) Decenary system:- It has 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 numbers. It is based on the powers of 10. It is the usual way of representing numbers. For example 121_10 = 121 and the number can be represented as the powers of 10 like [(1x10^2) + (2x10^1) + (1x10^0)]. But generally we are not representing the subscript ‘10’ since decenary system is the common method of numbering system. Here 10 is called as radix of the number. Radix is the first higher number obtained by using the digits of the numbering system.

(b) Binary system: - It has 0 and 1 numbers. Like in decenary system the binary system can also be represented as the powers of 2. Conversion of decenary number system to binary number system is as follows. For example the number 121 can be converted into binary number or binary digit (bit) as follows.

121/2 = 60 remainder is 1

60/2 = 30 remainder is 0

30/2 = 15 remainder is 0

15/2 = 7 remainder is 1

7/2 = 3 remainder is 1

3/2 = 1 remainder is 1

Now the binary number equivalent of 121 is 1111001_2.

Conversion of binary number to decenary number is as follows.

1 1 1 1 0 0 1

1 1 1 1 1 1 1 1×2^0 = 1

1 1 1 1 1 1 0×2^1 = 0

1 1 1 1 10×2^2 = 0

1 1 1 11×2^3 = 8

1 1 11×2^4 =16

1 11×2^5 =32

11×2^6 =64

121

Now the decenary equivalent of 1111001_2 is 121.

(c) Ternary system: - It has 0, 1 and 2 numbers. This system of representation is based on powers of 3. Here the radix is 3.

(d) Pental number system: - It has 0, 1, 2, 3 and 4 numbers. For example 121 is converted into pental system as follows.

121/5 = 24 remainder is 1

24/5 = 4 remainder is 4

Now the pental number equivalent of 121 is 441_5.

Conversion of pental number into decenary number is as follows.

4 4 1

1 1 11×5^0 = 1

1 14×5^1 = 20

14×5^2=100

121

Now the decenary equivalent of 441_5 is 121.

(e) Octal number system: - It has 0, 1, 2, 3, 4, 5, 6 and 7 numbers. For example 121 is converted into octal number as follows.

121/8 = 15 remainder is 1

15/8 = 1 remainder is 7

Now the octal number equivalent of 121 is 171_8.

Conversion of octal number into decenary number is as follows.

1 7 1

1 1 11×8^0 = 1

1 17×8^1 = 56

11×8^2 = 64

121

Now the decenary equivalent of 171_8 is 121.

(f) Hexadecimal number system:- It has 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F digits.

For example 121 is converted into hexadecimal system as follows.

121/16 = 7 remainder is 9

Now the hexadecimal number equivalent of 121 is 79_16.

7 9

1 19×16^0 = 9

17×16^1 =112

121

Now the decenary equivalent of 79_16 is 121.

(g) The concepts and ideologies which are applicable to one numbering system need not be applicable to other numbering systems.

(h) It is easy to convert from octal to binary system and vice versa since three binary digits make one octal digit. To convert from octal to binary digit, replace all octal digits by their binary equivalents. For example (347)8 = (011 100 111)2 = (11100111)2

(i) To convert binary to octal digit, partitions the binary digits in groups of three, starting from the right and then replaces each group by its octal digit.

For example (10111011)2 = (010 111 011)2 = (273)8

(j) Dozenal system or duodecimal system: - This system can also be called as duodecimal system. The radix of the system is 12. It has 1, 2, 3, 4, 5, 6, 7, 8, 9, A and B digits.

(k) A group of 4 bits can be represented by an equivalent hexadecimal digit as follows.

(1101 0010 1010 1100)2 = (D2AC)16

Numbering system

Binary

Ternary

Pental

Octal

Decimal

Dozenal

Hexadecimal

Radix value

2

3

5

8

10

12 or C

16 or G

Equivalent for 0_10

0000

0000

0

0

0

0

0

Equivalent for 1_10

0001

0001

1

1

1

1

1

Equivalent for 2_10

0010

0002

2

2

2

2

2

Equivalent for 3_10

0011

0010

3

3

3

3

3

Equivalent for 4_10

0100

0011

4

4

4

4

4

Equivalent for 5_10

0101

0012

10

5

5

5

5

Equivalent for 6_10

0110

0100

11

6

6

6

6

Equivalent for 7_10

0111

0101

12

7

7

7

7

Equivalent for 8_10

1000

0102

13

10

8

8

8

Equivalent for 9_10

1001

0110

14

11

9

9

9

Equivalent for 10_10

1010

0111

20

12

10

A

A

Equivalent for 11_10

1011

0112

21

13

11

B

B

Equivalent for 12_10

1100

1000

22

14

12

10

C

Equivalent for 13_10

1101

1001

23

15

13

11

D

Equivalent for 14_10

1110

1002

24

16

14

12

E

Equivalent for 15_10

1111

1010

30

17

15

13

F

(l) A comparison of binary, ternary, pental, octal, decimal, dozenal and hexadecimal system is as follows.

(1.19) Different types of mentioning the whole numbers and their corresponding equivalents are tabulated as follows.

Name

0

1

2

3

4

5

6

7

8

9

10

100

1000

Bengali

-

১

২

৩

৪

৫

৬

৭

৮

৯

-

-

-

Gujarathi

-

૧

૨

૩

૪

૫

૬

૭

૮

૯

-

-

-

Gurmukhi

-

੧

੨

੩

੪

੫

੬

੭

੮

੯

-

-

-

Hindi or Devanagiri

-

१

२

३

४

५

६

७

८

९

-

-

Japanese

零

一

二

三

四

五

六

七

八

九

十

百

千

Kannada

-

೧

೨

೩

೪

೫

೬

೭

೮

೯

-

-

-

Malayalam

-

൦

൧

൨

൩

൪

൫

൬

൭

൮

൯

-

-

Roman

-

I

II

III

IV

V

VI

VII

VIII

IX

X

C

M

Tamil

-

௧

௨

௩

௪

௫

௬

௭

௮

௯

௰

௱

௲

Telugu

-

౧

౨

౩

౪

౫

౬

౭

౮

౯

-

-

-

Thai

-

๑

๒

๓

๔

๕

๖

๗

๘

๙

-

-

-

(1.20) Arithmetic operations: - The four important basic operations of arithmetic are addition, subtraction, multiplication and division. Multiplication can also be called as consecutive addition and similarly division can also be called as consecutive subtraction. Apart from these basic arithmetic operations or properties the following other arithmetic operations can also be studied and compared with other mathematical operations.

(a) Additive property: - If A+A=2A then the property is known as additive property.

For example 9+3=12. This is called as arithmetic addition.

(b) Subtractive property: - If A-A=0 then the property is known as subtractive property.

For example 9-3=6. This is called as arithmetic subtraction.

(c) Multiplicative property: - If A×A=A^2 then the property is known as multiplicative property.

For example 9×3=27. This is called as arithmetic multiplication.

(d) Divisive property: - If A/A=1 then the property is known as divisive property.

For example 9/3=3. This is called as arithmetic division.

(e) Determinative property: - If A, B are two criterions and if A=B or A≠B then the property is known as determinative property.

For example √9 = 3 and √9 ≠ 2. This is called as arithmetic powers.

(f) Reflux property: - If A’ is obtained from A by mirror then the property is known as reflux property. Note that A’ may be equal to A or may not be equal to A.

For example the reflux number of 176 is 671

(g) Palindrome property: - If A’ is equal to A by mirror then the property is known as palindrome property.

In the case of 121 the reflux number of 121 is the same number itself and so it is a palindrome number. Note that a system obeying palindromic property will also obey reflux property but the converse is not true.

(h) Symmetric property: - If A*B = B*A then the property is known as symmetric property.

For example 3^2 = 9; 9 = 3^2. This is arithmetic powers

(i) Transitive property: - If A=B and B=C then A=C is known as transitive property.

For example 3^2 = 9; 9 = 5 + 4 then 3^2 = 5 + 4

(j) Commutative property: - If A*B = B*A (Note: - * means an arithmetic operation)

then the property is known as commutative property.

For example 3×2 = 6 = 2×3. This is the property of arithmetic multiplication.

(k) Anti commutative property: - If A*B = − (B*A) (Note: - * means an arithmetic operation) then the property is known as anti commutative property.

For example 5 – 2 = − (2 – 5). This is arithmetic subtraction.

(l) Left Associative property: - If A*(B*C) = (A*B)*C (Note: - * means an arithmetic operation) then the property is known as left associative property.

For example 3 × (2×5) = (3×2) ×5 = 30. This is arithmetic multiplication.

(m) Right associative property: - If (A*B)*C = A*(B*C) (Note: - * means an arithmetic operation) then the property is known as right associative property.

For example (3×2) ×5 = 3 × (2×5) = 30. This is arithmetic multiplication.

(n) Left cancellation property: - If A*B = A*C (Note: - * means an arithmetic operation) then B = C is known as left cancellation property.

For example 5×9 = 5×3×3 then canceling the common terms we get 9 = 3×3.

(o) Right cancellation property: - If A*B = C*B (Note: - * means an arithmetic operation) then A = C is known as right cancellation property.

For example 2^2×5 = 4×5 then canceling the common terms we get 2^2 = 4.

(p) Identity property: - If A*I = A (Note: - * means an arithmetic operation) then the property is known as identity property. Here I is called as identity element.

For example 5 + 0 = 5 Here 0 is the identity element.

(q) Inverse property: - If A*A-1 = 0 (Note: - * means an arithmetic operation) then the property is known as inverse property. Here A-1 is called as inverse element.

For example 2 – 2 = 0 Here -2 is the inverse element.

(r) Left distributive property: - If A (B*C) = AB * AC (Note: - * means an arithmetic operation) then the property is known as left distributive property.

For example 3(10 – 4) = 3(10) – 3(4) = 18.

(s) Right distributive property: - If (A*B) C = AC * BC (Note: - * means an arithmetic operation) then the property is known as right distributive property.

For example (10 – 4)3 = (10)3 – (4)3 = 18.

(t) Closure property: - If A*B (Note: - * means an arithmetic operation) is also the element of same order or in the same number system is known as closure property.

For example real number system 3 + 2 = 5

(u)Upset reflux property: - If by mirror then the property is known as upset reflux property.

For example where the symbol represents a mirror.

(v) Upset property: - If A*(1/A) = 1 (Note: - * means an arithmetic operation) then the property is known as upset property.

For example 2 × (1/2) = 1. This is arithmetic multiplication.

A comparison of these mathematical properties is listed as an annexure.

(1.21) Branches of arithmetic

(a) Statistics: - It is the arithmetic approach of arrangement and classification of the data collected.

(b) Mods: - It is the arithmetic study of systematic repetitive processes.

(c) Matrix: -It is the arithmetic study of array of numbers in rows and columns.

(d) Determinants: -It is the arithmetic study of square matrices.

(e) Sets: - It is the arithmetic study of well collection of defined set of objects.

(f) Algebra: - It is the arithmetic study of powers.

(g) Vectors: -It is the arithmetic study of properties involving both magnitude and direction.

(1.22)Polygonal numbers: -When the real numbers are expressed as dots based on the geometrical shapes formed they are classified as triangular numbers, square numbers, pentagonal numbers, and so on. The general terminology of these numbers is polygonal numbers and is discussed in chapter 10.

2. ADDITION

(2.1) Addition can also be called as summation and it is denoted by the symbol “+” and is called as plus. For example 5+3=8. Here 5 is called as addendum, 3 is called as adduct and 8 is called as the resultant.

(2.2) The number digit of the resultant is equal to the sum of the number digits of the addendum and adduct involved in the addition.

For example 3989 number digit 2

(+)2403 number digit 0 or 9

6392 number digit 2

Now the sum of the number digits of addendum and adduct is equal to 11 whose number digit is 2.

(a)The addition of numbers which are having number digits either of 3, 6 or 9 among themselves will give the resultant either of 3, 6 or 9 as their number digits.

For example 6 + 6 will give the resultant with number digit 3 while 6 + 9 will give the resultant with number digit 6.

(2.3)Addition is commutative.

For example 5+3=8 and 3+5=8

(2.4)

Addition

Odd number

Even number

Odd number

Even number

Odd number

Even number

Odd number

Even number

For example, odd number + even number = odd number

5+ 4=9

In broader sense, even + even + even +…+ even = even

even + even + even +…+ odd = odd

odd + odd + odd +…+odd = even if ‘n’ number of “even” odd numbers are present.

odd + odd + odd +…+odd = odd if ‘n’ number of “odd” odd numbers are present.

(2.5)If the sum of any three numbers is zero then the sum of their cubes is equal to

three times their product.

For example 3, 4 and -7

Sum of the above numbers 3+4+ (-7) =0

Sum of their cubes is 27+64+ (-343) = -252

Product of the above numbers 3×4× (-7) = -84

Three times the product is 3× (-84) =-252

(2.6)

Addition

Positive number

Negative number

Positive number

+

±

Negative number

±

-

For example Negative number + Positive number = Negative or positive number

(-4) + 2 = -2 and another example is 4 + (-2) = 2.

(2.7)Addition table

Addition

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

11

2

3

4

5

6

7

8

9

10

11

12

3

4

5

6

7

8

9

10

11

12

13

4

5

6

7

8

9

10

11

12

13

14

5

6

7

8

9

10

11

12

13

14

15

6

7

8

9

10

11

12

13

14

15

16

7

8

9

10

11

12

13

14

15

16

17

8

9

10

11

12

13

14

15

16

17

18

9

10

11

12

13

14

15

16

17

18

19

10

11

12

13

14

15

16

17

18

19

20

For example from the table we can see that 8+3=11

(2.8)The number who’s three times or third multiple’s sum of the digits is equal to

the sum of the digits itself is 27.

The number is 27 and its digits sum of the digits is 2+7=9

And three times the sum of the digits is 9×3=27

This can be extended to the multiples of nine.

(2.9)Rule of addition: - The first decimal number is to be added with first decimal

number followed by 10th or second decimal number with the 10th or second decimal

number and then 100th or third decimal number with the corresponding 100th or third

decimal number and so on from right to left.

For example 5132

132

30

5294

Note: - In 5132, ‘2’ is the first or unit decimal number, ‘3’ is the 10th decimal number, ‘1’ is the 100th decimal number and ‘5’ is the 1000th decimal number.

(2.10) 123456789×1 123456789×2 123456789×3 123456789×4

987654321×1 987654321×2 987654321×3 987654321×4

+1 +2 +3 +4

1,111,111,111 2,222,222,222 3,333,333,333 4,444,444,444

Likewise we can extend the series up to 9.

(2.11) (7+4)0 = 1

(7+4) = 11

57+54 = 111

557+554 = 1111

5557+5554 = 11111

……………………….

(2.12)Three digit miracle: - Take any three digit number except the palindrome

numbers, write it backwards and subtract the smaller number from the bigger number.

Now write the resultant backwards and add this to the resultant itself and surprisingly we will get the summation will always be 1089 irrespective of the three digit number taken.

For example 398 and if it is written backwards 893.

Subtract the smaller 893-398=495 and if it is written backwards 594

Now add 495+594=1089.

(2.13) Take any two digit number except the palindrome numbers, write it backwards and subtract the smaller number from the bigger number.

Suppose if it comes as single digit number put zero before the number and consider it as a double digit number. Now write the resultant backwards and add this to the resultant itself and surprisingly we will get the summation will always be 99 irrespective of the two digit number taken.

For example 23 and if it is written backwards 32.

Subtract the smaller 32-23=9 and put zero before the resultant and now if the

resultant is written backwards 90.

Now add 9+90=99

(2.14)The number zero is called as additive identity. It can also be called as subtractive identity.

That is n+0=n where n ≠ 0

For example 33+0=33 = 33-0

(2.15)Place value of a digit is the value of the digit in the place where it is occupying

the number. For example the place value of 7 in the number 607354 is 7000.

(2.16)Face value: -

(a)Face value of a digit is the value of the digit as such seen in the number. For example the face value of 7 in the number 607354 is 7.

(b) Zero has no face value but it has the place value. In the above case the face value of 0 is 0 and the place value of 0 is 10,000.

(c) Infinity has no place value but it has the face value.

(2.17)Take a sequence of consecutive number or a sequence of multiples of numbers

say 5,6,7,8,9,10,11,12,13,14,15 and 16. They can be put in sequences as

(5, 6, 7)(8, 9, 10)(11, 12, 13) and (14, 15, 16). Now write 5, 6, 7 as such and below them 8,9,10 in the reverse order then below them 11, 12, 13 as such and then below them 14, 15, 16 in the reverse order and do the column wise addition.

5 6 7

10 9 8

11 12 13

16 15 14

42 42 42

Surprisingly the column totals are the same. If the series is stopped with 3 rows only then the column totals come in a sequence skipping by 1.

5 6 7

10 9 8

11 12 13

26 27 28

Note: The common difference in the column totals is one.

(a)It is seen that when the count of rows is even the column totals are same. When the count of rows is odd then the column totals form a sequence.

5 6 7

10 9 8

15 15 15

(b)If we interchange any two adjacent numbers in a row then the column totals

form the same sequence of consecutive numbers 41, 42 and 43 but not in the same order. When the two numbers interchanged in a row are not adjacent then the column totals form the sequence 40, 42 and 44 with a column gap of two.

6 5 7 5 6 7 5 6 7 7 6 5

10 9 8 10 9 8 8 9 10 10 9 8

11 12 13 11 12 13 11 12 13 11 12 13

16 15 14 16 15 14 16 15 14 16 15 14

42 42 42 41 43 42 40 42 44 44 42 40

Note: - When the numbers interchanged are adjacent then the column totals have the common difference of one and when the numbers interchanged are alternate then the column totals have the common difference of two.

3. SUBTRACTION

(3.1) Subtraction is the reverse process of addition. Subtraction is denoted by the symbol “-” and is pronounced as minus. For example 11-9=2. Here 11 is called as subtrahend, 9 is called as minuend and 2 is called as the resultant.

(3.2)Digital root rule of subtraction: -

(i)The difference of the digital roots of subtrahend and minuend should be equal to the digital root of the resultant.

For example 653 Digital root is 5 (The difference in digital root is -1. Since the digital

(-) 186 Digital root is 6 root should not be negative, take complement

467 Digital root is 8 by subtracting from 9. We get 8 and is the

digital root of the resultant).

(ii) The subtraction of numbers which are having number digits either of 3, 6 or 9 among themselves will give the resultant either of 3, 6 or 9 as their number digits. For example 222(number digit 6) – 201(number digit 3) = 21(number digit 3).

(3.3) Rule of subtraction: -

(i)The first decimal number is to be subtracted with first decimal number followed by 10th or second decimal number with the 10th or second decimal number and then 100th or third decimal number with the corresponding 100th or third decimal number and so on from right to left.

For example 4320

(-)1120

3200

(ii) If any of the decimal number in the subtrahend is smaller than its corresponding decimal number in the minuend then we have to borrow 10 from its immediate left neighbor decimal number and do the subtraction.

For example 2320

(-)1129

191

(iii) If the whole number of the subtrahend itself is smaller than the minuend then we will get negative number.

For example 234

(-)916

-682

(3.4)Subtraction is not commutative.

11-9 ≠ 9-11

(3.5)

Subtraction

Odd number

Even number

Odd number

Even number

Odd number

Even number

Odd number

Even number

For example even number – odd number = odd number

12 – 3 = 9

Note:-For both addition and subtraction the operations to odd number and even number are same.

(3.6) Special subtractions

(i) 987654321 sum of the decimals in this number is 45

(-)123456789 sum of the decimals in this number is 45

864197432 sum of the decimals in this number is 45

Similarly 87654321 sum of the decimals in this number is 36

(-)12345678 sum of the decimals in this number is 36

75308643 sum of the decimals in this number is 36

(ii) 12345678 – 12345 = 123.

(3.7)Subtraction table

Subtraction

1

2

3

4

5

6

7

8

9

10

1

0

1

2

3

4

5

6

7

8

9

2

-1

0

1

2

3

4

5

6

7

8

3

-2

-1

0

1

2

3

4

5

6

7

4

-3

-2

-1

0

1

2

3

4

5

6

5

-4

-3

-2

-1

0

1

2

3

4

5

6

-5

-4

-3

-2

-1

0

1

2

3

4

7

-6

-5

-4

-3

-2

-1

0

1

2

3

8

-7

-6

-5

-4

-3

-2

-1

0

1

2

9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

From the table we can see that 3-8 = -5.

(3.8)

Subtraction

Positive number

Negative number

Positive number

±

±

Negative number

±

±

From the table we can find that negative number – negative number = ±

For example -4-(-2) = -2 and -4-(-6) = 2.

4. MULTIPLICATION

(4.1) Multiplication is denoted by the symbol “×” or simply by * and is pronounced as into. For example 3×5=15. Here 3 is called as multiplicand, 5 is the multiplier and 15 is called as the product or resultant.

(4.2) Lesser the difference between the multiplicand and multiplier higher the value of product. For example 9×1=9 but 9×9=81.

(4.3) Rule of multiplication: - The first decimal number of the multiplier is to be multiplied with each of the decimal numbers of the multiplicand individually from right to left followed by 10th or second decimal number of the multiplier with each of the decimal numbers of the multiplicand but placing the resultant below the 10th decimal place from right to left and then 100th or third decimal number of the multiplier with each of the decimal numbers of the multiplicand but placing the resultant below the 10th decimal place from right to left and so on. Then finally add to get the product.

For example 532×234

2128

1596

1064

124488

(4.4) Digital root rule of multiplication: -

(a)The number digit of the product of two numbers is equal to the product of the number digits of the numbers involved in the multiplication.

For example 6347 number digit 2

×206 number digit 8

1307482 number digit 7

Now the product of the number digits of multiplicand and multiplier is 16 whose number digit is 7.

(b) The multiplication of numbers which are having number digits either of 3, 6 or 9 among themselves will give the resultant either of 3, 6 or 9 as their number digits. Also it is to be noted that the number digit of any number multiplied by 9 will always be 9 only.

For example 27(number digit 9) × 21(number digit 3) = 567(number digit 9).

(c) The pattern of digital roots for 4 table is 4, 8, 3, 7, 2, 6, 1, 5 and 9. This pattern repeats itself if we continue the table on for higher numbers. Similarly we can find the patterns of digital roots for 2, 3, 5, 6, 7 and 8 tables which repeats themselves uniquely. But the digital roots for 9 table is one and the same that is 9 only.

(4.5) Binary method of multiplying two numbers: -

Step 1: Divide the multiplicand by 2, omitting fractions and multiply the multiplier by 2 and continue the operations till the multiplicand becomes 1.

For example 41×57

20×114

10×228

5×456

2×912

1×1824

Step 2: Delete all the even numbers of the multiplicands and the corresponding multipliers on the right hand side and retain the remaining ones.

41×57

5×456

1×1824

Step 3: Add the multipliers of all the numbers in the right hand side of step 2 and the resultant will be the required product.

That is 57+456+1824=2337 and this is the product of 41×57.

Logic of the method: Consider the numbers of the multipliers in step 2

57=2^0×57

456=2^3×57

1824=2^5×57 and now

2337= (2^0+2^3+2^5) ×57 = (1+8+32) × 57=41×57.

(4.6) An easy method of multiplying two numbers which gives the totaling of the unit digits to ten and the tenth digits of both the numbers is same is as follows.

For example 43×47(Here the total of the units is 10 and the tenth digits is same that is 4).

Step 1: -Multiply the unit digits and write the product in the right hand side as 21.

Step 2: -Multiply the tenth digit with its following digit and write the product in the left hand side and now the resultant of 43 × 47 = 2021.

Another example 61 × 69

Step 1: - Right hand side product is 1 × 9 = 09 (In this method of multiplication 100 is taken as the base so we should always have two digits on the right hand side.)

Step 2: - Left hand side product is 6 × 7 = 42 and now the resultant is 4209.

(4.7) Multiplication is commutative.

3×5 = 15 = 5×3.

(4.8)

Multiplication

Odd number

Even number

Odd number

odd number

even number

Even number

even number

even number

From the table we can tell that even number × odd number = even number

2 × 11 = 22

In broader sense, odd × odd × odd ×…× odd = odd (all odd numbers)

even × even × even ×…×even = even (all even numbers)

even × even × even ×…×odd = even (at least one even number).

(4.9) 1, 2 and 3 are the only three consecutive numbers whose summation and product

are same. That is 1 × 2 × 3 = 6 = 1 + 2 + 3

and similarly (-1)×(-2)×(-3)=(-6)=(-1)+(-2)+(-3)

(4.10)

Multiplication

Positive number

Negative number

Positive number

+

-

Negative number

-

+

From the table we can tell that (-3) × (-6) =18.

(4.11) Multiplication table

Multiplication

1

2

3

4

5

6

7

8

9

10

1

1

2

3

4

5

6

7

8

9

10

2

2

4

6

8

10

12

14

16

18

20

3

3

6

9

12

15

18

21

24

27

30

4

4

8

12

16

20

24

28

32

36

40

5

5

10

15

20

25

30

35

40

45

50

6

6

12

18

24

30

36

42

48

54

60

7

7

14

21

28

35

42

49

56

63

70

8

8

16

24

32

40

48

56

64

72

80

9

9

18

27

36

45

54

63

72

81

90

10

10

20

30

40

50

60

70

80

90

100

From the table we can tell that 6×7=42.

(4.12) The multiplication of a single digit number by 11 is equal to write the same

number two times adjacently. Similarly the multiplication of a two digit number by 101 is equal to write the same number two times adjacently and likewise for a three digit number 1001 and so on.

For example 9× 11=99

23×101=2323

321×1001=321321

(4.13) (a)The specialty of 15,873 is

15,873× 7 = 111,111 = 1(7) ×15,873 = 1(111,111)

15,873×14= 222,222 = 2(7) ×15,873 = 2(111,111)

15,873×21= 333,333 = 3(7) ×15,873 = 3(111,111)

15,873×28= 444,444 = 4(7) ×15,873 = 4(111,111)

15,873×35= 555,555 = 5(7) ×15,873 = 5(111,111)

15,873×42= 666,666 = 6(7) ×15,873 = 6(111,111)

15,873×49= 777,777 = 7(7) ×15,873 = 7(111,111)

15,873×56= 888,888 = 8(7) ×15,873 = 8(111,111)

15,873×63= 999,999 = 9(7) ×15,873 = 9(111,111)

(b)The specialty of 65359477124183 when multiplied by 17 and multiples of 17 is

65359477124183 × 17 = 1,111,111,111,111,111

65359477124183 × 34 = 2,222,222,222,222,222

65359477124183 × 51 = 3,333,333,333,333,333

65359477124183 × 68 = 4,444,444,444,444,444

65359477124183 × 85 = 5,555,555,555,555,555

65359477124183×102 = 6,666,666,666,666,666

65359477124183×119 = 7,777,777,777,777,777

65359477124183×136 = 8,888,888,888,888,888

65359477124183×153 = 9,999,999,999,999,999

(4.14) (a) 19×0.95=18.05 (b) 1⅓ × 4 = 16/3 (c) 6.25 ÷ 5 = 1.25

19−0.95=18.05 1⅓ + 4 = 16/3 6.25 – 5 = 1.25

(d) 1.2 × 6 = 7.2

1.2 + 6 = 7.2

(4.15) Features of number 8: -

(a) The number eight is a Fibonacci number.

(b) 12345679×8=98765432

(c) 8 is the first even cube number.

(d) 1×8+1=9

12×8+2=98

123×8+3=987

1234×8+4=9876

12345×8+5=98765

123456×8+6=987654

1234567×8+7=9876543

12345678×8+8=98765432

123456789×8+9=987654321

(e) 9×0+8=8

9×9+7=88

98×9+6=888

987×9+5=8888

9876×9+4=88888

98765×9+3=888888

987654×9+2=8888888

9876543×9+1=88888888

98765432×9+0=888888888

(f) Enigma of 8

12345679× 9 =111111111

12345679×18=222222222

12345679×27=333333333

12345679×36=444444444

12345679×45=555555555

12345679×54=666666666

12345679×63=777777777

12345679×72=888888888

12345679×81=999999999

Further the enigma of 8 can be extended for 0.12345679× 9 =1.11111111,

0.12345679×18=2.22222222 and so on up to 0.12345679×81=9.99999999

(4.16) Play with 3 and 4

4×4=16

34×34=1156

334×334=111556

3334×3334=11115556

33334×33334=1111155556

………………………………

(4.17) Number 9:– A marvel

(a) Number 9 is the first odd square number.

(b) The multiplication of any number with 9 will give a product whose summation of the digits (digit sum) are always 9 or multiples of 9. For example 1234×9=11,106 (sum of the digits of the product 1+1+1+0+6=9)

(c) The converse of the above fact is also true. That is when we subtract any number from its sum total of its digits or digit sum the resultant is always divisible by 9. For example 8247 and the sum of its digits is 8+2+4+7=21 and now 8247-21=8226 this is divisible by 9.

Note:-Digit sum is the resultant we get when we add all the digits in a number.

(d) Sum of any nine consecutive numbers will give the resultant with number digit 9. For example 21+22+23+24+25+26+27+28+29=225 and the number digit of the resultant is 9.

(e) When we subtract any number from any of its transposed number (if the number transposed is greater than the original subtract the original number) then the resultant will be a divisor of 9. For example 213 and the transposed numbers are 132, 321, 231, 123, and 312. Now 213-132=81 this is divisible by 9. Again 312-213=99 this is divisible by 9 and like this we can extend for other numbers.

(f) Magic with nines multiplication

9×9=81

99×99=9801

999×999=998001

9999×9999=99980001

99999×99999=9999800001

……………………………..

Note: - To get the square of a number which contains only 9s, write from right to left first 1, then zeros, one less than the number of 9s and then write 8. Finally write as many 9s as the zeros.

(g) 0×9+1=1

1×9+2=11

12×9+3=111

123×9+4=1111

1234×9+5=11111

12345×9+6=111111

123456×9+7=1111111

1234567×9+8=11111111

12345678×9+9=111111111

123456789×9+0=1111111111

(h) Any number when added to 9 will have the resultant having the sum of the

digits equal to the sum of the digits of the addendum added to 9. For example 1132+9=1141. Here the sum of the digits of addendum 1132 is 7 and the sum of the digits of resultant 1141 is 7.

Note:-For single digit numbers from 1 to 9 when added to 9 the resultant will have the

sum of the digits equal to the addendum that is the number itself. For example 8+9=17.

Here the addendum is 8 and the sum of the digits of the resultant is 8.

(i) 1×9-1=08

21×9-1=188

321×9-1=2888

4321×9-1=38888

54321×9-1=488888

654321×9-1=5888888

7654321×9-1=68888888

87654321×9-1=788888888

987654321×9-1=8888888888

From the resultant we can see that the first digit is obtained by subtracting 1 from the first digits of the multiplicand followed by writing first digit’s time of multiplicand by 8.

(j) 111111111×9=0999999999

222222222×9=1999999998

333333333×9=2999999997

444444444×9=3999999996

555555555×9=4999999995

666666666×9=5999999994

777777777×9=6999999993

888888888×9=7999999992

999999999×9=8999999991

From the resultant obtained we can see that the sum of the first and last digit is same and is 9. For example consider the product 5999999994 and the sum of the first and last digit is 5+4=9. Also the first digit is increasing and the last digit is decreasing while the middle digits remain the same.

(k) 123456789× 9=1111111101

123456789×18=2222222202

123456789×27=3333333303

123456789×36=4444444404

123456789×45=5555555505

123456789×54=6666666606

123456789×63=7777777707

123456789×72=8888888808

123456789×81=9999999909

(l) Also 987654321× 9=08888888889

987654321×18=17777777778

987654321×27=26666666667

987654321×36=35555555556

987654321×45=44444444445

987654321×54=53333333334

987654321×63=62222222223

987654321×72=71111111112

987654321×81=80000000001

(m) Multiplication of a number with series of 9: -

Case I: - When a number is multiplied by 9 or 99 or 999 and so on and if the number of digits in the multiplicand is equal to the number of digits of the multiplier then to get the product write 1 less than the multiplicand in the left hand side and subtract each digit of the LHS from 9 then write respectively in the right hand side.

For example 345×999=344654

Case II: - When a number is multiplied by 9 or 99 or 999 and so on and if the number of digits in the multiplicand is less than the number of digits of the multiplier then add zero or zeros to get the number of digits same. Then to get the product of the multiplication proceed as per case I.

For example 345×9999=0345×9999=03449655

Case III: - When a number is multiplied by 9 or 99 or 999 and so on and if the number of digits in the multiplicand is more than the number of digits of the multiplier then put as many zero or zeros as there in the multiplier and from the result if we subtract the multiplicand we will get the resultant.

For example 345×99=34500-345

=34155

111×9=1110-111

=999

(n) The number 18 can be called as brother of 9. The product obtained by multiplying

with 18 by any number will have the sum of the digits as 9 or multiples of 9. For example 18×11=198(sum of the digits of the resultant 198 is 18)

Another example18×18=324(sum of the digits of the resultant 324 is 9)

Likewise further powers of 18 also give the resultant whose sum of the digits equal to 9 or multiples of 9.

That is 18^4=104976 (sum of the digits of the resultant is 1+0+4+9+7+6=27.

(o) Consider the four numbers 58132764, 76125483, 72645831 and 81274365.

Multiply all these numbers by 9, 58132764×9=523194876

72645831×9=653812479

76125483×9=685129347

81274365×9=731469285

In each of the four numbers mentioned above has eight digits and they are from 1 to 8.

No digit is repeated. When these numbers are multiplied by 9, the products have digits

from 1 to 9 and here also no digit is repeated.

Now multiply the above numbers with 18. The product will have ten digits from 0 to 9

and again no digits are repeated.

58132764×18=1046389752

72645831×18=1307624958

76125483×18=1370258694

81274365×18=1462938570

(p) 142857×7=999999. Also the addition of digits of the multiplicand is

(1+4+2+8+5+7=27 and again 2+7=9).

(q) 0×9+8 = 8

9×9+7 = 88

98×9+6 = 888

987×9+5 = 888 8

9876×9+4 = 888 88

98765×9+3 = 888 888

987654×9+2 = 888 888 8

9876543×9+1 = 888 888 88

98765432×9+0 = 888 888 888

987654321×9-1 = 888 888 888 8

9876543210×9-2 = 888 888 888 88

(r) The multiplication of 1089 with 9 is the reverse of the number and similarly

the multiplication of 10989 with 9 is the reverse of the number and this can be extended

to 109989, 1099989…

That is 1089×9=9801

10989×9=98901

109989×9=989901

1099989×9=9899901

………………………

(s) An easy way of dividing a number by 9 is keep the left hand digit of the number as such and go on add the digits from left side onwards. Now in the resultant number the right hand digit is the remainder of the division and the left out digits will give the quotient. For example 1302/9 Now keep left digit 1 followed by 4 (that is 1+3) followed by 4 (that is 1+3+0) and then finally by 6 (that is 1+3+0+2) Now 144 is the quotient and 6 is the remainder.

(t) Multiplication table for 9

1 × 9 = /9 = (1-1)/9-(1-1) = 9

2 × 9 = 1/8 = (2-1)/9-(2-1) = 18

3 × 9 = 2/7 = (3-1)/9-(3-1) = 27

4 × 9 = 3/6 = (4-1)/9-(4-1) = 36

5 × 9 = 4/5 = (5-1)/9-(5-1) = 45

6 × 9 = 5/4 = (6-1)/9-(6-1) = 54

…………………………………….

In the products we see the unit digits goes like 9,8,7,…,0 while the tens digits goes like 0,1,2,3,…9 This pattern repeats further and is characteristic for multiplication table of 9. The same argument can be extended for multiplication of a number with 99, 999, 9999…

For example 11 × 99 = 10/89 = (11-1)/99-(11-1) = 1089

12 × 99 = 11/88 = (12-1)/99-(12-1) = 1188

13 × 99 = 12/87 = (13-1)/99-(13-1) = 1287

14 × 99 = 13/86 = (14-1)/99-(14-1) = 1386

15 × 99 = 14/85 = (15-1)/99-(15-1) = 1485

…………………………………………………

Note: - The above method is applicable only if the multiplicand and multiplier have equal number of digits.

(u) The set of fractions 1/9, 1/99, 1/999… will give infinite recurring fractions 0.1, 0.01, 0.001… respectively. Based on this method a simple way of converting a decimal into a recurring infinite decimal is as follows. 7/9, 8/99, 24/999… will give infinite fractions 0.7, 0.08, 0.024… respectively.

(v) The division of 1÷9=0.111…

2÷9=0.222…

3÷9=0.333…

4÷9=0.444…

5÷9=0.555…

6÷9=0.666…

7÷9=0.777…

8÷9=0.888…

9÷9=0.999…

Of course 9÷9=1 but is actually equal to 0.999… and this is proved as follows.

Consider N=0.999…

Then 10N=9.999…

Now 10N-N=9

That is 9N=9 or N=1=0.999…

(w)The recurring decimal (1/19) gives a special phenomenon where if we split the recurrence into two parts and add we get nine 9’s.

* *

(1/19) = 0.052631578947368421

The asterisk is put to indicate the recurrence. Now split the decimal into two parts and add

052631578

947368421

999999999

(4.18) Specialty of 142857: -

(a) The number 142857 is the recurring decimal obtained from the resultant of the infinite fraction 1/7. The recurring decimal of this infinite fraction has the number digit 9. That is 1/7=0.142857… and the number digit of recurring decimals is 9.

(b) The recurring decimal 142857 has 6 digits, the same as the difference between the denominator and the numerator of the fraction.

(c) Multiply 142857 by any number from 1 to 6. In each answer you find the

same digits. If the products are written in the form of a circle, the order of the digits

remains the same and so the number 142857 is called as cyclic number.

142857×1=142857

142857×2=285714

142857×3=428571

142857×4=571428

142857×5=714285

142857×6=857142

Another oddity is if we add the products of the number when multiplied by 1, 2, 3, 4, 5 and 6 horizontally and vertically, the digits all add up to 27.

1

4

2

8

5

7

2

8

5

7

1

4

4

2

8

5

7

1

5

7

1

4

2

8

7

1

4

2

8

5

8

5

7

1

4

2

For example if we add vertically 5 1 7 2 8 4 we get 27.

(d) Also 142857÷2 = 71428.5

142857÷5 = 28571.4

(e) The summation of the first three digits 142 and the second three digits 857 gives 999.

(f) The multiplication of 142857×7=999999

(g) 857^2 – 142^2 = 734449 – 20164 = 714285 again the same set of digits are repeating.

(h) 142857^2 = 20408122449 and now 20408 + 122449 = 142857

(i)The addition 142857 143

285714 286

571428 572

1000000001

(4.19) Special multiplications

(a) 5363222357×6215169=33333333333333333

(b) 10726444714×2071723=22222222222222222

(c) 1763664903 × 7 = 12345654321

(d) 135 = (1+3+5)(1×3×5)

(e) 144 = (1+4+4)(1×4×4)

(4.20) (a) The eighteen decimal number 421,052,631,578,947,368 when multiplied by 2 gives the resultant where the last digit of the multiplicand comes first followed by the same set of digits as in the multiplicand. That is 421,052,631,578,947,368×2=842,105,263,157,894,736

(b)The eighteen decimal number 263,157,894,736,842,105 when multiplied by 2 gives the resultant where the last digit of the multiplicand that is 5 comes first followed by the set of digits in the multiplicand. That is 526,315,789,473,684,210

(c)The twenty eight decimal number 1,034,482,758,620,689,655,172,413,793 when multiplied by 3 gives the resultant where the last digits of the multiplicand comes first followed by the same set of digits in the multiplicand.

(d) The thirteen decimal number 4,109,589,041,096 when multiplied by 83 gives the resultant where the second digit of the multiplier comes first followed by the same set of digits as in the multiplicand and finally the first digit of the multiplier comes last. That is 4,109,589,041,096 × 83 = 341,095,890,410,968

(4.21) The number one is called as multiplication identity. It can also be called as divisive identity. That is any number either multiplied or divided by 1 will give the same number.

That is n×1=n provided n≠0.

(4.22) The successive multiplications of real numbers from 1 onwards is known as the

factorial of the particular numbers. It is denoted by the symbol “!”

For example 5! = 1×2×3×4×5 = 120 and 120 is the factorial of 5.

For “n” numbers n! = 1×2×3×4×…. ×n.

(4.23) One way of multiplying by 11 is to write the last digit of the number followed

by the addition of the last digit number with the penultimate digit and so on like an

arithmetic addition from right to left side.

For example 5826×11=(5+1)(5+8+1)(8+2)(2+6)6

↓

(Remainder is one and is carried over)

=64086.

(4.24) One way of multiplying by 12 is to double the digit from right side and add to

its neighbor.

For example 74238×12

and is 074238×12=(0×2+7+1)(7×2+4+1)(4×2+2)(2×2+3+1)(3×2+8+1)(8×2)

=890856.

(4.25) The common factor in the following multiplications is that each of the digits

from 1 to 9 appears once and only once in the resultant, multiplicand and multiplier.

(1)483×12=5796

(2)138×42=5796

(3)297×18=5346

(4)198×27=5346

(5)159×48=7632

(6)186×39=7254

(7)157×28=4396

(8)1738×4 = 6952

(9)1963×4 = 7852

We can able to find only nine examples of the above type.

(4.26) Specialty of 3rd multiples

(a) 3×37= 111= 1(3)×37= 1(111)

6×37= 222= 2(3)×37= 2(111)

9×37= 333= 3(3)×37= 3(111)

12×37= 444= 4(3)×37= 4(111)

15×37= 555= 5(3)×37= 5(111)

18×37= 666= 6(3)×37= 6(111)

21×37= 777= 7(3)×37= 7(111)

24×37= 888= 8(3)×37= 8(111)

27×37= 999= 9(3)×37= 9(111)

This is called as triplication.

Thus the prime number 37 can divide 111,222,333,444,555,666,777,888,999 and

multiples of 10 of these numbers like 1110, 2220, etc. without any remainder.

3×37037=111111

6×37037=222222

9×37037=333333

12×37037=444444

15×37037=555555

18×37037=666666

21×37037=777777

24×37037=888888

27×37037=999999

This is called as hexation. The same can be extended to nonation, dodecation and so on by multiplying the third multiple numbers correspondingly with 37037037, 37037037037…

(b) The number 12345679 when multiplied by multiples of three numbers it gives special type of characteristic numbers as below.

12345679×3=37037037

12345679×6=74074074

12345679×9=111111111

12345679×12=148148148

12345679×15=185185185

12345679×18=222222222

12345679×21=259259259

12345679×24=296296296

12345679×27=333333333

12345679×30=370370370

12345679×33=407407407

12345679×36=444444444

12345679×39=481481481

12345679×42=518518518

12345679×45=555555555

12345679×48=592592592

12345679×51=629629629

12345679×54=666666666

12345679×57=703703703

12345679×60=740740740

12345679×63=777777777

12345679×66=814814814

12345679×69=851851851

12345679×72=888888888

12345679×75=925925925

12345679×78=962962962

12345679×81=999999999

12345679×84=1037037036

12345679×87=1074074073

12345679×90=1111111110

and so on.

If the number 12345679×999999999=12345678987654321

(c) 3 is the only natural number whose sum of its preceding numbers is equal to 3 itself.

(d) The 3rd multiple of 3 that is the square of 3 is in fact a marvel and its characteristics are described in (4.17).

(4.27) 7×7=49

67×67=4489

667×667=444889

6667×6667=44448889

66667×66667=4444488889

………………………………

(4.28) The product of consecutive numbers: -

(a)The product of two consecutive numbers is always an even number.

6×7=42

5×6=30

8×9=72

From the above products subtract the multiplicand that is (42-6), (30-5) and (72-8) respectively. We get 36, 25 and 64 and these are the squares of the multiplicands.

To the above products add the multiplier that is (42+7), (30+6) and (72+9) respectively.

We get 49, 36 and 81 and these are the squares of multipliers. The reason can be explained as follows.

N×(n+1)=n^2 + n

n^2 + n = n^2 + n (subtract n from both sides)

n^2 = n^2

Similarly (n-1)×n= n^2-n (add n to both sides)

n^2-n+n= n^2-n+n

n^2= n^2

(b) The product of two or more positive consecutive integers will not be a power number. For example 99x100=9900; 105×106×107 will not be a power number.

(c) Now take three consecutive numbers and see the product.

For example 13×14×15=2730 and now add the middle consecutive number to this

Product, we get 2730+14=2744 and this is the cube of the middle consecutive number.

That is 14^3=2744 This can be explained as follows.

(n-1)n(n+1)=(n^3-n)(n+1)

=n^3-n^2+n^2-n

=n^3-n

Now add n to the product we get n^3

(d) Now consider four consecutive numbers and see the product.

For example 4×5×6×7=840 and add 1 to this product. We get 841.

Now 841 is a square number and the square root of this number is 29.

Now add 1 to 29. We get 30 and this is the product of the two middle consecutive numbers. That is 5×6=30.

(4.29) The numbers 46 and 96 are peculiar, their product does not change if the digits are interchanged.

That is 46 × 96 = 64 × 69 = 4416. In the similar manner we have the following thirteen equations.

(1)12 × 42 = 21 × 24 = 504

(2)12 × 63 = 21 × 36 = 756

(3)12 × 84 = 21 × 48 = 1008

(4)13 × 62 = 31 × 26 = 806

(5)13 × 93 = 31 × 39 = 1209

(6)14 × 82 = 41 × 28 = 1148

(7)23 × 64 = 32 × 46 = 1472

(8)23 × 96 = 32 × 69 = 2208

(9)24 × 63 = 42 × 36 = 1512

(10)24 × 84 = 42 × 48 = 2016

(11)26 × 93 = 62 × 39 = 2418

(12)34 × 86 = 43 × 68 = 2924

(13)36 × 84 = 63 × 48 = 3024

5. POWERS

(5.1) The multiplication of one number with the same number is called as ‘to the power of two’ or the square of the number.

For example if 3 is the number then 3×3=9 is its square. This equation can be written as 3^2 = 9 and is read as 3 to the power of 2 is equal to 9 or 3 squared is equal to 9. Here 3 is called as the square root of 9 and it is denoted as ₂√9 =3. In this equation √ is called as radical 9 is called as radicand. The above equation can also be written as 9½ = 3.

The square number series is 4, 9, 16, 25, 36, 49, 64, 81…

Note: - Although 1 is a square number it cannot be considered as a perfect square since

1n = 1 and so 1 can be called as a power number rather than a square number.

Similarly the product of one number with the same number three times is called as

‘to the power of three’ or the cube of the number.

For example if 3 is the number then 3×3×3=27 is its cube. This equation can be written as 3^3 = 27 and is read as 3 to the power of 3 is equal to 27 or 3 cubed is equal to 27. Here 3 is called as cube root of 27 and is denoted as ₃√27 = 3 and also can be written as 271/3 = 3. Likewise the multiplication of a number with the same number four times is called as the fourth power of the number and so on.

(5.2) The digit sum of the square root when multiplied by itself (that is the square of the digit sum’s square root) should equal to the digit sum of the square.

For example 23^2 = 529. The digit sum of 23 is 5 and when 5 is multiplied by itself the answer is 25. The digit sum of 25 is 7 and the digit sum of 529 is also 7.

The digit sum of the cube root when multiplied by itself and once again by itself (that is the cube of the digit sum’s cube root) should equal to the digit sum of the cube.

For example 13^3 = 2197 The digit sum of 13 is 4 and when 4 is multiplied by itself and again by itself the answer is 64. The digit sum of 64 is 1 and the digit sum of 2197 is also 1. Likewise we can extend the concept for higher power numbers.

(5.3) The only number which has the same value of square and its sum value is 2.

That is 2×2=4 and 2+2=4

Similarly 4÷2=2 and 4-2 =2

(5.4) To find out the square of a number which ends with 5 such as 25,325,etc an easy method is put the square of 5 as 25 at the right end and the product of the left out number with its consecutive number in the left side as such.

For example the square of 105 is put 25 at the right end and the product of left out number 10 with its consecutive number 11 that is 110 at the left side and we get 11025.

Note: - We can apply the same technique for multiplication of any two numbers provided the following two conditions are satisfied. One is the resultant of last digit of the two numbers should be 10 and the second is the remaining digits should be same.

For example 91 × 99 = 9009

32 × 38 = 1216

(5.5) To find out the square of a number which ends with 3, an easy method is put the square of 3 as 9 at the right end and the product of the left out number by 6 penultimate to 9. If the product by 6 is a two digit number and above, put the unit decimal number only and carry over the remaining number and add to the square of the left out number and put this resultant in the extreme left side as such.

For example the square of 123 is put 9 at the right end and the product of left out number by 6 that is 72. Here put the unit decimal number 2 penultimate to 9 and carry over the remaining number 7. Then add the carry over number with the square of left out number that is 144 and put the resultant 151 at the extreme left to get the result 15129.

(5.6) To find out the square of a number which ends with 4 the same method described to find out the square of a number ending with 3 can be adopted. Only differences is put the square of 4 as 6 at the right end and carry over 1 and add the product of left out number by 8. If the addition of 1 and the product is a two digit number and above, put the unit decimal number only penultimate to 6 and carry over the remaining number and add to the square of the left out number and put this resultant in the extreme left side as such.

For example the square of 74 is put 6 at the right end and the product of left out number by 8 is 56 added with 1 is 57. Here put 7 penultimate to 6 and carry over the remaining 5 and add to the square of the left out number to get 54. Now put this number at the extreme left to get the square of 74. That is 5476.

(5.7) The only number whose power values like square, cube and nth power (where n is a real number) same is 1. So the number 1 is called as a unit number.

That is 1^2 = 1n = 1.

(5.8) For a number to be a square number the digital root of the number should be 1, 4 or 9. For example 102110022102101100 is not a square number since its digital root is 6.

(5.9) The summation of the cubic numbers is correspondingly equal to the sum of the arithmetic progressive squared.

1 = 1=1^3 = (1)2

1+8 = 9=1^3+2^3 = (1+2)2

1+8+27 = 36=1^3+2^3+3^3 = (1+2+3)2

1+8+27+64=100=1^3+2^3+3^3+4^3 = (1+2+3+4)2

1+8+27+64+125=225=1^3+2^3+3^3+4^3+5^3=(1+2+3+4+5)2

……………………………………………………………

(5.10) The product of a square number with the same or another square number is always a square number. That is the product of nth power number with the same or another nth power number is always a nth powered number.

For example 16×16=256

4^2×4^2 =16^2

(5.11) The multiples of a square number with a non square number two times is always an alternate square number. Similarly the multiples of a cubic number with a non cubic number three times are always a cubic number. Likewise the multiples of a fourth powered number with a non fourth powered number four times are always a fourth powered number and so on.

For example 27×2=54; 54×2=108; 108×2=216

That is 3^3×2^3=6^3

(5.12) Every odd cube number n^3 is a sum of three consecutive odd numbers whose middle term is n^2 with (n^2-2) as first term and (n^2+2) as last term.

For example 3^3 = 27 = 7 + 9 + 11

(5.13) If the square of a number has 2n (where n indicates the number of digits) digits, that is even number of digits then its square root will have ‘n’ digits. If the square number has odd number of ‘n’ digits, then its square root will have [(n+1)/2] digits. Similarly if the cubic number has 3n digits then its cubic root will have n digits and so on.

For example 4096=8^4

In the above case the fourth powered number has 4 digits and its fourth power root has single digit.

(5.14) The numbers which are both squares and cubes will have the peculiar property of the cubed square number series or the squared cube number series.

That is 1, 64, 729, 4096, 15625…

=1^3, 4^3, 9^3, 16^3, 25^3….

=1^2, 8^2, 27^2, 64^2, 125^2…

(5.15) If the difference of cubes of two consecutive integers is equal to square of an integer, then the resultant integer is the sum of the squares of two consecutive integers. For example 8^3 – 7^3 = 169 and this is the square of 13 which can be represented as 2^2 + 3^2.

(5.16) The difference between the squares of consecutive numbers is (n+1)2 – n^2 = 2n+1 where ‘n’ is a positive integer. For example the squares of the numbers 50 and 51 differ by 101.

(5.17) The real numbers which are not square numbers between n^2 and (n+1)2 is 2n. For example between 45^2 and 46^2 there are 90 non square numbers are present.

(5.18) If the square of a number is equal to the sum of two squares, then the number itself is a sum of two square numbers. For example 5^2 = 3^2 + 4^2 ; 5 = 1^2 + 2^2

(5.19) The square number when represented as dots will form a square. Similarly for cubic numbers it will form a cube and for a fourth powered number it will form a tetrahedron and so on.

For example 1 4 9 16

Diagrammatically it can be represented as follows.

Note:-Each row or column or diagonal of the square or cube or tetrahedron etc represents its square root or cubic root or fourth powered root respectively.

(5.20) Table of the end digit of powers and its properties: -

Construction of a table with the real number series against the ending digit of the number rose to the power gives some salient features.

Number

square

cube

4th

5th

6th

7th

8th

9th

10th

1

1

1

1

1

1

1

1

1

1

2

4

8

6

2

4

8

6

2

4

3

9

7

1

3

9

7

1

3

9

4

6

4

6

4

6

4

6

4

6

5

5

5

5

5

5

5

5

5

5

6

6

6

6

6

6

6

6

6

6

7

9

3

1

7

9

3

1

7

9

8

4

2

6

8

4

2

6

8

4

9

1

9

1

9

1

9

1

9

1

10

00

000

0000

00000

000000

0000000

0_8

0_9

0_10

Note: - The ending digits for other decades 11 to 20, 21 to 30… are the same as for the above decade.

(a) The last digit or unit digit of the square of a number will always end with 1, 4, 9, 6, 5 or 0 only. Also the sequence repeats for the squares of higher numbers.

Note: - The digits 2, 3, 7 and 8 do not appear as the last digit of a square number.

(b) In the cube column all the digits 1 to 9 appear in the order 1, 8, 7, 4, 5, 6, 3, 2, 9 corresponding to the numbers 1 to 9. The extreme pairs (1, 9), (8, 2), (7, 3) and (4, 6) are such that one digit in each pair is the complement of 10 of the other digit in the same pair.

(c) The powers of 2 series is 2, 4, 8, 16, 32, 64, 128…

The last digit of the powers of 2 repeats as 2, 4, 8, 6, 2, 4, 8… and it repeats in the same order as we move to the higher powers of 2.

Note: - For a number to be a sum of consecutive integers, it is necessary and sufficient that it shall not be a power of 2.

128 is the only multidigit power of 2 in which each digit is also a power of 2. That is 128 = 2^7 and 1 = 2^0; 2 = 2^1; and 8 = 2^3

(d) The powers of 3 series is 3, 9, 27, 81, 243, 729…

The last digit of the powers of 3 repeats as 3, 9, 7, 1, 3, 9, 7, 1… and it repeats in the same order as we move to the higher powers of 3.

(e) For the fourth power the endings are 1, 6, 1, 6, 5, 6, 1, 6, 1 so that there are only two digits involved that is 1 and 6 which bear symmetry on either side of 5.

(f) Fifth powers of numbers have the end digit from 1 to 9 and then 0 in an order. This type of sequence is recurring as we proceed further. That is

1^5=1; 2^5=32; 3^5=243; 4^5=1024; 5^5=3125; 6^5=7776; 7^5=16807; 8^5=32768; 9^5=59049; 10^5=100000; 11^5=161051; 12^5=248832; 13^5=371293; 14^5=537824; 15^5=759375; …….

(g) The endings in respect of 1st, 5th, ninth, thirteenth, seventeenth, twenty first and twenty fifth roots are like fifth roots in that the power and the root number have the same terminal digit number. This can be extended of the powers of 6th, 7th and 8th powers are exactly the same as those under 2nd, 3rd and 4th powers respectively.

From this we can find the generalized statement that the ending digits of the powers of the real number ‘n’ will be same as that of the powers of its ‘n+4’ number.

(h) The powers of the numbers ending with 4 will have their ending digits repeating in pairs (6,4) and for the powers of the numbers ending with 9 will have their ending digits repeating in pairs of (1,9), one digit in the pair being the complement of the other.

(i) Consider the power numbers 1^1, 2^2, 3^3, 4^4, 5^5, 6^6, 7^7, 8^8, 9^9…nn. On expansion these numbers can be written as 1, 4, 27, 256, 3125, 46656, 823543, 1677216, 387420489… The digits in the unit place of these numbers are 1, 4, 7, 6, 5, 6, 3, 6, 9, 0… The sequence repeats after 20th term. The twenty terms are 1, 4, 7, 6, 5, 6, 3, 6, 9, 0, 1, 6, 3, 6, 5, 6, 7, 4, 9 and 0. This is verified by expanding 11^11, 12^22, 13^33, 14^44…

(j) If xp = q and q has ‘r’ number of digits then the number of digits in ‘x’ is

[(p+r-1)/p]. For example x^4 = 332 150 625. Here p = 4 and q = 9, applying the above formula we get x = 3. In fact 135^4 = 332 150 625.

(k) No cube number can be written as a sum of two other cube numbers and vice versa. For example the sum of 12^3+11^3 cannot be written as another cube number.

(l) 9n ends with the unit digit as 1 or 9 depending on whether ‘n’ is odd or even. If ‘n’ is odd 9n ends with the unit digit 9 and vice versa.

(5.21) All odd squares leave a remainder of 1 when divided by 8.

For example 121/8 leaves a remainder of 1.

(5.22) The sum of two odd squares will not be a square number. This rule is used to find out the Pythagorean triples.

For example 121 + 49 = 170 and the resultant 170 is not a square number.

(5.23) Even numbered powers will have the sum of the digits that is number digits as 1, 4, 7 or 9 only. For example 4^4 = 256 and the sum of the digits of 256 is 4.

(5.24) Sum of the digits of a cubic number that is number digit will always end with 1, 8 or 9 and also it is seen that the order of 1, 8 or 9 is coming serially.

(5.25) No power number’s sum of the digits that is number digit will end with 3.

(5.26) The digits in the tens place of any cubic number (power of 3) is always an even number. For example 4^3 = 64 ; 9^3 = 729

The vice versa of the above concept that is the tens place of 3n (where ‘n’ is a positive integer) is always an even number. For example 3^8 = 6561.

(5.27) A square number cannot end with odd number of zeros.

For example 20^2 = 400

(5.28) A perfect square cannot end in 6 with its previous digit being even.

For example 196.

The converse of the above concept is also true. That is a number could not be a square number with 6 in the unit decimal place followed by an even number in the tenth place.

(5.29) Every perfect square has an odd number of factors whereas every non perfect square has an even number of factors.

For example 16 has 1, 2, 4, 8 and 16 factors (Total number of factors is 5)

24 has 1, 2, 3, 4, 6, 8, 12 and 24 factors (Total number of factors is 8)

(a) Also it is to be stated that the sum of the factors of the perfect square number is an odd number and the sum of the factors of the non perfect square number is an even number. That is in the above case the sum of the factors of 16 is 31 and the sum of the factors of 24 is 60.

(5.30) The number of digits in a cube root of a number is the same as the number of 3 digit groups in the given number (from left to right for decimals and from right to left for non decimals) including a single or two digit group left out in the 3 digit grouping.

For example 437,240 will have two digits in its cubic root.

25,072,306 will have three digits in its cubic root

45 will have single digit in its cubic root

Also the opening digit of a cube root of a number can be found from the cubic root of the first three digit group of the number.

Note: - The above rule is applicable for other powers also. For example the square root of a number can be found by grouping the number in 2’s and the fourth power root of a number is found by grouping the digits in 4’s and so on.

(5.31) Sum of the digits of the powers of the 3rd multiples is always 9.

3rd multiples

3

3rd multiples power

(3)2 =9

Sum of the digits of 3rd multiples power

9

6

(3+3)2=36

3+6=9

9

(3+3+3)2=81

8+1=9

12

(3+3+3+3)2=144

1+4+4=9

15

(3+3+3+3+3)2=225

2+2+5=9

…..

…………………….

……………

(5.32) Any number to the power of a square number is always a square number. Similarly for cubes and nth powers.

For example 21^4= 194,481 =441^2

5^9=1,953,125=125^3

(5.33) (a)Every positive integer can be written as the sum of four perfect squares.

For example 75=64+9+1+1

(b) Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes:

23 = 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3.

(5.34) If from the sequence 1,2,3,4,5,6,…we delete every nth term, then every (n-1)stterm, then every (n-1)ndterm, … and lastly every 2nd term, computing the partial sums after each round of deletions, then the final sequence of partial sums is the sequence of nth powers.

For example fifth power sequence is calculated as follows.

Step 1:- write the real number sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14…

Step 2:- delete every fifth number we get 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14…

Step 3:- partial sums of the above sequence 1, 3, 6, 10, 16, 23, 31, 40, 51, 63, 76, 90…

Step 4:- delete every fourth term of the above sequence 1, 3, 6, 16, 23, 31, 51, 63, 76…

Step 5:- partial sums of the above sequence 1, 4, 10, 26, 49, 80, 131,194, 270…

Step 6:- delete every third term of the above sequence 1, 4, 26, 49, 131, 194…

Step 7:- partial sums of the above sequence 1, 5, 31, 80, 211, 405…

Step 8:- delete every second term of the above sequence 1, 31, 211…

Step 9:- partial sums of the above sequence 1, 32, 243…

Now the above sequence is the fifth power sequence. Likewise we can calculate for the other power sequences.

(5.35) (a)The continuous square root of any number will lead to one.

For example square root of 21 = 4.58; square root of 4.58 =2.14; square root of 2.14=1.46; square root of 1.46= 1.21; square root of 1.21=1.09; square root of 1.09=1.04; square root of 1.04=1.02; square root of 1.02=1.01; square root of 1.01=1.006; square root of 1.006=1.002; square root of 1.002=1.001; square root of 1.007, square root of 1.0003, square root of 1.0002, square root of 1.00009, square root of 1.000046, square root of 1.0002, square root of 1.00001, square root of 1.000005, square root of 1.000003, square root of 1.000001, 1.0000007, square root of 1.0000004, square root of 1.0000002, square root of 1.0000001=1.

(b) The square root of 0.11111… is 0.33333…

(5.36) A comparative study of ending digits of square and its square root: -

(a)If the square ends in 1 then the last digits of its square root will either be 1 or 9.

(b) If the square ends in 4 then the last digits of its square root will either be 2 or 8.

(c) If the square ends in 5 then the last digits of its square root will either be 5 or 00.

(d) If the square ends in 6 then the last digits of its square root will either be 4 or 6.

(e) If the square ends in 9 then the last digits of its square root will either be 3 or 7.

(f) The square of any integer is either a multiple of 4 or one more than multiple of 4.

(5.37) A number formed with 3n identical digits is divisible by 3n. For example if n=2 then 3^2 = 9. So the numbers with nine identical digits is divisible by 9. That is 222,222,222÷9 = 24,691,358.

(5.38) The power of an odd number will be an odd number and the power of an even number will be an even number. For example 3^2 = 9 and 4^2 = 16.

(5.39) A comparative study of ending digits of cube and its cube root: -

(a)If a cubic number ends in 1, 4, 5, 6, 9 or 0 its cubic root will also end in the same number respectively. For example the cubic root of 64 is 4.

(b) If a cubic number ends in 2, the unit digits of the cubic root will end in 8 and vice versa.

(c) If a cubic number ends in 7, the unit digits of the cubic root will end in 3 and vice versa.

(5.40) 2 to the power of 2 again to the power of ‘n’ followed by the addition with 1, ends in the digit 7 for all the positive integers where n>1.

(5.41) Automorphic number:-

(a)Let ‘x’ be a number with the property that xn ends in the same digit as ‘x’ that is if the number of digits in ‘x’ is k, then the last k digits of xn are the same as those of ‘x’, then ‘x’ is called as an automorphic number. Or simply we can say that if the ending digit of a number is 0, 1, 5 or 6, the power of that number too ends in 0, 1, 5 or 6 respectively.

For example any number ends with twenty five to the power ‘n’ also ends with twenty five. That is 225^3=11390625

Single digit automorphic numbers are 0, 1, 5 and 6. For example 10^2=100; 11^3=1331; 125^3=1953125 and 126^3=2000376.

Two digit automorphic numbers are 00, 01, 25 and 76.

(b) Like the two digit number 25 which is giving this type of power phenomenon, three digit number 625 gives the same type of phenomenon. For example 3625^2 = 13140625

Similarly the strings of following numbers express this type of power phenomenon.

………….2890625

(c) From the above type of phenomenon we can say that any number ends with seventy six to the power ‘n’ also ends with seventy six. That is 276^2=76176

Like the two digit number 76 which is giving the power phenomenon, three digit

number 376 gives the same type of phenomenon. For example 3376^2 = 11397376

Similarly the strings of following numbers express this type of power phenomenon.

………….7109376

(5.42) The only square number which is having its sum of the digits equal to the square root of that number is 81. That is the sum of the digits of 81 is 9 which is nothing but the square root of 81.

(5.43) For every positive integral value of ‘n’ the number 3(1^5+2^5+3^5+…+n^5) is divisible by 1^3+2^3+3^3+…+n^3

For example if n = 3 then 3(1+32+243) = 828 and 1+8+27 = 36. Now 828÷36 = 23.

(5.44) Special triangle or Power triangle

Number of decimals ‘n’

Number of decimals (2n-1)

1^2

1

11^2

121

111^2

12321

1111^2

1234321

11111^2

123454321

111111^2

12345654321

1111111^2

1234567654321

11111111^2

123456787654321

111111111^2

12345678987654321

(a) The resultant in this pyramid are palindrome numbers. But further proceeding of the number 1111111111^2 will not give palindrome number.

(b) The maximum number coming in the resultant is equal to the sum of the

digits of its square root.

(c)The powers of 11 up to 4th power is palindrome. For example 11^4=14641 and 111^3=1367631. The number 11^5 is not a palindrome number.

(d)The resultant of the power numbers contain odd digits only.

(5.45) Armstrong number: - If the addition of the cubes of each individual digit in a number is equal to the number itself means then it is called as Armstrong number.

For example (a) 153=1^3+5^3+3^3=1+125+27

(b) 370=3^3+7^3+0^3=27+343+0

(c) 371=3^3+7^3+1^3=27+343+1

(d) 407=4^3+0^3+7^3=64+0+343

(5.46) Special numbers:-

(a) 89 = 8^1+9^2 = 8+81

(b) 135 = 1^1+3^2+5^3 = 1+9 +125

(c)175 = 1^1+7^2+5^3 = 1+49+125

(d) 2^2-1^2 = 3 = 2+1

(e) 36 = 1^3+2^3+3^3

(f) 6^3 = 3^3 + 4^3 + 5^3 That is 216 = 27 + 64 + 125

(g) 100 = 1^3 + 2^3 + 3^3 + 4^3

(h) 4913 = 17^3; 4+9+1+3 = 17

(i) 3435 = 3^3+4^4+3^3+5^5 = 27+256+27+3125

(j) 1634 = 1^4+6^4+3^4+4^4 = 1+1296+81+256

(k) 8208 = 8^4+2^4+0^4+8^4 = 4096+16+0+4096

(l) 9474 = 9^4+4^4+7^4+4^4 = 6561+256+2401+256

(m) 4150 = 4^5+1^5+5^5+0^5 = 1024+1+3125+0

(n)548834=5^6+4^6+8^6+8^6+3^6+4^6=15625+4096+262144+262144+729+4096

(o)2^9+5^9 =512 × 1953125 = 1000 000 000

(5.47) If x = y × z then xn = yn × zn

For example 10 = 2 × 5 then 10^9 = 2^9 × 5^9 = 512 × 1,953,125

6 = 3 × 2 then 6^3 = 3^3 × 2^3 = 27 × 8

(5.48) When an ‘n’ digit number is squared and in the square number if the right hand ‘n’ digits are added to the left hand ‘n ’ digits and is equal to the original number that is the square root itself means, then it is called as Kaprekar number.

The Kaprekar number series is 1, 9, 45, 55, 703, 2223….

For example 9 and the square of it is 81. Now the addition of the right hand digit and the left hand digit in the squared number is 8+1=9.

Other examples are 45^2=2025 and 20+25=45

55^2=3025 and 30+25=55

703^2=494209 and 494+209=703

(5.49) The sequence of squares and the sequence of odd numbers have a mysterious relationship as follows.

1^2= 1 =1

2^2= 4 =1+3

3^2= 9 =1+3+5

4^2=16=1+3+5+7

5^2=25=1+3+5+7+9

………………………..

(5.50) The relationship between the arithmetic sum of odd numbers and the sequence of arithmetic cubes is as follows.

Odd number series 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,…..

The above series can be arranged as 1; 3+5; 7+9+11; 13+15+17+19; 21+23+25+27+29…

Now 1 = 1^3 =1

8 = 2^3=3+5

27=3^3 =7+9+11

64=4^3 =13+15+17+19

125=5^3=21+23+25+27+29

.…..………………………

(a) Another relationship between the arithmetic sum of discrete numbers and the sequence of cubes is as follows.

1 = 1

2 + 3 + 4 = 1 + 8

5 + 6 + 7 + 8 + 9 = 8 + 27

10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64

…………………………………………………..

(b) The sequence of cubes is equal to the sequence multiplication of arithmetic progression.

(1×1)=1^3=1

(2×1)+(2×3)=2^3=8

(3×1)+(3×3)+(3×5)=3^3=27

(4×1)+(4×3)+(4×5)+(4×7)=4^3=64

(5×1)+(5×3)+(5×5)+(5×7)+(5×9)=5^3=125

………………………………………………

(5.51) If the natural numbers are arranged into groups like

(1), (2,3), (4,5,6), (7,8,9,10), (11,12,13,14,15),…

Now strike out the second groups alternately we get

(1), (4,5,6), (11,12,13,14,15),…

From the above sequence we can find out the fourth power of any number as follows.

For getting 1^4 add the first group number only that is 1

For getting 2^4 add the first and second group numbers only that is 1+4+5+6 = 16

For getting 3^4 add the first, second and third group numbers only and so on.

(5.52) Another relationship between the fourth power of numbers and the arithmetic sum of odd numbers is as follows.

1^4=1

2^4=1+3+5+7

3^4=1+3+5+7+9+11+13+15+17

4^4=1+3+5+7+9+11+13+15+17+19+21+23+25+27+29+31

………………………………………………………………..

(5.53) Multiply any four consecutive numbers and add number 1 to the product. The resultant would be a complete square.

For example (3×4×5×6)+1=360+1=361=19^2

(11×12×13×14)+1=24024+1=24025=155^2

(5.54) 7 × 2^1 = 14 × 2^0 = 7 × 2 = 14

7 × 2^2 = 14 × 2^1 = 14 × 2 = 28

7 × 2^3 = 14 × 2^2 = 28 × 2 = 56

7 × 2^4 = 14 × 2^3 = 56 × 2 = 112

7 × 2^5 = 14 × 2^4 =112 ×2 = 224

7 × 2^6 = 14 × 2^5 =224 ×2 = 448

7 × 2^7 = 14 × 2^6 =448 ×2 = 896

7 × 2^8 = 14 × 2^7 =896 ×2 =1792

7 × 2^9 = 14 × 2^8 =1792×2=3584

…………………………………

Likewise we can extend for the multiplication of power numbers with a constant number.

(5.55) (1^5+2^5+3^5+…+n^5) + (1^7+2^7+3^7+…+n^7) is always equal to 2(1+2+3+…+n)4

For example if n = 4 then (1^5+2^5+3^5+4^5) + (1^7+2^7+3^7+4^7) = 2(1+2+3+4)4

1300 + 18,700 = 20,000

20,000 = 20,000

(5.56) √n√n√n+…={[1+(√4n+1)]/2} For example √6√6√6+…=3

(5.57) Pythagorean triples: - If x^2+y^2=z^2 where x, y and z are real numbers then x, y and z are called as Pythagorean triples.

For example 9, 12 and 15 are called as Pythagorean triples.

9^2+12^2=15^2

81+144=225

(a) Pythagorean triples 3^2+4^2=5^2 ; 5^2+12^2=13^2 and the general equation is

(2n+1)2 + [2n(n+1)]2 = [2n(n+1)+1]2 For example the third equation is n=3 and substituting in the formula we get 7^2+24^2=25^2.

(b) Also it is to be noted that z^2 = m^2 + n^2 That is if the square of a number is equal to the sum of two square numbers, then the number itself is a sum of two square numbers.

For example in the above case 15^2 = 12^2 + 9^2

225 = 144 + 81

225 = 225

(c) Further it is to be noted that either of x or y in x^2+y^2=z^2 and either of m or n in

z^2 = m^2 + n^2 is odd and another should be even. It is not possible to have both the numbers in (m, n) pair or (x, y) pair as odd number or even number.

(d) Pythagorean theorem gives the relationship between the sides and hypotenuse of the right angled triangle. Thus 3^2+4^2=5^2; 5^2+12^2=13^2 and so on. Similarly we can relate the edges of a cuboid and its diagonal. For example replace 5^2 = 3^2 + 4^2 in the second equation we get 3^2 + 4^2 + 12^2 = 13^2

(5.58) Rachnisky sequence: - If (x-1)2 + x^2 + (x+1)2 = (x+2)2 + (x+3)2 then the sequence is known as Rachinsky sequence.

For example 10^2 + 11^2 + 12^2 = 13^2 + 14^2

(-2)2 + (-1)2 + 0^2 = 1^2 + 2^2

(5.59) 3^2=09

33^2=1089

333^2=110889

3333^2=11108889

33333^2=1111088889

…………………….........

(5.60) The square numbers 25 and 36 have special characteristic behavior that they are having the last digit of the number 5 and 6 as their square roots respectively.

(5.61) Power table

N

n^1

n^2

n^3

n^4

n^5

n^6

n^7

n^8

n^9

n^10

1

1

1

1

1

1

1

1

1

1

1

2

2

4

8

16

32

64

128

256

512

1024

3

3

9

27

81

243

729

2187

6561

19683

59049

4

4

16

64

256

1024

4096

18384

65536

262144

1048576

5

5

25

125

625

3125

15625

78125

390625

1953125

9765625

6

6

36

216

1296

7776

46656

279936

1679616

10077696

60466176

7

7

49

343

2401

16807

117649

823543

5764801

40353607

282475249

8

8

64

512

4096

32768

262144

2697152

16777216

134217728

1073741824

9

9

81

729

6561

59049

531441

4782969

43046721

387420489

3486784401

10

10

100

1000

10000

100000

1000000

10000000

100000000

1000000000

10000000000

Note:- For 10x put 1 followed by 1 × x number of zeros in the power number. For 100x put 1 followed by 2 × x number of zeros and for 1000x put 1 followed by 3 × x number of zeros and so on.

(5.62) Powers against digits of the number

Number

square

cube

4th power

5th power

6th power

7th

power

8th

power

9th power

10th

power

Single digit

2

(4,9)

1

(8)

--

--

--

--

--

--

--

Double digit

6

(16,25,36,49

64,81)

2

(27,

64)

2

(16,

81)

1

(32)

1

(64)

--

--

--

--

Triple digit

22(100,121,

144,169,196,

225,256,289,

324,361,400,

441,484,529,

576,625,676,

729,784,841,

900,961)

5

(125,

216,

243,

512,

729)

2

(256,

625)

1

(243)

1

(729)

1

(128)

1

(256)

1

(512)

--

(5.63) Chenshuwen’s equal sums of like powers

(i)1a+19a+20a+51a+57a+80a+82a = 2a+12a+31a+40a+69a+71a+85a

where a=1, 2,3,4,5 and 6

(ii)975b+224368b+300495b+366448b = 37648b+202575b+337168b+344655b

where b=2, 3 and 4

(iii) 1c+12c+25c+66c+91c+130c+174c+213c+238c+279c+292c+303c

= 4c+6c+31c+58c+105c+117c+187c+199c+246c+273c+298c+300c

where c=1,2,3,4,5,6,7,8,9,10 and 11

(6.64) In 1x + 2x + 3x +…+ 10x for all values of x from 1 to 9 the resultant is divisible by 11. For example if x = 1 then we get the resultant 55 which is divisible by 11.

(5.65) For real number series, the first common difference is 1 that is 0, 1, 2, 3,4,5,6,7,…

the first common difference is 1, 1, 1, 1…

For squares, the second common difference is 2 that is 0, 1, 4, 9, 16, 25, 36…

the first common difference is 1, 3, 5, 7, 9, 11…

the second common difference is 2, 2, 2, 2…

Likewise we can extend for the other power number sequences.

For cubes the common third difference is 6

For fourth powers the common fourth difference is 24

For fifth powers the common fifth difference is 120

………………………………………………………..

From the above data 1, 2, 6, 24, 120, 720…..

That is (1×1), (1×2), (1×2×3), (1×2×3×4), (1×2×3×4×5) …

(5.66) The square number series are 1,4,9,16,25,36,49,64,81,100,121,144,,……..

The multiplication of 1×1= 1 =1^2

1×4= 4 =2^2

1×4×9=36=6^2

1×4×9×16=576=24^2

1×4×9×16×25=14400=120^2

………………………………..

From the above data 1^2, 2^2, 6^2, 24^2, 120^2, 720^2…..

That is (1×1)2, (1×2)2, (1×2×3)2, (1×2×3×4)2, (1×2×3×4×5)2………

(5.67) Ramanujan number: - Ramanujan number is defined as the number which can be represented as the summation of two positive numbers or with their powers in two different ways.

(a) It can be classified into first order Ramanujan number, second order Ramanujan number, third order Ramanujan number and so on based on power raised on the number.

(b) First order Ramanujan’s number is defined as the number which can be represented as the summation of real numbers in two different ways.

For example 1+4 = 2+3 = 5

In general terms the above equation can be represented as L_1+L_2 = R_1+R_2

(c) Now place the digits of R_1 and R_2 adjacent to L_1 and L_2 and similarly place the digits of L1 and L_2 adjacent to R_1 and R_2 to get again the Ramanujan number.

That is L_1R_1+L_2R_2 = R_1L_1+R_2L_2

In the above case 12+43 = 21+34 = 55

Also it is seen that the components of the number in the left hand side of the equation is palindrome to the components of the number in the right side of the equation. That is 12 is palindrome to 21.

(d) For second order Ramanujan’s number summation of squares is considered.

For example 338 = 7^2+17^2 = 13^2+13^2

(e) The second order Ramanujan’s number can be found by the formula 13[(n+1)2+1] where n = 1, 2, 3….

For example if n = 1 means 65 = 1^2+8^2 = 4^2+7^2

(f) Another test for second order Ramanujan’s number is the summation of the numbers in the series can be divided by 10. For example in the above case 1+8+4+7=20 and the resultant 20 can be divided by 10.

(g) For third order Ramanujan’s number summation of cubes is considered. That is Ramanujan’s number of third order is defined as the summation of cubes of the two different numbers in two different ways. The first such number is 1729.

That is 1729 = 9^3+10^3 = 1^3+12^3

= 789+1000 = 1+1728

The next such number is 4104 and it can be represented as 2^3 + 16^3 = 9^3 + 15^3.

(5.68) Features of the number 1729:-

(a)The number 1729 is an odd number and peculiarly the divisors of the numbers are odd numbers. That is 1, 7, 13, 19, 91, 133, 247 and 1729 are the divisors of 1729 and are all odd numbers. Also in these divisors, except 91 all are prime numbers.

(b) The divisors of 1729 except 247 can be expressed as the summation of cubes of two different numbers or the difference of cubes two different numbers.

1=1^3+0^3

7=2^3-1^3

19=3^3-2^3

91=3^3+4^3

133=2^3+5^3

(c) The difference between the first four divisors of the number 1729 is 6. That is 1,7,13 and 19.

(d) The multiplication of the first four divisors of the 1729 is equal to 1729.

That is 1×7×13×19=1729.

(e) The multiplication of all the divisors of 1729 is

1×7×13×19×91×133×247×1729 = 1729^4.

And deleting the first four divisors and keeping the multiplication of the remaining divisors of 1729 is 91×133×247×1729 = 1729^3

And still further deleting the last divisor and keeping the multiplication of the remaining divisors of 1729 is 91×133×247 = 1729^2

(f) The resultant obtained by the summation of the divisors of 1729 can also be expressed as the sum of the cubes of the two different numbers or the difference of the cubes of two different numbers.

That is 1+7+13+19+91+133+247 = 511 = 8^3-1^3

1+7+13+19+91+133+247+1729 =2240= 12^3+8^3 =14^3+2^3-8^3

(g) We can state that 9^3 = 25^2+10^2 +2^2 So the basic Ramanujan’s equation can be written as 1729 = 9^3+10^3 = 10^3+25^2+10^2 +2^2

(h) Other cube relatives for the number 1729 are

1729 = 24^3+11^3+9^3-27^3

= 33^3+16^3+10^3-34^3

= 16^3+10^3+2^3-15^3

(i) The divisors of 1729 that is 1, 7,13,19,91,133,247 and 1729 when arrange in increasing order and multiplied as follows will give the resultant 1729.

1×1729= 1729

7× 247 = 1729

13×133= 1729

19 ×91 = 1729

(j) The divisors of 1729 have one extra to 6 or one extra to multiples of 6.

That is 7 = (6×1) +1

13 = (6×2) +1

19 = (6×3) +1

91 = (6×15) +1

133 = (6×22) +1

1729 = (6×288) +1

(k) 1729 is a Harshed number. That is the resultant of the sum of the digits of 1729 is a divisor of 1729 itself. That is 1+7+2+9 = 19 and 19 is one of the divisors of 1729.

(l) When the number 1729 is divided by the numbers from 1 to 9 except 5 and the divisors 1 and 7, the remainder is one only.

1729÷2 = 864 and the remainder is 1

1729÷3 = 576 and the remainder is 1

1729÷4 = 432 and the remainder is 1

1729÷6 = 288 and the remainder is 1

1729÷8 = 216 and the remainder is 1

1729÷9 = 192 and the remainder is 1

(m) 1729 can be expressed as the difference of squares of two numbers.

That is 1729= 55^2 – 36^2

= 73^2 – 60^2

=127^2 -120^2

=865^2 -864^2

Also we can see that the difference of numbers used in the above equations gives the divisor of 1729. That is 55 – 36 = 19; 73 – 60 = 13; 127 – 120 = 7; 865 – 864 = 1

(n) 1729 cannot be related by the difference of cubes of two numbers.

(o) 1729 can be related by the squares of three different numbers as follows.

That is 1729 = 37^2+19^2-1^2

= 41^2+ 7^2-1^2

= 38^2+17^2-2^2

= 42^2+ 1^2-6^2

(5.69) Ramanujan’s equation:-

Consider the following equations

1^2+1=2×1^2…………Equation (1)

7^2+1=2×5^2…………Equation (2)

41^2+1=2×29^2………Equation (3)

Now call the numbers in square form as x and y and generalize the above equations as follows x^2+1=2×y^2

Take any equation, multiply x of it by 6 and subtract the x of the previous equation.

For example x of the second equation is 7 and multiplication with 6 gives 42. Now subtract 1, the x value of the previous equation from 42 will give 41 which is x of the next equation.

It applies to y also. That is 6×5-1=29

We can get any number of equations by using the above method.

[(41×6)-7]2+1 = 2× [(29×6)-5]2

[246-7]2+1 = 2 × [174-5]2

239^2+1 = 2×169^2… Equation (4)

Another method of getting the similar type of equations is as follows.

3x+4y = x of the next equation

2x+3y = y of the next equation

For example the equation (4) can be obtained as follows.

[(3×41) + (4×29)]2+1 = 2 × [(2×41) + (3×29)]2

[123+116]2+1 = 2 × [82+87]2

239^2+1 = 2×169^2… Equation (4)

We can keep an equation where x = y = 1 as the first equation and further we can develop the equations.

(5.70) Arithmetic approach of powers involving addition, subtraction, multiplication and division using the alphabets is called as algebra.

For example (a+b)2 = a^2+2ab+b^2

Let us consider a=2 and b=3

(2+3)2 = 2^2+(2×2×3)+3^2

5^2 = 4+12+9

25= 25

(5.71) A Cullen number is a number of the form 2nn + 1. For example 3, 9, 25, 65, 161, 385...

6. DIVISION

(6.1) Division is the reverse process of multiplication. Division is denoted by the symbol ÷ or simply by /. There are two types of division namely perfect division and imperfect division or fraction division.

For example 30/3=10 Here 30 is called as dividend or numerator, 3 is called as divisor

or denominator and 10 is called as quotient or resultant. Here there is no remainder left over. This type of division is called as perfect division.

But in 30/4=7(2/4) or 7.5 In this case 2 is called as the remainder. This type of division

is called as imperfect division or fractional division.

Dividend = Quotient + Remainder

Divisor

(6.2) Rule of division: - Unlike addition, subtraction and multiplication we have to take last decimal number of dividend and divide from left to right side of the number by the divisor. Suppose the last decimal number is smaller than the divisor then we have to take both the last decimal number and its next right side decimal number and so on.

For example 12812÷2=6406

(6.3) The digital root of the divisor multiplied by that of the quotient, added to the digital root of the remainder is equal to the digital root of the dividend. That is in the formula

Dividend = Quotient + Remainder use digital root instead of the actual numbers.

Divisor

For example 6432÷23 gives 279 as the quotient and 15 as the remainder. Now (5×9)+6=51 whose digital root 6 is the digital root of the dividend 6432.

(6.4) The remainder theorem: -The remainder obtained from the product of numbers when divided by another number say ‘n’ is same as the remainder of the product of the remainders of the numbers divided individually by the same number ‘n’.

For example 20x41x19 when divided by 6 the remainder of the product is 4.

The remainder of 20÷6 is 2; 41÷6 is 5 and 19÷6 is 1. The product of the remainders is 2x5x1=10 and the remainder of the product 10÷6 is 4.

(6.5) If two numbers when divided by another number say ‘n’ give the same remainder means then the difference between the two numbers will be perfectly divisible by ‘n’.

For example the remainder of 41÷2 is 1 and the remainder of 21÷2 is 1. Now the difference of 41 and 21 is divisible by 2.

(6.6) The fractional division can be further divided into two types namely definite fraction or proper fraction and indefinite fraction or improper fraction. If the fractional division ends with a real number then it is called as a definite fraction. For example 30/4=7.5 is a definite fraction or finite fraction. But if the fractional division ends with a number and it continues and repeats itself means it is called as indefinite fraction or infinite fraction.

**

For example 30/11=2.72727272…or simply 2.72

*

Another example 10/3=3.333…… or simply 3.3

* *

One more example 1/7=0.142857

Note:-The asterisk above the digit represents the recurrence of the remainder. Usually the first asterisk above the digit represents the number from where the recurring starts and another asterisk above the number indicates where the recurring ends and repeats further.

(6.7) In a fractional division except 2/5 when both the numerator and denominator contains prime number then it will give an infinite fraction.

*

For example 3/7=0.428571

(6.8) Another example for infinite fractions are fractions whose denominators are 3’s like 1/3, 1/33, 1/333… and the corresponding infinite fractions are 0.333, 0.030, 0.003… respectively. This infinite fraction can be represented by 0.3+0.03+0.003+…..Here 1/3 is called as rational number and 0.333 as a repeating decimal. Conversion of this type of repeating decimal number to rational number is as follows.

*

0.3=1(0.3+0.03+0.003+….)

=1(0.3/0.9) or 1(3/9)

=1/3

Another example to convert the repeating decimal to rational number

*

4.76=4(0.76+0.076+0.0076+….)

=4(0.76/0.99) or 4(76/99)

=472/99

Further infinite fractions are obtained from the fractions whose denominators are 1’s like 1/11, 1/111, 1/1111… and the corresponding infinite fractions are 0.09, 0.009, 0.0009… respectively.

Note: - The above rule is applicable to all fractions whose denominators are 1’s, 3’s or 9’s. For example 8/99=0.08, 17/333=0.051 and 107/111=0.963 recurring fractions obtained.

A decimal point of a number borders between the real number’s units, tens, hundreds… from the left side to right side and the rational number’s units, tens, hundreds… from the right side to left side. That is

…hundreds, tens, units . tens, hundreds…

For example 6537.3451

(6.9) Infinite fraction is further classified into two types namely recurring infinite fraction like (30/11) and non recurring infinite fraction or unending fraction. For example √2 = 1.41421356… and (22/7) or π value = 3.14159265358… are non recurring infinite fractions.

Diagrammatically the classification of division can be mentioned as follows.

(6.10) A number which is divisible by another number without any remainder apart from 1 and the number we started itself means it is called as a composite number or divisible number. For example 21 and this number can be divided by 1, 3, 7 and 21 without any

remainder and these numbers are called as factors or divisors of 21.

Divisible number series is 4, 6, 8, 9, 12, 14, 15, 16, 18, 20…

(6.11) For any ‘n’ positive integers there exists at least a subset of them whose sum is divisible by ‘n’. For example let n = 6 and the integers be 7, 9, 10, 11, 13 and 14. Now (10, 14) a subset of them whose sum is 24 is divisible by 6. Another example let n = 8 and the numbers be 1, 7 ,9, 13, 17, 27, 2 and 30. Now (7, 9) a subset of them whose sum is 16 is divisible by 8.

(6.12) Highest common factor is shortly called as HCF. For example HCF for 63 and 84 can be calculated as follows.

3 1 63,84

7 1 21,28

3, 4 Now multiply 3 and 7 which are the common divisors of 63 and 84. The resultant obtained is 21 is called as the HCF of 63 and 84.

(6.13) Least common multiplier is shortly called as LCM. LCM for 63 and 84 can be calculated as follows.

3 1 63,84

7 1 21,28

3, 4 Now multiply 3 and 7 followed by the multiplication of 3 and 4, the remainders. 3×7×3×4 = 252 and the resultant obtained is 252 is called as the LCM of 63 and 84.

(a) Whenever if you take LCM leave or cancel the common factors of the denominator and then take the LCM.

For example 1/ (a+x)2 + 1/(a+x)(a-x) = 2a/(a^2 – x^2)

(6.14) If a, b and c are the three consecutive numbers and x, y and z are their corresponding lowest possible divisors then the product of x, y and z will be a divisor for the product of a, b and c.

For example 110,111 and 112

2 is the divisor for both 110 and 112.

3 is the divisor for 111.

Now the product of 2×3 = 6 is the divisor of the product of 110×111×112=1367520. That is the resultant 1367520 is divisible by 6.

(6.15) Prime number: -

(a)Prime number is a number which does not have a factor or divisor apart from 1 and the number itself. This number can also be called as indivisible number. For example 31 is a prime number because it is not having any number to divide it without any remainder apart from 1 and 31 itself.

The prime number series is 1,2,3,5,7,11,13,17,19,23,29,31,37,41,47,53,59,61,67,71…

Note: - The number 1 is called as unit number rather than a prime number.

(b) Except 2 prime numbers are always odd numbers.

(c) A conventional method of finding out a prime number is by using the Eratosthenes sieve as follows. Starting from 2 cross out every second number (or multiples of 2) but not 2 itself. Move to 3 and cross out every third number but not 3 itself. Then with 4, cross out every fourth number and continue crossing out of numbers up to 10. This will eliminate all the numbers which are not prime numbers.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

Now the prime numbers up to 100 are the numbers left out in the table. Likewise we can find the prime numbers by extending the table.

(d) The following formula x^2 – x + 41 where x = 0 to 40 gives prime number.

For example if x = 2 means 2^2 – 2 + 41 = 2 + 41 = 43 and 43 is a prime number.

(e) Another formula 2x^2 + 29 where x = 0 to 28 gives prime number.

For example if x = 3 means 2(3)2 + 29 = 18 + 29 = 47

(f) If ‘n’ is a positive integer then 2n – 1 is a prime number.

For example 2^6 – 1 = 64 – 1 = 63 is a prime number and the prime numbers formed are called as Mersenne prime numbers.

(g) Every even integer greater than 2 can be written as the sum of two primes.

For example 200=7+193

(h) Co prime or relatively prime numbers: - Two numbers a and b are said to be co prime or relatively prime numbers if they share no common factor between them.

For example 10 and 21 are co prime numbers because they don’t have a common factor

But 25 and 35 are not co prime numbers because they have a common factor 5.

Note: - Since 1 is a factor of all numbers, the number one cannot be considered as a common factor between two numbers.

(i) Twin primes: - Prime numbers that differ by 2 are called as twin primes.

For example (3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59,61), (71,73)…

(j) The numbers 9, 98, 987, 9876, 98765, 987654, 9876543, 98765432 and 987654321 are all composite numbers. But the reverse of these numbers leaving 9, will give prime numbers.

(k) There are 25 primes between 1 and 100. But there are only 9 primes between 9,999,900 and 10,000,000 and there are only 2 primes between 10,000,000 and 10,000,100. Those two primes are 10,000,019 and 10,000,079. The distribution of primes among natural numbers is very irregular.

(l) The reflux number of the following prime numbers are also prime numbers. That is 13, 31; 17, 71; 37, 73 and 79, 97

(m) 3, 5 and 7 are unique primes where these consecutive primes can be written as 3, 3 + 2 and 3 + 4.

(n) Every prime number ‘p’ except 2 and 5 will divide a number formed with (p-1) nines. For example 99 is divisible by 3 and similarly 999999 is divisible by 7 and so on.

(o) The sum of two prime numbers is a composite number provided that one prime number should not be 2. If one of the prime number is 2 its addition with another prime number may give again a prime number or a composite number. For example 2 + 11 = 13 and 13 is a prime number but 2 + 13 = 15 give a composite number.

(p) A safe prime is a prime number of the form 2p + 1, where p is also a prime. (Conversely, the prime p is a Sophie Germain prime.) The safe prime sequence is 3, 5, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907...

(6.16) Division is non commutative.

For example 2/5 ≠ 5/2

(6.17)

Division

Odd number

Even number

Odd number

odd number

even number

Even number

fraction

odd or even number

From the table it can be seen that 10/2=5 16/4=4

From the tables it can be seen that the operations with odd number and even number for addition and subtraction is same. But the operation with odd number and even number for multiplication and division is not same. Division obeys uniquely.

(6.18)

Division

Positive number

Negative number

Positive number

+

-

Negative number

-

+

For example -10/(-5)=2

(6.19) Division Table

Division

1

2

3

4

5

6

7

8

9

10

1

1

2

3

4

5

6

7

8

9

10

2

1/2

1

3/2

2

5/2

3

7/2

4

9/2

5

3

1/3

2/3

1

4/3

5/3

2

7/3

8/3

3

10/3

4

1/4

2/4

3/4

1

5/4

6/4

7/4

2

9/4

10/4

5

1/5

2/5

3/5

4/5

1

6/5

7/5

8/5

9/5

2

6

1/6

2/6

3/6

4/6

5/6

1

7/6

8/6

9/6

10/6

7

1/7

2/7

3/7

4/7

5/7

6/7

1

8/7

9/7

10/7

8

1/8

2/8

3/8

4/8

5/8

6/8

7/8

1

9/8

10/8

9

1/9

2/9

3/9

4/9

5/9

6/9

7/9

8/9

1

10/9

10

1/10

2/10

3/10

4/10

5/10

6/10

7/10

8/10

9/10

1

(6.20) Divisibility tests of numbers

Divisibility by 2:- A number is divisible by 2 if and only if the last digit is an even number or zero. For example 220 is divisible by 2 but not 221.

Divisibility by 3:- If the summation of digits in a number is divisible by 3 or multiples of 3 means the number also will be divided by 3. For example 729. Also the numbers formed by all the digits from 0 to 9 each once are divisible by 3.

For example (9013245768/3)=3004415256

Divisibility by 4:- When a number ends with its last two digits as zero or if the last two digits is divisible by 4 means the number is divisible by 4. For example 2748, 2000

Divisibility by 5:- If a number ends with 5 or zero then it must be divided by 5 without any remainder. For example 6025.

Divisibility by 6:- If a number is divisible by six means the last digit will be even and the sum of the digits is divisible by 3. That is a number which is divisible by 2 and 3 is also divisible by their multiplication product. For example 726.

Divisibility by 7:- Divide the digits of the number into three digit blocks, starting from the right. Compute the alternating three digit sum (the right most block receiving a positive sign). Now the number is divisible by 7 if and only if the alternating sum is divisible by 7. For example 371469 Split the number into 371/469 Now 469-371=98 which is divisible by 7 and so the number 371469 is divisible by 7.

Divisibility by 8:- When a number’s last three digits is divisible by 8 means the number is divisible by 8. For example 2432, 7176.

Divisibility by 9:- If the summation of digits in a number is divisible by 9 means the number also will be divided by 9. For example 927.

Note: - If a number is divisible by 9 means then the number can also be divided by 3, but the numbers which are divisible by 3 may not be divided by 9. Because 3 is the factor of 9. From this we can arrive the fact that a factor of a number x can divide all the numbers which are divisible by x or factors of x.

For example 18/3=6 18/9=2

21/3=7 21/9=fraction

Divisibility by 10:- If a number is divisible by 10 means it will have 0 in the unit decimal place. For example 910.

Divisibility by 11:- When a number’s difference between the converse digits or alternate digits become zero or 11 or multiples of 11 means then the number is divisible by 11. For example 4268 and the converse digits are 4, 6 and 2, 8. Now (4+6)-(2+8) =0

The number 4268 is divisible by 11.

Another example 87635064. Now (8+6+5+6)-(7+3+0+4) =25-14=11

If two numbers have the same digits, but every digit which occurs in an even place in one number is in an odd place in the other, then the sum of the numbers is divisible by 11. For example the resultant of 710534 + 405317 is divisible by 11.

Divisibility by 12:- Check for divisibility by both 3 and 4

For example 4944 can be divided by both 3 and 4 and so it can be divided by 12 also.

Divisibility by 13:- Delete the last digit from the given number. Then subtract 9 times the deleted digit from the remaining number. If what is left is divisible by 13, then so is the original number.

For example 5928 deleting the last digit number we will get 592 and when this number is subtracted by 9 times 8 we get 520. Now the number 520 is divisible by 13 and so the original number is divisible by 13.

In the same way we can continue the divisibility test for higher numbers.

Divisibility by 14: - Check for divisibility by both 2 and 7.

Divisibility by 15: - Check for divisibility by both 3 and 5.

Divisibility by 16: - Check for divisibility by both 2 and 8.

For some of the bigger numbers and prime numbers the divisibility tests are as follows.

Divisibility by 17: - Take the last digit of the number and multiply it by 5. Subtract this 5th multiple from the original number leaving the last digit. If the resultant obtained after subtraction is divisible by 17 means the original number is divisible by 17.

For example 1105 The last digit of this number is 5 and its 5th multiple is 25. Subtract this from the original number then we get 110-25=85 and the resultant is divisible by 17 and so the original number is divisible by 17.

Divisibility by 18: -A number is divisible by 18 when it is divisible both by 2 and 9. In other words if the sum of the digits of an even number is divisible by means then the number is divisible by 18 also.

Divisibility by 19:- A number is divisible by 19 if and only if the number of tens (not the tenth digit of the number but the total number of integral tens in the whole number) added to twice the number of units is divisible by 19.

For example 475 Here [47 + (5×2)] = 57 The resultant 57 is divisible by 19 and so for 475. Another example 4704590 Here [470459+0] = 470459 is divisible by 19 and so the number 4704590 can be divided by 19.

Divisibility by 20: -A number is divisible by 20 when it ends in zero and the second last digit is even.

Divisibility by 24: -A number is divisible by 24 when it is divisible both by 3 and 8.

Divisibility by 25:- When a number ends with its last two digits as zero or if the last two digits are divisible by 25 means the number is divisible by 25. For example 6075.

Divisibility by 30: -A number is divisible by 30 when it is divisible by 3 and 10.

Divisibility by 37:- A number is divisible by 37 if and only if the resultant of the three digit blocks, starting from right is divisible by 37.

For example 47109362 Split the number into 47/109/362 and now 362+109+47=518 which is divisible by 37 and so the number 47109362 is divisible by 37.

If a number of the form abc is divisible by 37 means its cyclic numbers bca and cab are also divisible by 37. For example 259 is divisible by 37 and so for 592 and 925.

Divisibility by 50:- When a number ends with its last two digits as zero or 50 then the number is divisible by 50. For example 1550.

Divisibility by 101:- Break the number into two digit blocks starting from the right. Compute its alternating two digit sum starting from right. Then the sum is divisible by 101 if and if only the alternating sum is divisible by 101.

For example 46763 The number is split into 4/67/63 and the alternate two digit sum yields 63-67+4=0 and is divisible by 101 and so the number 46763 is divisible by 101.

(6.21) The number 2520 can be divided by 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 without any remainder.

(6.22) Specialty of 2519 is when the number is divided by 1 the remainder is 0

when the number is divided by 2 the remainder is 1

when the number is divided by 3 the remainder is 2

when the number is divided by 4 the remainder is 3

when the number is divided by 5 the remainder is 4

when the number is divided by 6 the remainder is 5

when the number is divided by 7 the remainder is 6

when the number is divided by 8 the remainder is 7

when the number is divided by 9 the remainder is 8

when the number is divided by 10 the remainder is 9

(6.23) The number 381,654,729 is a special number in which the number formed by the first two digits from left side onwards is divisible by 2, the number formed by the first three digits is divisible by 3 and so on until the entire nine digit number is divisible by 9.

(6.24) The number 27,216 is divisible by 2, 7, 6, 27, 72, 21, 16, 162 and 216. All these numbers are in the given number.

(6.25) If the number of digits in a number is six and the number is divisible by 7, then the new number formed by shifting the unit digit to the extreme left is also divisible by 7.

For example 251,734 is a six digit number and is divisible by 7. So the number 425,173 is also divisible by 7. That is 425,173÷7 = 60,739

(6.26) Write any three digit number and write it twice. For example 123 and if it is written twice 123123. Divide this number by 7 followed by 11 and 13. There will be no remainder and the resultant is the starter itself. 123123/7 = 17589; 17589/11 = 1599; 1599/13 = 123.

Also it is seen that the product of any two of the above numbers (7, 11 and 13) followed by the third number division also give the same number started.

123123/91 = 1353; 1353/11 = 123

Also if we divide the number by 1001 we will get the same number started.

123123/(7×11×13) = 123123/1001 = 123

The reverse is true. That is to get a three digit number twice multiply the number by 11 and the product by 91 or simply multiply by 1001. For example 666 and to write it twice 666 × 1001 = 666,666 (or)

666 × 11 = 7326 followed by

7326 × 91 = 666666

Note:- The above rule is applicable for every number of the form ABCABC. Similar type of rule is applicable for every number of the form ABABAB is divisible by 3, 7, 13 and 37. For example to get a number of the form ABABAB multiply any number by 10101.

(6.27) A perfect number is defined as the number whose summations of the factors or divisors (excluding the number itself) give the same number or it is the number whose summations of the factors including the number itself is twice the original number.

For example 6 and the summation of the factors of this number 1+2+3=6

28 and the summation of the factors of this number 1+2+4+7+14=28

496 and the summation of the factors of this number

1+2+4+8+16+31+62+124+248 = 496

8128 and the summation of the factors of this number

1+2+4+8+16+32+64+127+254+508+1016+2032+4064 = 8128

33550336 and the summation of the factors of this number 1+2+4+8+16+32+64+128+262112+256+131056+512+65528+1024+32764+2048+4096+8191+16382+524224+1048448+2096896+4193792+8387584+16775168=33550336. Other examples are 8589869056, 137438691328 and 2305843008139952128.

Note: - For ‘n’ digit number only one perfect number is present.

(6.28) A harshad number is defined as the number whose summation of the digits is one of the divisors of the number or simply it is a number which is divisible by the sum of its own digits. For example 1458 whose summation of the digits is 1+4+5+8=18 and 18 is a divisor of the number 1458.

Harshad number series is 12, 20, 24, 27, 30, 36,40,42,45,48…

Note: - All the n! is a harshad number except for n = 432.

(6.29) BODMAS: -Suppose if the sum involves several operations the order of preference to do the operations is given by the Rule of BODMAS. The expansion is Bracket, Of, Division, Multiplication, Addition and Subtraction. (Note: - Of means multiplication)

For example 75% of 220×5(7+3)/10-100=75% of 220×5(10)/10-100

=(75/100)×220×5×(10/10)-100

=165×5×(10/10)-100

=165×5-100

=825-100

=725

(6.30) Play with two

2+2-2-(2/2) =1

2+2+2-2-2 =2

2+2-2+(2/2) =3

2×2×2-2-2 =4

2+2+2-(2/2) =5

2+2+2+2-2 =6

(22/2)-2-2 =7

2×2×2+2-2 =8

2×2×2+(2/2) =9

2 – 2 =0

(6.31) Specialty of 13:- 1, 3, 4, 9, 10, 12 when divided by 13, the resultant form a group of numbers.

1/13 = 0.076923 076923

3/13 = 0.230769 230769

4/13 = 0.307692 307692

9/13 = 0.692307 692307

10/13=0.769230 760230

12/13=0.923076 923076

In all the above six fractions 0, 7, 6, 9, 2 and 3 have come in the same order. In the above resultant numbers after 6 nine has come and behind six is seven.

Another set of fractions 2/13=0.153846 153846

5/13 = 0.384615 384615

6/13 = 0.461538 461538

7/13 = 0.538461 538461

8/13 = 0.615384 615384

11/13=0.846153 846153

In these fractions 1, 5, 3, 8, 4 and 6 have come in the same order. In all the above resultant numbers after 3, eight has come and behind 3 is five.

(6.32) Rules of fractions:-

(a) (a±b)/c = (a/c) ± (b/c)

For example consider a=6; b=9 and c=3 then (6+9)/3 = (6/3)+(9/3)

15/3 = 2 + 3

5 = 5

(b) a/ (b+c) ≠ (a/b) + (a/c)

For example consider a=6; b=9 and c=3 then 6/(9+3) ≠ (6/9) + (6/3)

6/12 ≠ 2/3 + 2

½ ≠ 8/3

(c) If a/b = c/d then (a+b)/b = (c+d)/d

For example consider a=2; b=3; c=4 and d=6 then 2/3 = 4/6

also (2+3)/3 = (4+6)/6

5/3 = 10/6

(d) If a/b = c/d then (a-b)/b = (c-d)/d

For example consider a=2; b=3; c=4 and d=6 then 2/3 = 4/6

also (2-3)/3 = (4-6)/6

(-1)/3 = (-2)/6

(e) If a/b = c/d then (a+b)/(c+d) = b/d

For example consider a=2; b=3; c=4 and d=6 then 2/3 = 4/6

also (2+3)/(4+6) = 3/6

5/10 = ½

(f) If a is less than b then a/b is less than 1.

For example consider a=2 and b=3 then 2/3 is less than 1.

(g) (b-c)/a = -(c-b)/a

For example consider a=2; b=3 and c=4

(3-4)/2 = -(4-3)/2

(-1)/2 = (-1)/2

(h) a/(b-c) = -a(c-b)

For example consider a=2; b=3 and c=4

2/(3-4) = -2(4-3)

-2 = -2

(i) (a±b)/ (c±d) = (b±a)/(d±c)

For example consider a=2; b=3; c=4 and d=5

(2+3)/(4+5) = (3+2)/(5+4)

5/9 = 5/9

(j) (a±b)/(c±d) = (a – or + b)/(c – or + d)

For example consider a=2; b=3; c=4 and d=5

(2-3)/(4-5) = (3-2)/(5-4)

(-1)/(-1) = 1/1

1 =1

Note: - In all the above examples a, b, c and d are real numbers.

(k) Multiplication of the numerator and denominator by the same quantity will not change the value of the fraction.

For example 2/5 when multiplied by 3 both the numerator and denominator will give 6/15 but the value remains the same.

(l) The value of the fraction increases with the increase of numerator and decreases with the increase of denominator.

For example 2/5= 0.4; 3/5=0.6; But 2/6=0.33

(m) To find the smaller fraction cross multiply the fraction

For example 2/3, 5/9, ½, 11/18

(18)2/3, 5/9(15) 5/9 is smaller than 2/3

(18)1/2, 11/18(22) ½ is smaller than 11/18

Now (10)5/9, ½ (9) ½ is smallest among these fractions

(n) Even numbers of fractions alone obey the cross multiplication rule.

For example ½ = 2/4 = 4/8 = 8/16

1×4×4×16 = 2×2×8×8

256 = 256

Odd numbers of fractions do not obey the cross multiplication rule.

For example ½ = 2/4 = 4/8

1×4×4 = 2×2×8

16 ≠ 32

(6.33) Mods:-

(a)If a system comes repetitively like 7 days a week, 24 hours per day then they all obey mod system. For example if a system comes repetitively after every 7 times then 8 can be indicated as follows. 8=1(mod 7)

(b) If a=b(mod m) and c=d(mod m) then a+c = b+d(mod m)

For example 8=1(mod 7) and 9=2(mod 7) then 8+9 = 1+2(mod 7)

That is 17 = 3(mod 7)

(c) If a=b(mod m) and c=d(mod m) then a-c = b-d(mod m) provided that a>c

For example 9=2(mod 7) and 8=1(mod 7) then 9-8 = 2-1(mod 7)

That is 1 = 1(mod 7)

(d) If a=b(mod m) and c=d(mod m) then a×c = b×d(mod m)

For example 8=1(mod 7) and 9=2(mod 7) then 8×9 = 1×2(mod 7)

72 = 2(mod 7)

(e) If a=b(mod m) and c=d(mod m) then a/c = b/d(mod m)

For example 8=1(mod 7) and 16=2(mod 7) then 8/16 = 1/2(mod 7)

1/2 = 1/2(mod 7)

(6.34) Ratio: -

(a) Ratio is the comparison of the two things in terms of fractions. For example if x and y earns Rs.2000/- and Rs.3000/- respectively then y is earning 3/2 times of x. This can be represented as a ratio and the symbol is “:” pronounced as “is to”. The above relationship is represented in terms of ratio as x:y :: 2:3 Here 2 is called as precedent and 3 is called as consequent.

(b) It is to be noted that the product of terminal numbers in the ratio will be equal to the product of middle numbers. This is called as the cross multiplication rule of ratios.

For example 2:3 :: 2000:3000

2×3000 = 3×2000

6000 = 6000

(c) The cross multiplication rule of ratios is obeyed by even number of ratios only.

For example 2:3 :: 4:6 :: 8:12 :: 16:24

2×6×8×24 = 3×4×12×16

2304 = 2304

For odd number of ratios the cross multiplication rule is not obeyed.

For example 2:3 :: 4:6 :: 8:12

2×6×8 ≠ 3×4×12

96 ≠ 144

(d) Sum of the numbers in a ratio must be a factor of the measurement. For example if a carton box containing a dozen mirror is dropped the ratio of broken mirrors to unbroken mirrors may be 2:1, 3:1, 5:1 and 7:5 but not 3:2 because in the former cases the sum of the ratios 2+1, 3+1, 5+1 and 7+5 are factors of 12. But the sum of 3+2 is not a factor of 12.

(6.35) Proportionality: -

(a) Proportionality is also a type of ratio which involves comparison between two things. Otherwise proportionality is an uniform ratio where if one property increases or decreases with respect to another property is known as proportionality.

(b) There are two types of proportionality namely direct proportionality and inverse proportionality.

(c)Direct proportionality: - If one property increases with other property or decreases with other property is known as direct proportionality.

For example working days and wages.

(d)Inverse proportionality: - If one property increases while other property decreases or vice versa is known as inverse proportionality.

For example number of workers and time of completion of work.

(e) In direct proportion the multiplication factor will be inversed while in the inverse proportion it will come as such.

For example if ‘x’ is getting Rs.100/- per day means what he will get for 15 days? This is a direct proportion (15/1)×100 = 1500/-

If 10 people can able to complete a construction work in 15 days means in how many days 5 people can able to complete the work? This is an inverse proportion.

(10/5)×15 = 30. The work will be completed in 30 days.

(6.36) Percentage: -

(a)Percentage is also a type of comparison of two things with respect to 100 or simply it is a ratio to 100. This is a direct proportion. This is denoted by ‘%’. Percentage can also be called as parts per hundred.

For example if the price of an edible oil is increased to Rs.200/- from Rs.150/- then what is the price increase in terms of percentage can be calculated as follows.

For 150 the increase is 200 then for 100(percentage) what is the increase

(100/150)×200 = (400/3) = 133.3%

(b) To convert fractions or decimals to percentage multiply by 100.

For example 4/3 can be represented in terms of percentage as follows.

(4/3)×100 = 80%

To convert percentage into fractions divide it by 100.

For example 25% can be represented in terms of fraction as follows.

(25/100) = 0.25

(c) If A is R% more than B, then B is less than A by [{R/(100+R)}×100] % Similarly if A is R% less than B, then B is more than A by [{R/(100-R)}×100]%

For example if A is getting 200% more than B, then B is less than A by 16.6%

If A is getting 20% less than B, then B is more than A by 25%

(d) Like percentages, per thousands (parts per thousand or ppt), per millions (parts per million or ppm), per billion (parts per billion) and so on can be expressed. The rules applicable for percentages are applicable for per thousands, per millions, per billions and so on.

(e) It is quite obvious that to convert percentages into ppm multiply by 10,000.

(6.37) Amicable number: -

(a)Two distinct positive integers (m, n) are said to be amicable when each integer is the sum of the proper divisors of the other. Proper divisors mean all divisors of a number except the number itself. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110. Their sum is 284. The proper divisors of 284 are 1, 2, 4, 71 and 142. Their sum is 220. Therefore (220, 284) is a pair of amicable numbers. Some other examples of amicable pairs are (1184, 1210), (2620, 2924), (5020, 5564) and (6232, 6368).

(b) Three distinct positive integers (m, n and p) are said to be amicable triples when each integer is the sum of the proper divisors of the other. For example (1980, 2016, 2556), (9180, 9504, 11556) and (21668, 22200, 27312) are the first three amicable triples.

(6.38) Demlo number: -

(a)Consider the number 136653. If you add the first two and last two digits that is (13+53) will give 66 which is the digits in between the numbers. This type of phenomenon is called as demlo number.

The demlo number series is 121, 253, 473…..

A demlo number has three parts. Addition of the first and last parts gives the digit or digits of the middle part.

(b) A test for demlo number is it is divisible by 11.

For example in the above case (136653/11) = 12423

(c) Demlo number is obtained by adding a two digit number two times, three digit number four times, four digit number by six times and five digit number eight times as follows.

For example for 42 the demlo number is 42 for 234 it is 234 for 1012 it is 1012

42 234 1012

462 234 1012

234 1012

259974 1012

1012

112444332

for 20150 (five digit number) the demlo number is 20150

20150

20150

20150

20150

20150

20150

20150

223888886650

7. PALINDROME NUMBERS

(7.1) When a number is written in the reverse order is equal to the original number itself means it is called as a palindrome number or symmetrical number. It obeys the reflux property or simply it is equal to the mirror image of the original number.

For example 101 when written in the reverse order is 101 and it is a palindrome number. The palindrome numbers sequence is 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191…

The palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151…

The palindromic square numbers are 0, 1, 4, 9, 121, 484, 676, 10201, 12321…

(7.2) All numbers in base 10 with one digit are palindromic. So including zero we have 10 single digit palindrome numbers.

(7.3) The number of palindromic numbers with two digits is nine. They are 11, 22, 33, 44, 55, 66, 77, 88 and 99.

(7.4) There are 90 palindromic numbers with three digits. (Using the rule of product: 9 choices for the first digit – which determines the third digit as well – multiplied by 10 choices for the second digit). That is {101, 111, 121, 131, 141, 151, 161, 171, 181, 191, …, 909, 919, 929, 939, 949, 959, 969, 979, 989, 999}

(7.5) 90 palindrome numbers with four digits: (Again, 9 choices for the first digit multiplied by ten choices for the second digit. The other two digits are determined by the choice of the first two). That is {1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, …, 9009, 9119, 9229, 9339, 9449, 9559, 9669, 9779, 9889, 9999}

(7.6) There are 199 palindrome numbers below 10^4. Below 10^5 there are 1099 palindromic numbers and for other exponents of 10n where n>5 we have the total number of palindrome numbers as 1999, 10999, 19999, 109999, 199999, 1099999… respectively.

(7.7) The number of palindromic numbers formed with the number 7537252 by changing its digits are 6. There are two twos, two fives two sevens and only one 3. So 3 must be in the middle of the palindromic numbers. 2, 5 and 7 can be arranged in 3! Ways. So we get 7523257, 2753572, 5723275, 7253527, 2573752 and 5273725.

(7.8) Palindromic numbers can be considered in other numbering systems also like in the decimal system. For example, the binary palindromic numbers are: 0, 1, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001…

(7.9) The multiplication of 1089 with 9 is the reverse of the number that is a palindrome number and similarly the multiplication of 10989 with 9 is the reverse of the number and this can be extended to 109989,1099989,…

That is 1089×9=9801

10989×9=98901

109989×9=989901

1099989×9=9899901

………………………

(7.10) The number 1089 gives palindrome number as follows

1089×1=1089 and its palindrome number 9801=1089×9

1089×2=2178 and its palindrome number 8712=1089×8

1089×3=3267 and its palindrome number 7623=1089×7

1089×4=4356 and its palindrome number 6534=1089×6

1089×5=5445 and 5445 is a palindrome number.

(a) The smallest number that is reversed on multiplication is 1089 followed by the second smallest number that is reversed on multiplication is 2178. That is 1089×9=9801 and 2178×4=8712.

(7.11) The sum of the digits of the palindrome number is equal to the sum of the digits in the original number. For example 1551 and its sum of the digits that the digital root is 3.

The palindrome of the above number is again 1551 and its digital root is 3.

(7.12) The multiplication of a number and its palindrome is a square number. For example 1089×9801=10673289=3267^2. Another example 169 × 961 = 403^2

(7.13) The product of the following multiplications is same whether read from right or left.

139 × 109 = 15151

14287143 ×7 = 100010001

142857143 ×7 = 1000000001

27994681 × 441 = 12345654321

333333666667 ×33 = 11000011000011

(7.14) The number 8 and its derivatives like 88, 888… are also palindrome numbers and these numbers have special characteristics in that they remain unchanged when they are written upside down. 88

88

Note: - We can include ‘0’ also in this category. 8008

8008

(7.15) 9 + 9 = 18 and its palindrome number 81 = 9 × 9

24 + 3 = 27 and its palindrome number 72 = 24 × 3

47 + 2 = 49 and its palindrome number 94 = 47 × 2

497 + 2 = 499 and its palindrome number 994=497×2

(7.16) A test for palindrome number is if we split the number into two parts the summation of the digits of the left side is equal to the summation of the digits of the right hand side.

For example 251152 and if we split into 251 and 152, the summation of the digits of the left and right hand side is 8.

Another example is 123454321 and if we split into two parts 1234/5/4321 the summation of the digits of the left and right hand side is 10.

(7.17) Any number when written in the reverse order and the reverse number is written adjacent to the original number then the resultant obtained will be a palindrome number.

For example 786 and the reverse number is 687. When the reverse number is written adjacent to the original number then we get 786687 and this is a palindrome number.

(7.18) A comparative study of divisibility and the reflux property of numbers: -

(a) When a number is divisible by 3 means the mirror image of that number is also divisible by 3. For example 225 and the mirror image of that number is 522. Both the numbers are divisible by 3.

Note: - All palindrome numbers are mirror image numbers but all mirror image numbers are not palindrome numbers.

(b) When a number is divisible by 9 means the mirror image of that number is also divisible by 9. For example 558 and the mirror image of that number is 855. Both the numbers are divisible by 9.

(c) When a number is divisible by 33 means the mirror image of that number is also divisible by 33. For example 330.

(d) When a number is divisible by 99 means the mirror image of that number is also divisible by 99. For example 990.

(e) When a number is divisible by 11 means the mirror image of that number is also divisible by 11. For example 220.

(7.19) The square of the palindrome number of the following

(a) 12^2 = 144 and 441 = 21^2 and again typically 12 is the mirror image of 21.

(b) 13^2 = 169 and 961 = 31^2 and again typically 13 is the mirror image of 31.

(c) 20^2 = 400 and 004 = 02^2 and again typically 20 is the mirror image of 02.

(d) 30^2 = 900 and 009 = 03^2 and again typically 30 is the mirror image of 03.

(e)102^2 = 10404 and 40401 = 201^2 and again typically 102 is the mirror image of 201.

(f)103^2 = 10609 and 90601 = 301^2 and again typically 103 is the mirror image of 301.

(g)110^2 = 12100 and 00121 = 011^2 and again typically 110 is the mirror image of 011.

(h)112^2 = 12544 and 44521 = 211^2 and again typically 112 is the mirror image of 211.

(i) 113^2 = 12769 and 96721 = 311^2 and again typically 113 is the mirror image of 311.

(j)120^2 = 14400 and 00441 = 021^2 and again typically 120 is the mirror image of 021.

(k)122^2 = 14884 and 48841 = 221^2 and again typically 122 is the mirror image of 221.

(l)130^2 = 16900 and 00961 = 031^2 and again typically 130 is the mirror image of 031.

(m)200^2 = 40000 and 00004 = 002^2 and again typically 200 is the mirror image of 002.

(n)201^2 = 40401 and 10404 = 102^2 and again typically 201 is the mirror image of 102.

(o)210^2 = 44100 and 00144 = 012^2 and again typically 210 is the mirror image of 012.

(p)211^2 = 44521 and 12544 = 112^2 and again typically 211 is the mirror image of 112.

(q)220^2 = 48400 and 00484 = 022^2 and again typically 220 is the mirror image of 022.

(r)221^2 = 48841 and 14884 = 122^2 and again typically 221 is the mirror image of 122.

Likewise we can extend for higher power palindrome numbers.

For example 20^3 = 800 and 008 = 02^3 and again typically 20 is the mirror image of 02.

(7.20) The Square of the palindrome number of the following is the same number itself.

(a) 11^2 = 121 and the mirror image of the resultant is 121 = 11^2.

(b) 101^2 = 10201 and the mirror image of the resultant is 10201 = 101^2.

(c )111^2 = 12321 and the mirror image of the resultant is 12321 = 111^2.

(d) 121^2 = 14641 and the mirror image of the resultant is 14641 = 121^2.

(e) 202^2 = 40804 and the mirror image of the resultant is 40804 = 202^2.

(f) 212^2 = 44944 and the mirror image of the resultant is 44944 = 212^2.

Likewise we can extend for higher power palindrome numbers.

For example 11^3 = 1331 and the mirror image of the resultant is 1331 = 11^3.

(7.21) Any ‘n’th digit number when subtracted from its palindrome number gives the resultant equal to 9 or multiples of 9.

For example 3142 2413 and if we subtract the smaller number from the bigger number 3142 – 2143 = 729 which is the divisor of 9.

Another example 26153 35162 and if we subtract the smaller number from the bigger number 35162 – 26153 = 9009 which is the divisor of 9.

(7.22) The numbers 11, 111, 1111… and the multiples of these numbers from 1 to 9 give perfect palindrome numbers.

For example 111 × 3 = 333. Now where represents a mirror and 333 if written in the reverse order is 333.

(7.23) The total number of palindrome numbers will be more in pental system when compared with decimal system. The total number of palindrome numbers will be highest in binary system as the number of repeating digits are 0 and 1 only.

(7.24) 11 is the only palindromic number in two digit numbers. There are no other palindromic prime number comprising even number of digits since they are divisible by 11. Conversely every palindromic prime must contain an odd number of digits with one exception that 11 is a palindromic prime having even number of digits.

(7.25) The smallest palindromic prime number containing all the digits from 0 to 9 is 1023456987896543201.

(7.26) The Repunit numbers are palindrome numbers.

(7.27) The number containing ‘n’ repunits separated by (n-2) zeros between the repunits is divisible by the number formed by ‘n’ repunits. For example 1001001001 can be divided by 1111 and 20202 can be divided by 222.

8. ARITHMETICS OF ZERO AND INFINITY

(8.1) Zero is really a typical number without which it is very difficult to express the bigger numbers. This is denoted as ‘0’ and can also be called as cipher.

(8.2) Zero is an even number.

(8.3) Zero is a neutral number in the number system. That is zero is neither a positive integer nor a negative integer. That is there is no +0 or -0.

(8.4) Zero is the additive identity and it can also be considered as subtractive identity.

(8.5) x + 0 = x where x is a real number

(8.6) x – 0 = x where x is a real number

(8.7) 0 – x = -x where x is a real number

(8.8) 0 + x = x (provided x ≠ 0)

(8.9) x – x = 0

(8.10) 0 + 0 = 0

(8.11) 0 – 0 = 0

(8.12) 0 × 0 = ∞ because 0 = ∞ ÷ 0

(8.13) 0 ÷ 0 = ∞ because 0 = ∞ × 0

(8.14) Factorial zero is 1.

(8.15) Zero has no face value but it has the place value.

(8.16) In any number the position of each digit or numerical from the right shows how many ones, tens, hundreds, thousands and so on are there in the number.

For example 1 0 0 2 place value of 2 is 1

---- place value of 0 is 10

-------place value of 0 is 100

---------place value of 1 is 1000

(8.17) Zero written in the left side of a number will not increase the value of the number. But if zero is written between the digits of a number or after the number from left side onwards will increase the value of the number.

For example 123 and 0123, 00123, 000123… (Equation 1) are same as that of 123.

But 1023, 10023, 100023… (Equation 2) has the value greater than their corresponding counterparts of equation 1 and 1203, 12003, 120003… (Equation 3) has still greater value corresponding to their counterparts of equations 1 and 2. Similarly 1230, 12300, 123000… (Equation 4) has still greater value when compared to the values of equations 1, 2 and 3.

Similar arguments can be extended to infinity also.

From the above statements we can say that unlike other integers which have the discrete values, 0 and ∞ have not discrete value at all. That is zero and infinity has a set of values with each individual integer themselves. That means 1 can be represented as 01, 001, 0001… which implies that integer 1 has zero in itself or the other way that is zero has 1 itself then we can represent 1 as 1zero (don’t confuse with the number 10). The same argument can be extended to the integer 2 and so on. Then we will have 2zero, 3zero… Similarly if we extrapolate the argument for ∞ we will get 1∞, 2∞, 3∞ …

In broader sense we can state a little bit confusing but still with authentic that zero has infinite sets of discrete values and infinity has infinite sets of discrete values.

From the above utopian statement we can derive one more value that is 0∞ which give the actual relationship between 0 and ∞.

If we relate zero to the birth of a species that is as the origin of species, then ∞ can be related to the death of the species.

(8.18) There are three classes of zeros. They are

(a) Leading zeros are zeros that precede all of the non zero digits. They never count as significant figures. For example 0.0025 the three zeros indicate the position of the decimal point. The number has only two significant figures 2 and 5.

(b) Captive zeros are zeros that fall between non zero digits. They always count as significant figures. For example 1.008 has four significant figures.

(c) Trailing zeros are zeros at right end of the number. They are significant only if the number is written with a decimal point. For example 100 has only one significant figure, but written as 100. , it has three significant figures.

(8.19) Scientific notation number: -It is the representation of any number as the powers of 10. Or it is the representation of larger number concisely as the powers of 10.

For example 0.00034 = 3.4×10-4 and 26000000 = 26×10^6

(8.20) Zero is an important number

S.No.

Number

Name

1.

10-18

Ioto

2.

10-15

Femto

3.

10-12

pico

4.

10-9

nano

5.

10-6

micro

6.

10-3

milli

7.

10-2

Centi

8.

10-1

deci

9.

10^1

Ten

10.

10^2

Hundred

11.

10^3

Thousand or kilo

12.

10^4

Ten thousand

13.

10^5

Lakh

14.

10^6

Ten Lakh or Million or mega

15.

10^7

Crore or Ten million

16.

10^8

Hundred million

17.

10^9

Billion or giga

18.

10^10

Ten billion

19.

10^11

Hundred billions

20.

10^12

Trillion

21.

10^13

Ten trillions

22.

10^14

Hundred trillions

23.

10^15

quadrillion

24.

10^16

Ten quadrillions

25.

10^17

Hundred quadrillions

26.

10^18

quintillion

27.

10^19

Ten quintillions

28.

10^20

Hundred quintillions

29.

10^21

sextillion

30.

10^22

Ten sextillions

31.

10^23

Hundred sextillions

32.

10^24

septillion

33.

10^25

Ten septillion

34.

10^26

hundred septillion

35.

10^27

octillion

36.

10^28

Ten octillions

37.

10^29

hundred octillions

38.

10^30

nonillion

39.

10^33

decillion

40.

10^36

undecillion

41.

10^39

duodecillion

42.

10^42

tredecillion

43.

10^45

quatuordecillion

44.

10^48

quindecillion

45.

10^51

sexdecillion

46.

10^54

septendecillion

47.

10^57

octodecillion

48.

10^60

novemdecillion

49.

10^63

vigintillion

50.

10^100

centillion

Note: - The superscript on zero indicates the number of zeros. For example 10^12 indicates the number 1 followed by 12 zeros. That is 1,000,000,000,000.

(8.21) The equation 1 = 2 can be proved as follows.

Multiply both sides with zero and is 1×0 = 2×0 = 0

Cancel the common number zero we get 1=2

This means at zero state all numbers are equal.

(8.22) Infinity: -

(a) Infinity is the imaginary number and is considered to be the last number of the whole number series. That is both zero and the infinity are imaginary numbers and are at the extreme end of the whole number system. It is denoted by the symbol ‘∞’.

(b) Infinity is a neutral number in the number system. That is infinity is neither a positive integer nor a negative integer. That is there is no -∞ or +∞.

(c) 0 + ∞ = ∞ because 0 is an additive identity.

(d) 0 - ∞ = ∞

(e) 0 × ∞ = 0

(f) 0 ÷ ∞ = 0

(g) ∞ - 0 = ∞

(h) x + ∞ = ∞

(i) x - ∞ = ∞

(j) x × ∞ = ∞

(k) x ÷ ∞ = ∞

(l) ∞ + ∞ = ∞

(m) ∞ - ∞ = ∞

(n) ∞ ÷ ∞ = 1

(o) x∞ = ∞ provided that x ≠ 0

(o) ∞! = ∞

(p) Infinity has no place value. But it has the face value.

(q) The equation 1 = 2 can be proved as follows.

Multiply both sides with infinity and is 1×∞ = 2 ×∞

Cancel the common number infinity we get 1 = 2

This means at infinity state all numbers are equal.

(r) Since at both ‘0’state and ‘∞’ state all numbers are equal, it doesn’t mean that 0 is equal to ∞. This can be confirmed by the following arguments.

(I)If we consider the rational number system as a big utopian circle 0 and ∞ will be the opposite members in the system.

0

-1 1

-2 2

-3 3

…… ∞……..

(II)If we consider the whole number system as a big utopian circle starting from 0 to ∞ then 0 and ∞ will be the neighbors in the system.

0

∞ 1

. 2

. 3

.......………

(III)Geometrical approach of 0 and ∞:- If we consider the arithmetic numbers in geometrical view 1 can be represented as .(dot), 2 is represented as straight line, 3 as triangle, 4 as square, 5 as pentagon, 6 as hexagon, 7 as heptagon and so on. If we extrapolate the same concept for infinity we will get a circle and the imaginary origin of the circle is zero.

(IV)Theosophical approach of 0 and ∞:- Hypothetically 0 and ∞ are polymorphs of God. Zero and infinity are super natural numbers similar to God. Skeptically at 0 and ∞ states all numbers are equal, but in reality it is not possible.

As such 0 has no value but if it is placed next to a number the value of the number increases. If the number is associated with infinity means it attains the eternal infinity state. Simply zero is nothing but it is everything and infinity is of everything with its existence is nothing.

Likewise each human being is a number of his own kind and must associate with God to add value in their life and attain the eternal state and they should not ask about the existence of Almighty like zero and infinity.

(s) Between the whole numbers 1 to 2 infinity numbers are there like 1.1, 1.11, 1.111…1.2, 1.22…1.21, 1.2121…But if we consider only the whole numbers means there is no numbers between 1 and 2. For 1.0, 1.00, 1.000…is equal to 1 only.

9. SEQUENCES AND SERIES

(9.1) A sequence or series of numbers is nothing but arrangement of numbers having common relationship or common difference among themselves.

For example 0,2,4,6,8,10,12,14,16,18…

(9.2) Fibonacci series: -

(a) The series 1, 1,2,3,5,8,13,21,34,55,89,144,233,..

In this series every term except the first and second term is the sum of two previous terms. That is 3+2=5 and so on. This series is called as Fibonacci series. The ‘n’th term in this series is usually denoted by Fn. Thus F_6 of Fibonacci series is 8.

(b) In this sequence if ‘n’ is divisible by 5, then Fn is divisible by 5. That is F_10 = 55. Here 10 is divisible by 5 and so the 10th term 55 is divisible by 5.

(c) If Fn is the ‘n’th Fibonacci number and if a = F2n-1 and b = F2n+1 then a^2 + 1 is divisible by b and b^2 + 1 is divisible by a.

For example let n = 6 then a = F12-1 = F_11 = 89 and b = F2n+1 = F_13 = 233. Now a^2 + 1 = 7922 and b^2 + 1 = 54290. Then 7922 is divisible by 233 and 54290 is divisible by 89.

(d) Take any four consecutive numbers in the Fibonacci sequence. The product of first and fourth numbers and the twice the product of the middle numbers form the sides a right triangle. For example 5,8,13 and 21 are the four consecutive numbers in the Fibonacci sequence. Now 5×21=105; 8×13×2=208. So the 105 and 208 are the sides of a right triangle. That is 105^2 + 208^2 = 11025 + 43264 = 54289 = 233^2.

(e) The product of F2n, F2n+2, and F2n+4 of three consecutive Fibonacci numbers is the product of three consecutive natural numbers.

For example if n=2 then F_4×F_6×F_8 = 3×8×21 = 504 = 7×8×9

If n=3 then F_6×F_8×F_10 = 8×21×55 = 9240 = 20×21×22.

(9.3) Pascal series: -

(a) The following series which starts from 1 and ends with 1 with specified difference is known as Pascal’s series. That is 1 9 36 84 126 84 36 9 1

(b) The triangle formed by using the Pascal’s series as depicted below is known as Pascal’s triangle.

1

24. 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

…………………

Diagrammatically it can be represented as follows.

………………………………………………….

Note: - Each line is split into two lines and the number of lines represent the intensity of lines.

(c) The coefficients of the nth power of the binomial (a+b) n will give Pascal’s triangle. That is (a+b)1 = (a+b) =a+b 1 1

(a+b)2 = (a+b)(a+b) =a^2+2ab+b^2 1 2 1

(a+b)3 = (a+b)(a+b)(a+b)=a^3+3a^2b+3ab^2+b^3 1 3 3 1

……………………………………………. …………...

(d) The summation of the two consecutive numbers in the series will be equal to the following number in the next series. That is it will form a triangle.

For example 3+3=6.

(e) The first angular number series in the Pascal’s triangle is uniformly 1.

(f) The second angular number series is a real number series. That is 1,2,3,4,5,…or in the other way as 1,(1+1),(1+1+1),(1+1+1+1),(1+1+1+1+1),…

(g) The third angular number series is nothing but the triangular number series. That is 1,3,6,10,15,… or 1,(1+2),(1+2+3),(1+2+3+4),(1+2+3+4+5),…One more method of writing the same series is 1,(1+2),(3+3) ,(6+4) ,(10+5),…

(h) Pascal’s triangle is useful in expressing the splitting pattern of hydrogen atoms in a compound in the NMR spectrum.

(i) The triangle is symmetric in nature. That is if we split the triangle into two halves the left side part is symmetrical to the right side part of the triangle.

(9.4) Arithmetic progression:-

(a)It is a symmetrical sequence having the property that each term after the first is obtained by adding a fixed number to the preceding term.

For example 100, 95, 90, 85, 80, 75…

In the above sequence if we add -5 to the preceding term, we get the following number in the series and this number is called as the common difference. The common difference is denoted by the symbol ‘d’.

(b) There are two types of arithmetic progressive series called as finite arithmetic progressive series and infinite arithmetic progressive series.

For example 100, 95, 90, 85, 80… is called as infinite arithmetic progressive series and

For example 100, 95, 90, 85, 80. is called as finite arithmetic progressive series.

(b) The ‘n’th number in the arithmetic progression is

n = a+(n-1)d where n = ‘n’th number in the series

a = first number in the series

d = common difference in the series

This formula is applicable to both finite arithmetic progressive series and infinite arithmetic progressive series.

For example using the above formula the 7th term of the series 100, 95, 90, 85, 80… can be calculated as follows. Here a=100, n=7 and d=5

Applying to the above formula 7th term=100+[(7-1)×(-5)]

7th term= 100+[6×(-5)]

7th term= 100+[-30]

7th term= 70

(c) The sum (S) of ‘n’ terms of arithmetic series is

S = {n/2[2a + (n-1) d]} where S = Sum of ‘n’ terms of arithmetic series

n = ‘n’th number in the series

a = first number in the series

d = common difference in the series

This formula is applicable to both finite arithmetic progressive series and infinite arithmetic progressive series.

For example using the above formula the sum of the arithmetic series up to 5th term of the series 100, 95, 90, 85, 80… can be calculated as follows.

Here a=100, n=5 and d=5

Now the sum of the series up to 5th term S = 5/2[(2×100) + (5-1) (-5)]

= 5/2[200 + (-20)]

= 5/2(180)

= 450.

(d) For finite arithmetic progressive series the sum (S) of ‘n’ terms of arithmetic series is calculated by the formula

S = [n/2(a+l)] where S = sum of ‘n’ terms of arithmetic series

n = ‘n’ termed arithmetic series

a = first number in the series

l = last number in the series

For example using the above formula the sum of the finite arithmetic series up to 5 terms of the series 100, 95, 90, 85 and 80 can be calculated as follows.

Here a=100, n=5 and l=80

S = 5/2(100+80)

= 5/2(180)

= 450.

(e) A sequence of real numbers is an arithmetic sequence of first order.

1, 2, 3, 4, 5…….. Primary sequence

1, 1, 1, 1……… First difference sequence

Similarly the sequence of square numbers is an arithmetic sequence of second order.

1, 4, 9, 16, 25……Primary sequence

3, 5, 7, 9…….. First difference sequence

2, 2, 2…… Second difference sequence

Likewise the sequence of cubic numbers is an arithmetic sequence of third order and so on. That is 1, 8, 27, 64, 125…Primary sequence

7, 19, 37, 61……First difference sequence

12, 18, 24………Second difference sequence

6, 6………….. Third difference sequence

A sequence of numbers whose ‘k’th difference is constant is called as an arithmetic sequence of ‘k’th order.

The sequence of fourth power numbers is an arithmetic sequence of fourth order and the constant can be calculated by multiplying the real numbers from 1 to 4.

That is 1×2×3×4 = 24 and similarly for the fifth power number series the constant is calculated by multiplying the real numbers from 1 to 5. That is 1×2×3×4×5 = 120.

(f) The arithmetic sum of real numbers starting from 1 to n (where n is a real number) is calculated by the formula S = [n (n+1)/2]

For example 1+2+3+4+5.

The sum of numbers up to 5 is S_5 = [5(5+1)/2]

= [(5×6)/2]

= [30/2]

= 15.

(g) The sum of even numbered real number arithmetic series starting from ‘0’ onwards is [2×n(n+1)/2] that is n(n+1). Similarly for 3rd multiples starting from ‘1’ onwards is [3×n (n+1)/2] and for 4th multiples [4×n (n+1)/2] that is 2n (n+1) and so on.

(h) The arithmetic sum of square numbers starting from ‘1’ onwards is [n(n+1)(2n+1)/6] where ‘n’ is a real number.

For example 1^2+2^2+3^2+… Here n=3

The sum of square numbers up to 3 is {3(3+1) [(2×3) + 1]/6}

= {(3×4×7)/6}

= 14.

(i) The arithmetic sum of cubic numbers starting from ‘1’ onwards is [n (n+1)/2]2 where ‘n’ is a real number.

For example 1^3+2^3+3^3+… Here n=3

The sum of cubic numbers up to 3 is [3(3+1)/2]2

= [(3×4)/2]2

= 36.

(j) The arithmetic sum of odd number series starting from ‘1’ onwards is [(n+1)/2]2 where ‘n’ is the last number of series.

For example 1, 3, 5, 7…… Here n=7

The arithmetic sum of odd number series starting from 1 to 7 is [(7+1)/2]2 = 16.

(k) The arithmetic sum of odd number series starting from ‘1’ onwards can also be calculated by another formula n^2 where ‘n’ is the number of term in the series.

For example 1, 3, 5, 7……… Here n=4

The arithmetic sum of odd number series starting from 1 to 7 is 4^2 = 16.

(l) The arithmetic sum of odd numbered squares is [n(4n^2 – 1)/3] where ‘n’ is the number of the term in the series.

For example 1^2+3^2+5^2+7^2+9^2 Here n=5

The arithmetic sum of odd numbered squares up to 9 is 5[4(5)2 – 1]/3= (5×99)/3 = 165.

(m) The arithmetic sum of addition of multiples of two from ‘1’ onwards can be calculated by multiplying the last number by 2 and then subtract 1 from the resultant.

For example 1+2+4+8+16+32+…

Or in other words 1+2+(2×2)+(2×2×2)+(2×2×2×2)+…

The sum up to 32 is (32×2)-1=63.

(n) The arithmetic sum of addition of multiples of three from ‘1’ onwards can be calculated by adding the last number with half of the last number minus 1.

For example 1+3+9+27+81+243+…

Or in other words 1+3+(3×3)+(3×3×3)+(3×3×3×3)+…

The sum up to 81 is [81+(81-1)/2]=121.

(o) If an arithmetic progression is added, subtracted, multiplied or divided by any constant number the resultant will also be an arithmetic progressive series.

(9.5) Geometric progression: - It is a sequence having the property that each term after the first is obtained by multiplying a fixed number to the preceding term.

For example 1, 2, 4, 8, 16…

(9.6) Harmonic progression: -

(a)If a sequence having the property that each reciprocals of the term form an arithmetic progression is known as harmonic progression.

For example ½, 1/4, 1/6, 1/8, 1/10…

(b) If the given sequence of numbers are both in arithmetic progression and harmonic progression then they will be also in geometric progression.

10. POLYGONAL NUMBERS

(10.1) The real numbers whose units or digits can be laid to form a specified geometrical figure are called as polygonal numbers. These are one type of 2-dimensional figurate numbers. Except 1 and 2, all real numbers when represented in digits can be laid to form a closed geometrical figure. Based upon the geometrical shape formed the numbers can be classified as triangular number, square number, pentagonal number and so on.

(10.2)Triangular number: -

(a) A triangular number is the one whose units or digits can be laid to form an equilateral triangle. The triangular number series is 1,3,6,10,15,21,28,36,45,55,…

(b) In the triangular number series if we categorize the numbers as odd(O) and even(E) the sequence is O,O,E,E,O,O,E,E,… and it repeats further as we move

(c) The lowest number which is behaving both as square number and as a

triangular number is 36. (Note: - 1 is considered as power number rather than a square number).

(d) No triangular number is a cube or a fourth power number.

(e) Relationship between triangular number and square number: - The multiplication of a triangular number by 8 followed by the addition with 1 will give a square number.

For example 21 is a triangular number and the multiplication with 8 gives 168. Now the addition of 168 with 1 will give 169 which is nothing but the square of 13.

This relationship can be taken as a rule for a triangular number. A number ‘x’ is a triangular number if and only if “8x + 1” is a square number.

(f) ‘n’th triangular number can be calculated by using the formula [n(n+1)/2]. That is half the sum of any positive integer and its square number is a triangular number.

For example if n=10 the 10th triangular number is 55.

(g) Number 3 is the only prime triangular number.

(h) Adding up the triangular number will give the tetrahedral number.

Numbers that are Added

The Sum of These(the new pattern)

1

1

1 + 3

4

1 + 3 + 6

10

1 + 3 + 6 + 10

20

1 + 3 + 6 + 10 + 15

35

1 + 3 + 6 + 10 + 15 + 21

56

1 + 3 + 6 + 10 + 15 +21 + 28

84

1 + 3 + 6 + 10 + 15 + 21 + 28 + 36

120

1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45

165

1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55

220

The numbers 1, 4, 10, 20, 35, 56, 84, 120, 165, 220… are tetrahedral numbers.

(i) Every positive integer can be written as a sum of three triangular numbers.

For example 50=28+21+1

(j)Centered triangular numbers: - 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, 1786, 1891, 1999, 2110, 2224, 2341, 2461, 2584, 2710, 2839, 2971...

(10.3) Rectangular number: - The numbers whose dots form a lattice work of rectangles are called as rectangular numbers. In other words any natural number that can be represented as the product of two numbers (not 1 or the number itself) is a rectangular number. For example 6, 8, 10, 12, 14, 16, 18…

Note: - square number, cube number and other power numbers are also rectangular numbers with special characteristic geometrical representation.

(10.4) Square number: -

(a)The numbers whose dots form a lattice work of squares are called as square numbers. If we tilt the lattice work of square, rhombus is formed. So the square numbers can also be called as rhombus numbers. They are discussed in chapter 5.

(b)Centered square numbers: -Centered square numbers are the sum of two consecutive square numbers. The first few centered Square Numbers are 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4325...

(10.5) Pentagonal number: -

(a)The numbers whose dots form a lattice work of pentagons are called as pentagonal numbers.

The pentagonal number series is 1, 5, 12, 22, 35, 51, 70, 92,117,145…

(b) A number ‘x’ is a pentagonal number if and only if “24x + 1” is a square number.

For example 22 is a pentagonal number because [(24×22)+1] = 529 is a square number.

(c) ‘n’th pentagonal number can be calculated by using the formula [n(3n-1)/2]

For example 5th pentagonal number is 35.

(d)A pentagonal number should not be a square number.

(e)Centered pentagonal numbers 1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, 526, 601, 681, 766, 856, 951, 1051, 1156, 1266, 1381, 1501, 1626, 1756, 1891, 2031,2176,2326,2481,2641,2806,2976...

(10.6) Hexagonal number or cornered hexagonal number:-

(a)A hexagonal number is a figurate number, The nth hexagonal number will be the number of points in a hexagon with n regularly spaced points on a side. So the hexagonal number can also be called as hexagonal triangular number.

For example the hexagonal number sequence is: 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946…

(b) The formula for the nth hexagonal number is hn = n(2n-1).

Where ‘n’ is a real number. Applying the formula the third hexagonal number is 15.

(c) Every hexagonal number is a triangular number, but only every other triangular number or alternate triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number.

(d)Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9.

(e) One can efficiently test whether a positive integer x is an hexagonal number by computing the formula n={[(square root of 8x+1)+1]/4}. If n is an integer, then x is the nth hexagonal number. If n is not an integer, then x is not a hexagonal number. Applying the formula if x=6 we can get n=2 and then 6 is a second hexagonal number.

(f) Centered hexagonal number: -A centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. For example the centered hexagonal number sequence is 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 9191027, 1141, 1261, 1387, 1519, 1657, 1801, 1951, 2107, 2269, 2437, 2611, 2791, 2977, 3169, 3367, 3571, 3781, 3997, 4219, 4447, 4681, 4921, 5167, 5419, 5677, 5941, 6211, 6487...

(g) The nth centered hexagonal number is given by the formula 3n(n-1)+1. Applying the formula if n=3 we get 19 and 19 is the third centered hexagonal number.

(10.7) Heptagonal number: - (a)A heptagonal number is a figurate number, the nth heptagonal number will be the number of points in a heptagon with n regularly spaced points on a side. For example the heptagonal number sequence is : 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, 1918, 2059, 2205, 2356, 2512, 2673, 2839, 3010, 3186, 3367, 3553, 3744, 3940, 4141, 4347, 4558, 4774, 4995, 5221, 5452, 5688… (b) The nth heptagonal number is given by the formula [(5n^2-3n)/2]. Applying the formula if n=4 we get 34 and 34 is the fourth heptagonal number in the sequence.

(c) The centered heptagonal number: -A centered heptagonal number is a centered figurate number that represents a heptagon with a dot in the center and all other dots surrounding the center dot in successive heptagonal layers. For example the sequence of centered heptagonal number is 1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, 547, 638, 736, 841, 953, 1072, 1198, 1331, 1471, 1618, 1772, 1933, 2101, 2276, 2458, 2647, 2843, 3046, 3256, 3473, 3697, 3928, 4166, 4411, 4663, 4922, 5188, 5461, 5741, 6028, 6322, 6623, 6931, 7246... (d) The centered heptagonal number for n is given by the formula [(7n^2-7n+2)/2]. Applying the formula if n=3 we get 22 and is the third centered heptagonal number in the sequence.

(10.8) Octagonal number or cornered octagonal number: -

(a)A octagonal number is a figurate number, The nth octagonal number will be the number of points in a octagon with n regularly spaced points on a side.

(b) Centered octagonal number: -Centered octagonal number is a centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers. For example the octagonal numbers are 1, 9, 25, 49, 81, 121, 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 10891225, 1369, 1521, 1681, 1849, 2025, 2209, 2401, 2601, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 4761, 5041, 5329, 5625…

(10.9) Nonagonal number: -

(a)A nonagonal number is a figurate number, the nth nonagonal number will be the number of points in a nonagon with n regularly spaced points on a side. The nonagonal number sequence is 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, 3075, 3286, 3504, 3729, 3961, 4200, 4446, 4699, 4959, 5226, 5500, 5781, 6069, 6364, 6666, 6975, 7291, 7614, 7944, 8281, 8625, 8976, 9334, 9699...

(b) The nonagonal number for n is given by the formula [(7n^2-5n)/2]. Applying the formula if n=3 we get 24 and is the third nonagonal number in the sequence.

(c) A centered nonagonal number is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal number sequence is 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946, 1081, 1225, 1378, 1540, 1711, 1891, 2080, 2278, 2485, 2701, 2926, 3160, 3403, 3655, 3916, 4186, 4465, 4753, 5050, 5356, 5671, 5995, 6328, 6670, 7021, 7381, 7750, 8128, 8515, 8911, 9316.....

(d) The centered nonagonal number for n is given by the formula [(3n-1)(3n-2)/2]. Applying the formula if n=4 we get 55 and is the fourth nonagonal number in the sequence.

(10.10) Decagonal number: -

(a)A decagonal number is a figurate number, the nth decagonal number will be the number of points in a decagon with n regularly spaced points on a side.

(b)A centered decagonal number is a centered figurate number that represents a decagon with a dot in the center and all other dots surrounding the center dot in successive decagonal layers. The centered decagonal number for n is given by the formula 5(n^2 − n) + 1.

(c)The centered decagonal numbers are 1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, 781, 911, 1051, 1202, 1361, 1531, 1711, 1901, 2101, 2311, 2531, 2761, 3001, 3251, 3511, 3781, 4061, 4351, 4651, 4961, 5281, 5611, 5951, 6301, 6661, 7031, 7411, 7801, 8201, 8611, 9031, 9461, 9901.. ...

(d)Like any other centered k-gonal number, the nth centered decagonal number can be reckoned by multiplying the (n - 1)th triangular number by k, 10 in this case, then adding 1. As a consequence of performing the calculation in base 10, the centered decagonal numbers can be obtained by simply adding a 1 to the right of each triangular number. Therefore, all centered decagonal numbers are odd and in base 10 always end in 1.

(e)A centered decagonal prime is a centered decagonal number that is prime. The first few centered decagonal primes are 1, 11, 31, 61, 101, 151, 211, 281, 661, 911, 1051, 1201, 1361, 1531, 1901, 2311, 2531, 3001, 3251, 3511, 4651, 5281, ...

(10.11) Hendecagonal number: -

(a)A hendecagonal number is a figurate number, the nth hendecagonal number will be the number of points in a hendecagon with n regularly spaced points on a side. The hendecagonal number sequence is 1, 11, 30, 58, 95, 141, 196, 260, 333, 415, 506, 606...

(b)Centered hendecagonal numbers are 1, 12, 34, 67, 111, 166, 232, 309, 397, 496, 606, 727, 859, 1002, 1156, 1321, 1497, 1684, 1882, 2091, 2311, 2542, 2784, 3037, 3301,3576,3862,4159,4467,4786…

(10.12) Dodecagonal number: -

(a)A dodecagonal number is a figurate number, the nth dodecagonal number will be the number of points in a dodecagon with n regularly spaced points on a side. The dodecagonal number sequence is 1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, 672...

(b)Centered dodecagonal numbers are 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, 5953, 6337, 6733, 7141, 7561, 7993, 8437, 8893, 9361,9841,10333,10837…

(10.13) Tridecagonal number: -

(a)A tridecagonal number is a figurate number, the nth tridecagonal number will be the number of points in a tridecagon with n regularly spaced points on a side. The tridecagonal number sequence is 1, 13, 36, 70, 115, 171, 238, 316, 405, 505, 616, 738...

(b)Centered tridecagonal numbers are 1, 14, 40, 79, 131, 196, 274, 365, 469, 586, 716, 859, 1015, 1184, 1366, 1561, 1769, 1990, 2224, 2471, 2731, 3004, 3290, 3589, 3901, 4226, 4564, 4915, 5279, 5656..

(10.14) Tetradecagonal number: -

(a)A tetradecagonal number is a figurate number, the nth tetradecagonal number will be the number of points in a tetradecagon with n regularly spaced points on a side. The tetradecagonal number sequence is 1, 14, 39, 76, 125, 186, 259, 344, 441, 550, 671, 804…

(b)Centered tetradecagonal numbers are 1, 15, 43, 85, 141, 211, 295, 393, 505, 631, 771, 925, 1093, 1275, 1471, 1681, 1905, 2143, 2395, 2661, 2941, 3235, 3543, 3865, 4201,4551,4915,5293,5685,6091…

(10.15) Pentadecagonal number: -

(a)A pentadecagonal number is a figurate number, the nth pentadecagonal number will be the number of points in a pentadecagon with n regularly spaced points on a side. The pentadecagonal number sequence is 1, 15, 42, 82, 135, 201, 280, 372, 477, 595, 726, 870…

(b)Centered pentadecagonal numbers are 1, 16, 46, 91, 151, 226, 316, 421, 541, 676, 826, 991, 1171, 1366, 1576, 1801, 2041, 2296, 2566, 2851, 3151, 3466, 3796, 4141, 4501, 4876, 5266, 5671, 6091, 6526..

(10.16) Hexadecagonal number: - (a)A hexadecagonal number is a figurate number, the nth hexadecagonal number will be the number of points in a hexadecagon with n regularly spaced points on a side. The hexadecagonal number sequence is 1, 16, 45, 88, 145, 216, 301, 400,513, 640, 781, 936… (b)Centered hexadecagonal number 1, 17, 49, 97, 161, 241, 337, 449, 577, 721, 881, 1057, 1249, 1457, 1681, 1921, 2177, 2449, 2737, 3041, 3361, 3697, 4049, 4417, 4801, 5201, 5617, 6049, 6497, 6961..

(10.17) Heptadecagonal number: - (a)A heptadecagonal number is a figurate number, the nth heptadecagonal number will be the number of points in a heptadecagon with n regularly spaced points on a side. The heptadecagonal number sequence is 1, 17, 48, 94, 155, 231, 322, 428, 549, 685, 836, 1002… (b)Centered heptadecagonal numbers 1, 18, 52, 103, 171, 256, 358, 477, 613, 766, 936, 1123, 1327, 1548, 1786, 2041, 2313, 2602, 2908, 3231, 3571, 3928, 4302, 4693, 5101, 5526, 5968, 6427, 6903, 7396..

(10.18) Octadecagonal number: - (a)A octadecagonal number is a figurate number, the nth octadecagonal number will be the number of points in a octadecagon with n regularly spaced points on a side. The octadecagonal number sequence is 1, 18, 51, 100, 165,246, 343, 456, 585, 730, 891, 1068… (b)Centered octadecagonal Numbers 1, 19, 55, 109, 181, 271, 379, 505, 649, 811, 991, 1189, 1405, 1639, 1891, 2161, 2449, 2755, 3079, 3421, 3781, 4159, 4555, 4969, 5401, 5851, 6319, 6805, 7309, 7831..

(10.19) Nonadecagonal number: - (a)A nonadecagonal number is a figurate number, the nth nonadecagonal number will be the number of points in a nonadecagon with n regularly spaced points on a side. The noctadecagonal number sequence is 1, 19, 54, 106, 175, 261, 364, 484, 621, 775, 946, 1134… (b)Centered nonadecagonal numbers are 1, 20, 58, 115, 191, 286, 400, 533, 685, 856, 1046, 1255, 1483, 1730, 1996, 2281, 2585, 2908, 3250, 3611, 3991, 4390, 4808, 5245, 5701, 6176, 6670, 7183, 7715, 8266..

(10.20) Icosagonal number: - (a)An icosagonal number is a figurate number, the nth icosagonal number will be the number of points in an icosagon with n regularly spaced points on a side. The icosagonal number sequence is 1, 20, 57, 112, 185, 276, 385, 512, 657, 820, 1001, 1200… (b)Centered icosagonal numbers are 1, 21, 61, 121, 201, 301, 421, 561, 721, 901, 1101, 1321, 1561, 1821, 2101, 2401, 2721, 3061, 3421, 3801, 4201, 4621, 5061, 5521, 6001, 6501, 7021, 7561, 8121, 8701…

(10.21) Icosihenagonal number: -

(a)An icosihenagonal number is a figurate number, the nth icosihenagonal number will be the number of points in an icosihenagon with n regularly spaced points on a side. The icosihenagonal number sequence is 1, 21, 60, 118, 195, 291, 406, 540, 693, 865, 1056, 1266… (b)Centered icosihenagonal numbers are 1, 22, 64, 127, 211, 316, 442, 589, 757, 946, 1156, 1387, 1639, 1912, 2206, 2521, 2857, 3214, 3592, 3991, 4411, 4852, 5314, 5797, 6301, 6826, 7372, 7939, 8527, 9136…

(10.22) Icosidigonal number: - (a)An icosidigonal number is a figurate number, the nth icosidigonal number will be the number of points in an icosidigon with n regularly spaced points on a side. The icosidigonal number sequence is 1, 22, 63, 124, 205, 306, 427, 568, 729, 910, 1111, 1332…

(b) Centered icosidigonal numbers are 1, 23, 67, 133, 221, 331, 463, 617, 793, 991, 1211, 1453, 1717, 2003, 2311, 2641, 2993, 3367, 3763, 4181, 4621, 5083, 5567, 6073, 6601, 7151, 7723, 8317, 8933, 9571…

(10.23) Icositrigonal number: -

(a)An icositrigonal number is a figurate number, the nth icositrigonal number will be the number of points in an icositrigon with n regularly spaced points on a side. The icositrigonal number sequence is 1, 23, 66, 130, 215, 321, 448, 596, 765, 955, 1166, 1398…

(b) Centered icositrigonal numbers are 1, 24, 70, 139, 231, 346, 484, 645, 829, 1036, 1266, 1519, 1795, 2094, 2416, 2761, 3129, 3520, 3934, 4371, 4831, 5314, 5820, 6349, 6901, 7476, 8074, 8695, 9339, 10006…

(10.24) Icositetragonal number: -

(a)An icositetragonal number is a figurate number, the nth icositetragonal number will be the number of points in an icositetragon with n regularly spaced points on a side. The icositetragonal number sequence is 1, 24, 69, 136, 225, 336, 469, 624, 801, 1000, 1221, 1464…

(b) Centered icositetragonal numbers are 1, 25, 73, 145, 241, 361, 505, 673, 865, 1081, 1321, 1585, 1873, 2185, 2521, 2881, 3265, 3673, 4105, 4561, 5041, 5545, 6073, 6625, 7201, 7801, 8425, 9073, 9745, 10441…

(10.25) Icosipentagonal number: -

(a)An icosipentagonal number is a figurate number, the nth icosipentagonal number will be the number of points in an icosipentagon with n regularly spaced points on a side. The icosipentagonal number sequence is 1, 25, 72, 142, 235, 351, 490, 652, 837, 1045, 1276, 1530…

(b) Centered icosipentagonal numbers are 1, 26, 76, 151, 251, 376, 526, 701, 901, 1126…

(10.26) Icosihexagonal number: -

(a)An icosihexagonal number is a figurate number, the nth icosihexagonal number will be the number of points in an icosihexagon with n regularly spaced points on a side. The icosihexagonal number sequence is 1, 26, 75, 148, 245, 366, 511, 680, 873, 1090, 1331, 1596…

(b) Centered icosihexagonal numbers are 1, 27, 79, 157, 261, 391, 547, 729, 937, 1171…

(10.27) Icosiheptagonal number: -

(a)An icosiheptagonal number is a figurate number, the nth icosiheptagonal number will be the number of points in an icosiheptagon with n regularly spaced points on a side. The icosiheptagonal number sequence is 1, 27, 78, 154, 255, 381, 532, 708, 909, 1135, 1386, 1662…

(b) Centered icosiheptagonal numbers are 1, 28, 82, 163, 271, 406, 568, 757, 973, 1216…

(10.28) Icosioctagonal number: -

(a)An icosioctagonal number is a figurate number, the nth icosioctagonal number will be the number of points in an icosioctagon with n regularly spaced points on a side. The icosioctagonal number sequence is 1, 28, 81, 160, 265, 396, 553, 736, 945, 1180, 1441, 1728…

(b) Centered icosioctagonal numbers are 1, 29, 85, 169, 281, 421, 589, 785, 1009, 1261, 1541, 1849…

(10.29) Icosinonagonal number: -

(a)An icosinonagonal number is a figurate number, the nth icosinonagonal number will be the number of points in an icosinonagon with n regularly spaced points on a side. The icosinonagonal number sequence is 1, 29, 84, 166, 275, 411, 574, 764, 981, 1225, 1496, 1794…

(b)Centered icosinonagonal numbers are 1,30, 88, 175, 291, 436, 610, 813, 1045, 1306, 1596, 1915…

(10.30) Triacontagonal number: -

(a)A triacontagonal number is a figurate number, the nth triacontagonal number will be the number of points in a triacontagonal with n regularly spaced points on a side. The triacontagonal number sequence is 1, 30, 87, 172, 285, 426, 595, 792, 1017, 1270, 1551, 1860…

(b)Centered triacontagonal numbers are 1, 31, 91, 181, 301, 451, 631, 841, 1081, 1351, 1651, 1981…

Further polygonal numbers are

31-gonal Numbers 1, 31, 90, 178, 295, 441, 616, 820, 1053, 1315, 1606, 1926, 2275…

32-gonal Numbers 1, 32, 93, 184, 305, 456, 637, 848, 1089, 1360, 1661, 1992, 2353...

33-gonal Numbers 1, 33, 96, 190, 315, 471, 658, 876, 1225, 1405, 1716, 2058…

34-gonal Numbers 1, 34, 99, 196, 325, 486, 679, 904, 1161, 1450, 1771, 2124...

35-gonal Numbers 1, 35, 102, 202, 335, 501, 700, 932, 1197, 1495, 1826, 2190…

36-gonal Numbers 1, 36, 105, 208, 345, 516, 721, 960, 1233, 1540, 1881, 2256...

37-gonal Numbers 1, 37, 108, 214, 355, 531, 742, 988, 1269, 1585, 1936, 2322...

38-gonal Numbers 1, 38, 111, 220, 365, 546, 763, 1016, 1305, 1630, 1991, 2388...

39-gonal Numbers 1, 39, 114, 226, 375, 561, 784, 1044, 1341, 1675, 2046...

40-gonal Numbers 1, 40, 117, 232, 385, 576, 805, 1072, 1377, 1720, 210…

41-gonal Numbers 1, 41, 120, 238, 395, 591, 826, 1100, 1413, 1765, 2156...

42-gonal Numbers 1, 42, 123, 244, 405, 597, 847, 1128, 1449, 1810, 2211...

43-gonal Numbers 1, 43, 126, 250, 415, 603, 868, 1156, 1485, 1855, 2266…

44-gonal Numbers 1, 44, 129, 256, 425, 618, 889, 1184, 1521, 1900, 2321...

45-gonal Numbers 1, 45, 132, 262, 435, 633, 910, 1212, 1557, 1945, 2376…

46-gonal Numbers 1, 46, 135, 268, 445, 648, 931, 1240, 1593, 1990, 2431…

47-gonal Numbers 1, 47, 138, 274, 455, 663, 952, 1268, 1629, 2035...

48-gonal Numbers 1, 48, 141, 280, 465, 678, 973, 1296, 1665, 2080…

49-gonal Numbers 1, 49, 144, 286, 475, 693, 994, 1324, 1701, 2125…

50-gonal Numbers 1, 50, 147, 292, 485, 708, 1015, 1352, 1737, 2170…

51-gonal Numbers 1, 51, 150, 298, 495, 723, 1036, 1380, 1773, 2215…

52-gonal Numbers 1, 52, 153, 304, 505, 738, 1057, 1408, 1809, 2260...

53-gonal Numbers 1, 53, 156, 310, 515, 753, 1078, 1436, 1845, 2305…

54-gonal Numbers 1, 54, 159, 316, 525, 768, 1099, 1464, 1881, 2350…

55-gonal Numbers 1, 55, 162, 322, 535, 783, 1120, 1492, 1917, 2395...

56-gonal Numbers 1, 56, 165, 328, 545, 798, 1141, 1520, 1953, 2440…

57-gonal Numbers 1, 57, 168, 334, 555, 813, 1162, 1548, 1989, 2485...

58-gonal Numbers 1, 58, 171, 340, 656, 828, 1183, 1576, 2025...

59-gonal Numbers 1, 59, 174, 346, 575, 843, 1204, 1604, 2061…

60-gonal Numbers 1, 60, 177, 352, 585, 858, 1225, 1632, 2097…

61-gonal Numbers 1, 61, 180, 358, 595, 873, 1246, 1660, 2133...

62-gonal Numbers 1, 62, 183, 364, 605, 888, 1267, 1688, 2169…

63-gonal Numbers 1, 63, 186, 370, 615, 903, 1288, 1716, 2205...

64-gonal Numbers 1, 64, 189, 376, 625, 918, 1309, 1744, 2241...

65-gonal Numbers 1, 65, 192, 382, 635, 933, 1330, 1772, 2277...

66-gonal Numbers 1, 66, 195, 388, 645, 948, 1351, 1800, 2313…

67-gonal Numbers 1, 67, 198, 394, 655, 963, 1372, 1828, 2349...

68-gonal Numbers 1, 68, 201, 400, 665, 978, 1393, 1856, 2385...

69-gonal Numbers 1, 69, 204, 406, 675, 993, 1414, 1884, 2421…

70-gonal Numbers 1, 70, 207, 412, 685, 1008, 1435, 1912, 2457…

71-gonal Numbers 1, 71, 210, 418, 695, 1023, 1456, 1940, 2493…

72-gonal Numbers 1, 72, 213, 424, 705, 1038, 1477, 1968, 2529...

73-gonal Numbers 1, 73, 216, 430, 715, 1053, 1498, 1966, 2565...

74-gonal Numbers 1, 74, 219, 436, 725, 1068, 1519, 2024...

75-gonal Numbers 1, 75, 222, 442, 735, 1083, 1540, 2052...

76-gonal Numbers 1, 76, 225, 448, 745, 1098, 1561, 2080...

77-gonal Numbers 1, 77, 228, 454, 755, 1113, 1582, 2108…

78-gonal Numbers 1, 78, 231, 460, 765, 1128, 1603, 2136…

79-gonal Numbers 1, 79, 234, 466, 775, 1143, 1624, 2164…

80-gonal Numbers 1, 80, 237, 472, 785, 1158, 1645, 2192...

81-gonal Numbers 1, 81, 240, 478, 795, 1173, 1666, 2220…

82-gonal Numbers 1, 82, 243, 484, 805, 1188, 1687, 2248…

83-gonal Numbers 1, 83, 246, 490, 815, 1203, 1708, 2276...

84-gonal Numbers 1, 84, 249, 496, 825, 1218, 1729, 2304...

85-gonal Numbers 1, 85, 252, 502, 835, 1233, 1750, 2332…

86-gonal Numbers 1, 86, 255, 508, 845, 1248, 1771, 2360…

87-gonal Numbers 1, 87, 258, 514, 855, 1263, 1792, 2388...

88-gonal Numbers 1, 88, 261, 520, 865, 1278, 1813, 2416…

89-gonal Numbers 1, 89, 264, 526, 875, 1293, 1834, 2444...

90-gonal Numbers 1, 90, 267, 532, 885, 1308, 1855, 2472...

91-gonal Numbers 1, 91, 270, 538, 895, 1323, 1876, 2500…

92-gonal Numbers 1, 92, 273, 544, 905, 1338, 1897, 2528...

93-gonal Numbers 1, 93, 276, 550, 915, 1353, 1918, 2556...

94-gonal Numbers 1, 94, 279, 556, 925, 1368, 1939, 2584…

95-gonal Numbers 1, 95, 282, 562, 935, 1383, 1960, 2612...

96-gonal Numbers 1, 96, 285, 568, 945, 1398, 1981, 2640...

97-gonal Numbers 1, 97, 288, 574, 955, 1413, 2002, 2668…

98-gonal Numbers 1, 98, 291, 580, 965, 1428, 2023...

99-gonal Numbers 1, 99, 294, 586, 975, 1443, 2044...

100-gonal Numbers 1, 100, 297, 592, 985, 1458, 2065...

101-gonal Numbers 1, 101, 300, 598, 995, 1473, 2086...

102-gonal Numbers 1, 102, 303, 604, 1005, 1488, 2107...

103-gonal Numbers 1, 103, 306, 610, 1015, 1503, 2128...

104-gonal Numbers 1, 104, 309, 616, 1025, 1518, 2149…

105-gonal Numbers 1, 105, 312, 622, 1035, 1533, 2170...

106-gonal Numbers 1, 106, 315, 628, 1045, 1548, 2191...

107-gonal Numbers 1, 107, 318, 634, 1055, 1563, 2212…

108-gonal Numbers 1, 108, 321, 640, 1065, 1578, 2233...

109-gonal Numbers 1, 109, 324, 646, 1075, 1593, 2254...

110-gonal Numbers 1, 110, 327, 652, 1085, 1608, 2275…

111-gonal Numbers 1, 111, 330, 658, 1095, 1623, 2296...

112-gonal Numbers 1, 112, 333, 664, 1105, 1638, 2317...

113-gonal Numbers 1, 113, 336, 670, 1115, 1653, 2338…

114-gonal Numbers 1, 114, 339, 676, 1125, 1668, 2359...

115-gonal Numbers 1, 115, 342, 682, 1135, 1683, 2380...

116-gonal Numbers 1, 116, 345, 688, 1145, 1698, 2401…

117-gonal Numbers 1, 117, 348, 694, 1155, 1713, 2422...

118-gonal Numbers 1, 118, 351, 700, 1165, 1728, 2443…

119-gonal Numbers 1, 119, 354, 706, 1175, 1743, 2464...

120-gonal Numbers 1, 120, 357, 712, 1185, 1758, 2485…

121-gonal Numbers 1, 121, 360, 718, 1195, 1773, 2506...

122-gonal Numbers 1, 122, 363, 724, 1205, 1788, 2527...

123-gonal Numbers 1, 123, 366, 730, 1215, 1803, 2548…

124-gonal Numbers 1, 124, 369, 736, 1225, 1818, 2569...

125-gonal Numbers 1, 125, 372, 742, 1235, 1833, 2590...

126-gonal Numbers 1, 126, 375, 748, 1245, 1848, 2611…

127-gonal Numbers 1, 127, 378, 754, 1255, 1865, 2632...

128-gonal Numbers 1, 128, 381, 760, 1265, 1878, 2653…

129-gonal Numbers 1, 129, 384, 766, 1275, 1893, 2674…

130-gonal Numbers 1, 130, 387, 772, 1285, 1908, 2695...

131-gonal Numbers 1, 131, 390, 778, 1295, 1923, 2716...

132-gonal Numbers 1, 132, 393, 784, 1305, 1938, 2737…

133-gonal Numbers 1, 133, 396, 790, 1315, 1953, 2758...

134-gonal Numbers 1, 134, 399, 796, 1325, 1968, 2779...

135-gonal Numbers 1, 135, 402, 802, 1335, 1983, 2800…

136-gonal Numbers 1, 136, 405, 808, 1345, 1998, 2821...

137-gonal Numbers 1, 137, 408, 814, 1355, 2013...

138-gonal Numbers 1, 138, 411, 820, 1365, 2028...

139-gonal Numbers 1, 139, 414, 826, 1375, 2043…

140-gonal Numbers 1, 140, 417, 832, 1385, 2058…

141-gonal Numbers 1, 141, 420, 838, 1395, 2073…

142-gonal Numbers 1, 142, 423, 844, 1405, 2088...

143-gonal Numbers 1, 143, 426, 850, 1415, 2103...

144-gonal Numbers 1, 144, 429, 856, 1425, 2118…

145-gonal Numbers 1, 145, 432, 862, 1435, 2133...

146-gonal Numbers 1, 146, 435, 868, 1445, 2148...

147-gonal Numbers 1, 147, 438, 874, 1455, 2163…

148-gonal Numbers 1, 148, 441, 880, 1465, 2178…

149-gonal Numbers 1, 149, 444, 886, 1475, 2193...

150-gonal Numbers 1, 150, 447, 892, 1485, 2208...

151-gonal Numbers 1, 151, 450, 898, 1495, 2223…

152-gonal Numbers 1, 152, 453, 904, 1505, 2238...

153-gonal Numbers 1, 153, 456, 910, 1515, 2253...

154-gonal Numbers 1, 154, 459, 916, 1525, 2268…

155-gonal Numbers 1, 155, 462, 922, 1535, 2283...

156-gonal Numbers 1, 156, 465, 928, 1545, 2298...

157-gonal Numbers 1, 157, 468, 934, 1555, 2313…

158-gonal Numbers 1, 158, 471, 940, 1565, 2328…

159-gonal Numbers 1, 159, 474, 946, 1575, 2343...

160-gonal Numbers 1, 160, 477, 952, 1585, 2358...

161-gonal Numbers 1, 161, 480, 958, 1595, 2373…

162-gonal Numbers 1, 162, 483, 964, 1605, 2388…

163-gonal Numbers 1, 163, 486, 970, 1615, 2403...

164-gonal Numbers 1, 164, 489, 976, 1625, 2418...

165-gonal Numbers 1, 165, 492, 982, 1635, 2433…

166-gonal Numbers 1, 166, 495, 988, 1645, 2448...

167-gonal Numbers 1, 167, 498, 994, 1655, 2463...

168-gonal Numbers 1, 168, 501, 1000, 1665, 2478...

169-gonal Numbers 1, 169, 504, 1006, 1675, 2493...

170-gonal Numbers 1, 170, 507, 1012, 1685, 2508…

171-gonal Numbers 1, 171, 510, 1018, 1695, 2523…

172-gonal Numbers 1, 172, 513, 1024, 1705, 2538…

173-gonal Numbers 1, 173, 516, 1030, 1715, 2553…

174-gonal Numbers 1, 174, 519, 1036, 1725, 2568...

175-gonal Numbers 1, 175, 522, 1042, 1735, 2583...

176-gonal Numbers 1, 176, 525, 1048, 1745, 2598…

177-gonal Numbers 1, 177, 528, 1054, 1755, 2613...

178-gonal Numbers 1, 178, 531, 1060, 1765, 2628...

179-gonal Numbers 1, 179, 534, 1066, 1775, 2643…

180-gonal Numbers 1, 180, 537, 1072, 1785, 2658…

181-gonal Numbers 1, 181, 540, 1078, 1795, 2673...

182-gonal Numbers 1, 182, 543, 1084, 1805, 2688...

183-gonal Numbers 1, 183, 546, 1090, 1815, 2703…

184-gonal Numbers 1, 184, 549, 1096, 1825, 2718...

185-gonal Numbers 1, 185, 552, 1102, 1835, 2733...

186-gonal Numbers 1, 186, 555, 1108, 1845, 2748…

187-gonal Numbers 1, 187, 558, 1114, 1855, 2763...

188-gonal Numbers 1, 188, 561, 1120, 1865, 2778...

189-gonal Numbers 1, 189, 564, 1126, 1875, 2793…

190-gonal Numbers 1, 190, 567, 1132, 1885, 2808...

191-gonal Numbers 1, 191, 570, 1138, 1895, 2823...

192-gonal Numbers 1, 192, 573, 1144, 1905, 2838…

193-gonal Numbers 1, 193, 576, 1150, 1915, 2853...

194-gonal Numbers 1, 194, 579, 1156, 1925, 2868...

195-gonal Numbers 1, 195, 582, 1162, 1935, 2883…

196-gonal Numbers 1, 196, 585, 1168, 1945, 2898…

197-gonal Numbers 1, 197, 588, 1174, 1955, 2913...

198-gonal Numbers 1, 198, 591, 1180, 1965, 2928...

199-gonal Numbers 1, 199, 594, 1186, 1975, 2943...

200-gonal Numbers 1, 200, 597, 1192, 1985, 2958...

201-gonal Numbers 1, 201, 600, 1198, 1995, 2973...

202-gonal Numbers 1, 202, 603, 1204, 2005, 2988…

203-gonal Numbers 1, 203, 606, 1210, 2015...

204-gonal Numbers 1, 204, 609, 1216, 2025...

205-gonal Numbers 1, 205, 612, 1222, 2035…

206-gonal Numbers 1, 206, 615, 1228, 2045...

207-gonal Numbers 1, 207, 618, 1234, 2055...

208-gonal Numbers 1, 208, 621, 1240, 2065...

209-gonal Numbers1, 209, 624, 1246, 2075…

210-gonal Numbers 1, 210, 627, 1252. 2085...

211-gonal Numbers 1, 211, 630, 1258. 2095...

212-gonal Numbers 1, 212, 633, 1264, 2105...

213-gonal Numbers 1, 213, 636, 1270, 2115…

214-gonal Numbers 1, 214, 639, 1276, 2125...

215-gonal Numbers 1, 215, 642, 1282, 2135...

216-gonal Numbers 1, 216, 645, 1288, 2145…

217-gonal Numbers 1, 217, 648, 1294, 2155...

218-gonal Numbers 1, 218, 651, 1300, 2165...

219-gonal Numbers 1, 219, 654, 1306, 2175…

220-gonal Numbers 1, 220, 657, 1312, 2185...

221-gonal Numbers 1, 221, 660, 1318, 2195...

222-gonal Numbers 1, 222, 663, 1324, 2205…

223-gonal Numbers 1, 223, 666, 1330, 2215...

224-gonal Numbers 1, 224, 669, 1336, 2225...

225-gonal Numbers 1, 225, 672, 1342, 2235…

226-gonal Numbers 1, 226, 675, 1348, 2245...

227-gonal Numbers 1, 227, 678, 1354, 2255...

228-gonal Numbers 1, 228, 681, 1360, 2565...

229-gonal Numbers 1, 229, 684, 1366, 2275…

230-gonal Numbers 1, 230, 687, 1372, 2285...

231-gonal Numbers 1, 231, 690, 1378, 2295...

232-gonal Numbers 1, 232, 693, 1384, 2305…

233-gonal Numbers 1, 233, 696, 1390, 2315...

234-gonal Numbers 1, 234, 699, 1396, 2325...

235-gonal Numbers 1, 235, 702, 1402, 2335…

236-gonal Numbers 1, 236, 705, 1408, 2345...

237-gonal Numbers 1, 237, 708, 1414, 2355...

238-gonal Numbers 1, 238, 711, 1420, 2365…

239-gonal Numbers 1, 239, 714, 1426, 2375...

240-gonal Numbers 1, 240, 717, 1432, 2385...

241-gonal Numbers 1, 241, 720, 1438, 2395…

242-gonal Numbers 1, 242, 723, 1444, 2405...

243-gonal Numbers 1, 243, 726, 1450, 2415...

244-gonal Numbers 1, 244, 729, 1456, 2425…

245-gonal Numbers 1, 245, 732, 1462, 2435...

246-gonal Numbers 1, 246, 735, 1468, 2445...

247-gonal Numbers 1, 247, 738, 1474, 2455…

248-gonal Numbers 1, 248, 741, 1480, 2465..

249-gonal Numbers 1, 249, 744, 1486, 2475…

250-gonal Numbers 1, 250, 747, 1492, 2485…

251-gonal Numbers 1, 251, 750, 1498, 2495…

252-gonal Numbers 1, 252, 753, 1504, 2505…

253-gonal Numbers 1, 253, 756, 1510, 2515...

254-gonal Numbers 1, 254, 759, 1516, 2525...

255-gonal Numbers 1, 255, 762, 1522, 2535…

256-gonal Numbers 1, 256, 765, 1528, 2545...

257-gonal Numbers 1, 257, 768, 1534, 2555...

258-gonal Numbers 1, 258, 771, 1540, 2565…

259-gonal Numbers 1, 259, 774, 1546, 2575...

260-gonal Numbers 1, 260, 777, 1552, 2585...

261-gonal Numbers 1, 261, 780, 1558, 2595…

262-gonal Numbers 1, 262, 783, 1564, 2605…

263-gonal Numbers 1, 263, 786, 1570, 2615...

264-gonal Numbers 1, 264, 789, 1576, 2625...

265-gonal Numbers 1, 265, 792, 1582, 2635...

266-gonal Numbers 1, 266, 795, 1588, 2645...

267-gonal Numbers 1, 267, 798, 1594, 2655…

268-gonal Numbers 1, 268, 801, 1600, 2665…

269-gonal Numbers 1, 269, 804, 1606, 2675...

270-gonal Numbers 1, 270, 807, 1612, 2685...

271-gonal Numbers 1, 271, 810, 1618, 2695…

272-gonal Numbers 1, 272, 813, 1624, 2705…

273-gonal Numbers 1, 273, 816, 1630, 2715...

274-gonal Numbers 1, 274, 819, 1636, 2725...

275-gonal Numbers 1, 275, 822, 1642, 2735…

276-gonal Numbers 1, 276, 825, 1648, 2745...

277-gonal Numbers 1, 277, 828, 1654, 2755...

278-gonal Numbers 1, 278, 831, 1660, 2765…

279-gonal Numbers 1, 279, 834, 1666, 2775...

280-gonal Numbers 1, 280, 837, 1672, 2785...

281-gonal Numbers 1, 281, 840, 1678, 2795...

282-gonal Numbers 1, 282, 843, 1684, 2805...

283-gonal Numbers 1, 283, 846, 1690, 2815…

284-gonal Numbers 1, 284, 849, 1696, 2825...

285-gonal Numbers 1, 285, 852, 1702, 2835...

286-gonal Numbers 1, 286, 855, 1708, 2845…

287-gonal Numbers 1, 287, 858, 1714, 2855...

288-gonal Numbers 1, 288, 861, 1720, 2865...

289-gonal Numbers 1, 289, 864, 1726, 2875…

290-gonal Numbers 1, 290, 867, 1732, 2885...

291-gonal Numbers 1, 291, 870, 1738, 2895...

292-gonal Numbers 1, 292, 873, 1744, 2905…

293-gonal Numbers 1, 293, 876, 1750, 2915…

294-gonal Numbers 1, 294, 879, 1756, 2925...

295-gonal Numbers 1, 295, 882, 1762, 2935...

296-gonal Numbers 1, 296, 885, 1768, 2945…

297-gonal Numbers 1, 297, 888, 1774, 2955...

298-gonal Numbers 1, 298, 891, 1780, 2965...

299-gonal Numbers 1, 299, 894, 1786, 2975…

300-gonal Numbers 1, 300, 897, 1792, 2985...

301-gonal Numbers 1, 301, 900, 1798, 2995...

302-gonal Numbers 1, 302, 903, 1804, 3005…

303-gonal Numbers 1, 303, 906, 1810, 3015...

304-gonal Numbers 1, 304, 909, 1816, 3025...

305-gonal Numbers 1, 305, 912, 1822, 3035…

306-gonal Numbers 1, 306, 915, 1828, 3045...

307-gonal Numbers 1, 307, 918, 1834, 3055...

308-gonal Numbers 1, 308, 921, 1840, 3065…

309-gonal Numbers 1, 309, 924, 1846, 3075...

310-gonal Numbers 1, 310, 927, 1852, 3085...

311-gonal Numbers 1, 311, 930, 1858, 3095…

312-gonal Numbers 1, 312, 933, 1864, 3105…

313-gonal Numbers 1, 313, 936, 1870, 3115…

314-gonal Numbers 1, 314, 939, 1876, 3125…

315-gonal Numbers 1, 315, 942, 1882, 3135...

316-gonal Numbers 1, 316, 945, 1888, 3145...

317-gonal Numbers 1, 317, 948, 1894, 3155...

318-gonal Numbers 1, 318, 951, 1900, 3165…

319-gonal Numbers 1, 319, 954, 1906, 3175...

320-gonal Numbers 1, 320, 957, 1912, 3185...

321-gonal Numbers 1, 321, 960, 1918, 3195…

322-gonal Numbers 1, 322, 963, 1924, 3205...

323-gonal Numbers 1, 323, 966, 1930, 3215...

324-gonal Numbers 1, 324, 969, 1936, 3225…

325-gonal Numbers 1, 325, 972, 1942, 3235...

326-gonal Numbers 1, 326, 975, 1948, 3245...

327-gonal Numbers 1, 327, 978, 1954, 3255…

328-gonal Numbers 1, 328, 981, 1960, 3265...

329-gonal Numbers 1, 329, 984, 1966, 3275...

330-gonal Numbers 1, 330, 987, 1972, 3285…

331-gonal Numbers 1, 331, 990, 1978, 3295...

332-gonal Numbers 1, 332, 993, 1984, 3305...

333-gonal Numbers 1, 333, 996, 1990, 3315…

334-gonal Numbers 1, 334, 999, 1996, 3325...

335-gonal Numbers 1, 335, 1002, 2002, 3335...

336-gonal Numbers 1, 336, 1005, 2008, 3345...

337-gonal Numbers 1, 337, 1008, 2014…

(10.31) Pyrimadal number: - (a)A pyramidal number is a figurate number that represents a pyramid with a base and a given number of sides. The pyrimadal number is named after the geometrical base on which the pyramid is built. (b)Triangular pyramidal number (three sides): - This is a figurate number which represents a pyramid with triangle base. (c) Square pyramidal number (4 sides): -This is a figurate number which represents a pyramid with square base. The square pyramid number sequence is 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819… A formula to find out the ‘n’th square pyramid number is [n(n+1)(2n+1)/6] where ‘n’ is a real number. Applying the formula the fifth square pyramid we will get 55. The sum of two consecutive square pyramidal numbers is an octahedral number. For example 5 + 14 = 19 and is an octahedral number. Besides 1, there is only one other number that is both a square and a square pyramid number is 4900, which is both the 70th square number and the 24th square pyramidal number. (d) Pentagonal pyramidal number (five sides): - This is a figurate number which represents a pyramid with pentagon base. The pentagonal pyramidal numbers are: 1, 6, 18, 40, 75, 126, 196, 288, 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126,10206, 11368, 12615, 13950, 15376, 16896, 18513, 20230, 22050, 23976, 26011, 28158, 30420, 32800, 35301... The formula for the nth pentagonal pyramidal number is [n^2(n+1)/2]. So the nth pentagonal pyramidal number is the average of n^2 and n^3. The nth pentagonal pyramidal number is also n times the nth triangular number. For example the 5th pentagonal pyramidal number is 75 and is equal to 5 times the 5th triangular number 15 that is 5×15=75. (e) Hexagonal pyramidal number (six sides): - This is a figurate number which represents a pyramid with hexagon base. The hexagonal pyramidal numbers are 1, 7, 22, 50, 95, 161, 252, 372, 525, 715, 946, 1222, 1547, 1925, 2360, 2856, 3417, 4047, 4750, 5530, 6391, 7337, 8372, 9500, 10725, 12051, 13482, 15022, 16675, 18445, 20336, 22352, 24497, 26775, 29190, 31746, 34447, 37297, 40300... (f)Heptagonal pyramidal number (seven sides): - This is a figurate number which represents a pyramid with heptagon base. The heptagonal pyramidal numbers are 1, 8, 26, 60, 115, 196, 308, 456, 645, 880, 1166, 1508, 1911, 2380, 2920, 3536, 4233, 5016, 5890, 6860, 7931, 9108, 10396, 11800, 13325, 14976, 16758, 18676, 20735, 22940, 25296, 27808, 30481, 33320, 36330, 39516, 42883, 46436, 50180, 54120...

(10.32) Cube number: -

(a)The numbers whose dots form a lattice work of cubes are called as cube numbers. They are discussed in chapter 5.

(b) Centered Cube Numbers are 1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2331, 3059, 3925, 4941, 6119, 7471, 9009, 10745, 12691, 14859, 17261, 19909, 22815, 25991, 29449, 33201, 37259, 41635, 46341, 51389, 56791, 62559, 68705, 75241, 82179, 89531, 97309, 105525...

(10.33) Tetrahedral number: - The numbers whose dots form a lattice work of tetrahedrons are called as tetrahedral numbers. For example the numbers 1, 4, 10, 20, 35, 56, 84, 120, 165, 220… are tetrahedral numbers.

(10.34) Octahedral number: -

(a)An octahedral number is a figurate number that represents an octahedron, or two square pyramids placed together, one upside-down underneath the other. For example the numbers 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726, 13131, 14644, 16269, 18010, 19871, 21856, 23969, 26214, 28595, 31116, 33781, 36594, 39559, 42680...
are octahedral numbers.

(b) The nth octahedral number can be obtained by adding the (n-1)th and nth square pyramidal numbers together, or by using the following formula: [n(2n^2+1)/3]

Multi-Polygonal Numbers: - Some numbers can be expressed in two and more geometrical shapes and these numbers are generally termed as multi-polygonal numbers.

(10.35) Square triangular number: - Some numbers can be expressed in triangle as well as in square shape. Such numbers are called as triangular square number or square triangular number. The square triangular number series is 1, 36, 1225, 41616, 1413721, 48024900…

(10.36) Pentagonal triangular number: - Some numbers can be expressed in triangle as well as in pentagon shape. Such numbers are called as pentagonal triangular number. The pentagonal triangular number sequence is 1, 210, 40755, 7906276, 1533776805…

(10.37) Heptagonal triangular number: - Some numbers can be expressed in triangle as well as in heptagon shape. Such numbers are called as heptagonal triangular number. The heptagonal triangular number sequence is 1, 55, 121771, 5720653, 12625478965…

(10.38) Octagonal triangular number: - Some numbers can be expressed in triangle as well as in octagon shape. Such numbers are called as octagonal triangular numbers. The octagonal triangular number sequence is 1, 21, 11781, 203841, 113123361…

(10.39) Nonagonal triangular number: - Some numbers can be expressed in triangle as well as in nonagon shape. Such numbers are called as nonagonal triangular numbers. The nonagonal triangular number sequence is 1, 325, 82621, 20985481, 5330229625, 1353857339341…

(10.40) Square Pyramidal triangular numbers: - Some numbers can be expressed in triangle as well as in square pyramid shape. Such numbers are called as square pyramidal triangular numbers. The square pyramidal triangular number sequence is 1, 55, 91, 208335…

(10.41)Tetrahedral triangular numbers: - Some numbers can be expressed in triangle as well as in tetrahedron shape. Such numbers are called as tetrahedral triangular numbers. The tetrahedral triangular number sequence is 1, 10, 120, 1540, 7140…

(10.42) Pentagonal square number: - Some numbers can be expressed in square as well as in pentagon shape. Such numbers are called as pentagonal square number. The pentagonal square number sequence is 1, 9801, 94109401, 903638458801, 8676736387298001…

(10.43) Hexagonal square number: - Some numbers can be expressed in square as well as in hexagon shape. Such numbers are called as hexagonal square number. The hexagonal square number sequence is 1, 1225, 1413721, 1631432881…

(10.44) Heptagonal square number: - Some numbers can be expressed in square as well as in heptagon shape. Such numbers are called as heptagonal square number. The heptagonal square number sequence is 1, 81, 5929, 2307361, 168662169, 12328771225…

(10.45) Octagonal square number: - Some numbers can be expressed in square as well as in octagon shape. Such numbers are called as octagonal square number. The octagonal square number sequence is 1, 225, 43681, 8473921, 1643897025…

(10.46) Nonagonal square number: - Some numbers can be expressed in square as well as in nonagon shape. Such numbers are called as nonagonal square number. The nonagonal square number sequence is 1, 9, 1089, 8281, 978121, 7436529…

(10.47)Square Pyramidal square Numbers: - Some numbers can be expressed in square as well as in square pyramid shape. Such numbers are called as square pyramidal square numbers. The square pyramidal square number sequence is 1, 4900…

(10.48)Tetrahedral square Numbers: - Some numbers can be expressed in tetrahedron as well as in square shape. Such numbers are called as tetrahedral square numbers. The tetrahedral square number sequence is 1, 4, 19600…

(10.49) Hexagonal pentagon number: - Some numbers can be expressed in hexagon as well as in pentagon shape. Such numbers are called as hexagonal pentagon number. The hexagonal pentagon number sequence is 1, 40755, 1533776805…

(10.50) Heptagonal pentagon number: - Some numbers can be expressed in pentagon as well as in heptagon shape. Such numbers are called as heptagonal pentagon numbers. The heptagonal pentagon number sequence is 1, 4347, 16701685, 64167869935, 246532939589097…

(10.51) Octagonal pentagon number: - Some numbers can be expressed in pentagon as well as in octagon shape. Such numbers are called as octagonal pentagon numbers. The octagonal pentagon number sequence is 1, 176, 1575425, 234631320, 2098015778145…

(10.52) Nonagonal pentagon number: - Some numbers can be expressed in pentagon as well as in nonagon shape. Such numbers are called as nonagonal pentagon numbers. The nonagonal pentagon number sequence is 1, 651, 180868051, 95317119801, 26472137730696901…

(10.53) Heptagonal hexagon number: - Some numbers can be expressed in hexagon as well as in heptagon shape. Such numbers are called as heptagonal hexagon numbers. The heptagonal hexagon number sequence is 1, 121771, 12625478965, 1309034909945503…

(10.54) Octagonal hexagon number: - Some numbers can be expressed in hexagon as well as in octagon shape. Such numbers are called as octagonal hexagon numbers. The octagonal hexagon number sequence is 1, 11781, 113123361, 1086210502741…

(10.55) Nonagonal hexagon number: - Some numbers can be expressed in hexagon as well as in nonagon shape. Such numbers are called as nonagonal hexagon numbers. The nonagonal hexagon number sequence is 1, 325, 5330229625, 1353857339341, 22184715227362706161…

(10.56) Octagonal heptagon number: - Some numbers can be expressed in heptagon as well as in octagon shape. Such numbers are called as octagonal heptagon numbers. The octagonal heptagon number sequence is 1, 297045, 69010153345…

(10.57) Nonagonal heptagon number: - Some numbers can be expressed in heptagon as well as in nonagon shape. Such numbers are called as nonagonal heptagon numbers. The nonagonal heptagon number sequence is 1, 26884, 542041975, 10928650279834…

(10.58) Nonagonal octagon number: - Some numbers can be expressed in octagon as well as in nonagon shape. Such numbers are called as nonagonal octagon numbers. The nonagonal octagon number sequence is 1, 631125, 286703855361, 130242107189808901…

(10.59)Square Pyramidal tetrahedral Numbers: - Some numbers can be expressed in square pyramid as well as in tetrahedron shape. Such numbers are called as square pyramidal tetrahedral numbers. The only number which is simultaneously tetrahedral and square pyramidal is 1.

11. MAGIC MATHEMATICS

(11.1) Magic triangles :-

(a)Magic triangles are the triangles having the peculiar property that the sum is constant regardless of adding the numbers, forming the triangle in any side.

(b)Magic triangles with consecutive numbers starting from 1 to n (where n>2)can be solved by applying the formula [n(n+1)/2] where ‘n’ is the end number or highest number in the magic triangle. For example if ‘n’ is 3, by applying the above formula, end number ‘E’ is 6. The magic triangle is formed as follows.

The magic triangle sum ‘M’ is 9, a constant regardless of adding the numbers in any side.

(c)The order of a magic triangle is determined by the number of entries between any two corners of the triangle. For example in the above case order ‘O’ = 1. In other words for forming a 1st order magic triangle, 6 consecutive numbers are required.

The relation between the order number and the end number of the triangle border puzzle is

Order number (O)

1

2

3

4

5

End number (E)

6

9

12

15

18

From the table we can arrive E = 3(O+1)

For example for order 6 the end number is 21.

(d) The sum of consecutive numbers that make up the magic triangle denoted as ‘S’, can be found by using the formula [n(n+1)/2]. For example in the above case,

sum ‘S’ = (6×7)/2 = 21.

(e) Any magic triangle puzzle will give four sums

(I) the sum of consecutive numbers that make up the magic triangle denoted as ‘S’

(II)the sum of corner numbers in the magic triangle denoted as ‘C’

(III)the sum of non corner numbers in the magic triangle denoted as ‘B’ and

(IV)the magic triangle sum denoted as ‘M’

For example in the magic triangle puzzle of order 1, starting from 1 to 6

8.1. S = [(6×7)/2] = 21

8.2. C = 1+2+3 = 6

8.3. B = 6+4+5 = 15

8.4. M = 1+6+2 = 2+4+3 = 1+5+3 = 9

(f) The relationship between S, C and M is S + C = 3M

C = 3M – S

M = ⅓(S+C)

S + C = 4M for magic square border puzzle and S + C = 5M for magic pentagonal border puzzle and so on.

(g) The relationship between B, S and M is B = 2S – 3M

(h) ‘M’ lies between ⅓S and ⅔S. The ‘S’ lies between ‘C’ and ‘B’.

(i) For magic triangle puzzle of order 1, there are four solutions available and they are

Now for the above magic triangle puzzles a, b, c and d

a

b

c

d

S

21

21

21

21

C

6

9

12

15

M

9

10

11

12

B

15

12

9

6

(j) There are four magic border sums for magic triangle of order 1 and seven magic border sums for magic triangle of order 2 and so on. The formula is 3n + 1 where ‘n’ is the order of the magic triangle. For magic triangle of order 3 the magic border sums possible are [(3×3)+1] = 10.

(k)The magic triangles of odd order magic triangles can be solved like this method. For example the magic triangle of order 3 is formed as follows. Here the magic triangle border sum is 28.

Similarly the higher odd ordered magic triangles are solved by filling up the consecutive numbers in clockwise and anticlockwise directions alternatively. The following magic triangle of order 5 has the magic triangle border sum 59.

(l) The even ordered magic triangles have irregular arrangements. The following magic triangle of order 2 has the magic triangle border sum 17.

This magic triangle can be written in another way, keeping the magic border sum same is as follows.

The above magic triangle of order 2 can be arranged in such a manner so that the sum becomes 20.

(11.2) Magic squares: -

(a)Magic squares are the squares having the peculiar property that the sum is constant regardless of adding the numbers in any side. For example a magic square can be formed by using the numbers from 1 to 9 is as follows.

6

1

8

7

5

3

2

9

4

In the above magic square the row sum is 15, the column sum is 15 and the diagonal sum is also 15 regardless of adding the numbers in any side. Here 15 is called as magic square sum.

(b)Magic squares are classified into two types namely odd magic squares and even magic squares based upon the number of squares present in the magic square. For example the above magic square is an odd magic square.

(c)The order of magic square puzzle is equal to the number of rows or number of columns present in the magic square puzzle. For example in the above case the order is 3.

(d) A table of order of magic square starting from 1 onwards against the magic square sum is as follows.

Order of magic square

3

4

5

6

7

Magic square sum

15

34

65

111

175

Note: - Magic square of order 1 and 2 is not possible.

(e) A magic square sum remains unchanged when rotated or reflected.

(f) A third order magic square comprising of prime numbers only is as follows.

71

5

101

89

59

29

17

113

47

The magic square sum is 177.

(g) If we delete the central square keeping the outermost border square only then we will get the magic square border puzzle of the same order. For example in the above case if we remove the central square of 5 then we will get the magic border puzzle of order 3.

8

3

4

1

9

6

7

2

Here except the diagonal sum the row sum and the column sum is continue to be equal. But the numbers are not consecutive as 5 gets left out from 1 to 9.

(h) Thrice the amount of central number of the magic square is equal to the magic square sum. In the above case 5×3=15.

(i) The sum of exact opposite numbers in the magic square is ⅔ of magic square sum. In the above case 9+1=10 which is ⅔ of 15.

(j) A simple way of arranging this type of magic square having nine consecutive numbers is as follows.

6th number

1st number

8th number

7th number

5th number

3rd number

2nd number

9th number

4th number

(k) The magic square of 4×4 starting from 1 onwards is as follows. Here the magic square sum is 34.

7

12

1

14

2

13

8

11

16

3

10

5

9

6

15

4

Likewise we can extend the magic squares for 5×5 starting from 1 onwards and so on.

(l) Examples of magic square of order 4×4 without the continuous sequence of

numbers are as follows.

Table 1 Table 2

44

51

2

7

Table 1:- The sum of the numbers along the row, column or diagonal is 104.

Table 2:- The sum of the numbers along the row, column or diagonal is 76.

30

37

2

7

6

3

48

47

6

3

34

33

50

45

8

1

36

31

8

1

4

5

46

49

4

5

32

35

(11.3) Magic square border puzzle:-

(a)The magic square puzzle with consecutive numbers from 1 to 9 is unique giving only one magic square sum whereas the magic square border puzzle with consecutive numbers starting from 1 to 8 of the third order give four magic square border sums as follows.

Square border puzzle A:-

6

4

2

5

XII

7

1

8

3

Square border puzzle B:-

8

3

2

4

XIII

6

1

7

5

Square border puzzle C:-

8

2

4

5

XIV

3

1

6

7

Square border puzzle D:-

8

1

6

4

XV

2

3

5

7

Note: - The magic square border sum is shown in Roman numerals.

(b) In the magic square border puzzles the sum of the middle numbers in the horizontal border is equal to the sum of the middle numbers in the vertical borders. Call these sum of the middle numbers as ‘S’.

In the square border puzzle A 4+8 = 7+5 = 12

In the square border puzzle B 3+7 = 4+6 = 10

In the square border puzzle C 5+3 = 2+6 = 8

In the square border puzzle D 4+2 = 1+5 = 6.

(c) The arithmetic sum ‘N’ of the consecutive numbers starting from 1 to 8 making up the square border puzzle of the third order is 36. That is [8(8+1)/2]=36.

The corner border sums in the four solutions of the 3rd order magic square border puzzles are, in the square border puzzle A 1+2+3+6 = 12

in the square border puzzle B 1+2+5+8 = 16

in the square border puzzle C 1+4+7+8 = 20

in the square border puzzle D 3+6+7+8 = 24

Call these corner sums as ‘C’ numbers.

The magic border sums are 12, 13, 14 and 15. Call these numbers as ‘M’ numbers. Now tabulate N, C and M numbers.

N

36

36

36

36

C

12

16

20

24

M

12

13

14

15

Draw another table

N+C

48

52

56

60

M

12

13

14

15

Now the sum of the consecutive numbers making up the square border ‘N’ with ‘C’ is equal to 4times ‘M’. That is N+C=4M.

From this we can derive the fact that C = 4M – N.

(d) Tabulate S, C and M numbers

S

12

10

8

6

C

12

16

20

24

M

12

13

14

15

From this table we can derive S+C=2M. Combining with the N+C=4M we get S = N-2M.

(e) The relationship between the magic square border sums and the magic square sum is 15 + 0 = 14 + 1 = 13 + 2 = 12 + 3. In other words the cumulative deduction figures are simply the 15 components of the magic border sums.

(f) Square border puzzle of first order is not possible and also the square border puzzle of second order is also not possible. So the square border puzzle starts from 3rd order onwards only.

(g) Step by step approach for building magic square border of order 3 is as follows. Write the sequence of consecutive numbers starting from 1 onwards

For example 1 2 3 4 5 6 7 8 ;

Pick out the numbers which make the corner border sum (C=16) Here 1+2+5+8 = 16 ;

Fix the corner numbers (1, 2, 5 and 8) in diagonal wise order. That is

8

2

1

5

Note the adjacent corner number sum on each border outside the border of the square. That is adjacent sum of 5 and 2 is 7; 8 and 1 is 9 ; and so on.

10

8

2

1

5

6

Fix the complements of the adjacent corner sum numbers with respect to the magic border sum (M=13) For example 13-7 = 6; 13-9 = 4; and so on.

Now we get the magic border puzzle of order 3 with magic border sum of 13.

(13-10)

8

3

2

4

6

1

7

5

(13-6)

(h) Magic square border puzzle of fourth order starting from 1 to 12 is as follows.

12

8

1

5

2

11

9

4

3

7

10

6

The sum is 26. Likewise we can extend the square border puzzles of higher orders.

A magic square border sum remains unchanged when rotated or reflected.

(i) Tabulate the order number, consecutive numbers and the total number in the consecutive numbers as follows.

Order

number

Consecutive numbers

Total number in the consecutive numbers

3

1,2,3,4,5,6,7,8

8 or 4(3-1)

4

1,2,3,4,5,6,7,8,9,10,11,12

12 or 4(4-1)

5

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16

16 or 4(5-1)

From the table we can that if the order number is given, take its precursor and multiply it by 4 to get the total number of consecutive numbers.

(11.4) Magic star puzzle: -

(a) Magic stars are the stars having the peculiar property that the sum is constant regardless of adding the numbers diagonal-wise in any side. Magic star starting from 1 to 5 can be drawn as follows. Here 1 is the center of the magic star. The magic star sum is found by adding the numbers diagonal-wise in any side.

(b) The order of the magic star is the number of integers present in a ray excluding the central number. In the above case it is a magic star of order 1 and since it has four rays it can also be called as four directional magic star of order one.

(c) The magic stars are further can be developed by increasing the number of directions or rays to 8, 16, 32 and so on.

(d) The eight directional magic star of order 1 is as follows. Here the magic star sum is 12.

(e) The eight directional magic star of order 1 can be drawn as follows.

(f)The eight directional magic star can be arranged in another way so that the magic star sum is 15 in all its diameters.

(g) Except 4, we can arrange the numbers from 3 to 12 to express eight directional magic star of order to have the magic star sum ‘20’ in all sides.

(h) Ray sum is the sum of the integers in a ray of the magic star including the central number. The ray sum cannot be arranged equal in the first order magic star puzzles.

(i) Magic star of order 2 with four directions is as follows.

│

The magic star sum is 23. Here the ray sums are different.

(j) The four directional magic star of order 2 can be arranged in such a way that the ray sum is also equal. In the following magic star puzzle the ray sum is 12 and star sum is 23.

(k) Eight directional magic star of order 2 is formed as follows.

(l) The ‘n’ order magic stars will contain ‘n’ number of sub magic stars. For example second order magic star will contain two sub magic stars. One is the even numbered inner ring magic star and the another one is the odd numbered outer ring magic star.

In the above case the inner ring sub magic star sum is 19 and the outer ring sub magic star sum is 21. Ray sum is the average of inner ring magic star sum and the outer ring magic star sum.

Inner ring sub magic star Outer ring sub magic star

(m) The even numbered sub magic star and the odd numbered magic sub star come alternatively.

(n) The summation of the inner ring sub magic star sum and the outer ring sub magic star when subtracted with central number will give the magic star sum.

In the above case 19 + 21 – 1 = 39.

(o) The steps for making eight directional 2nd order magic star puzzle is

Step 1 Step 2

Step 3

(p) The magic star of order 2 can also be arranged so that the ray sums also become equal.

(q) Another way of arranging the ray sum to equal is to position alternatively the even and odd numbers about the inner ring and the outer ring clockwise and anticlockwise respectively. In the following magic star puzzle the ray sum is 20 and the magic star sum is 39 in all sides.

(r) The six pointed magic star from 1 to 12 can be arranged in such a manner so that the sum of the numbers in every row and at the points are same. In the following six pointed magic star the sum is 26 in all sides and at the points (vertices).

The sum of the numbers in the points is 26.

(s) The eight pointed magic star from 1 to 16 has total of 34 in all sides except in the vertices.

(11.5) Applying arithmetic addition only 28 ones can be used to make 1000, 23 twos can be used to make 1000, 21 fours can be used to make 1000, 20 fives can be used to make 1000 and 8 eights can be used to make 1000.

111 222 444 555 888

111 222 444 55 88

111 222 44 55 8

111 222 44 55 8

111 22 44 55 8

111 22 44 55 1000

111 22 44 55

111 22 44 55

111 22 44 55

1 2 4 5

1000 1000 1000 1000

ANNEXURES

Annexure 1:- Glossary

Abscissa: - X-axis or X-co ordinate.

Aliter: - Alternate method of doing a sum.

Amicable number: - Two distinct positive integers (m, n) are said to be amicable when each integer is the sum of the proper divisors of the other.

Armstrong number: - If the addition of the cubes of each individual digit in a number is equal to the number itself means then it is called as Armstrong number.

Automorphic number: - If ‘x’ be a number with the property that xn ends in the same digit as ‘x’ then it is an automorphic number.

Average: - Arithmetic mean

Binary number: - It is the real number represented as the power of 2 and with the radix 2.

Cardinal number: - Counting of objects in terms of numbers like 1, 2, 3, 4… is called as cardinal number.

Centered triangular number: - A centered triangular number is a centered figurate number that represents a triangle with a dot in the center and all other dots surrounding the center dot in a triangular lattice.

Centered square number: -A centered square number is a centered figurate number that represents a square with a dot in the center and all other dots surrounding the center dot in a square lattice.

Centered pentagonal number: -A centered heptagonal number is a centered figurate number that represents a pentagon with a dot in the center and all other dots surrounding the center dot in a pentagonal lattice.

Centered hexagonal number: -A centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice.

Centered heptagonal number: -A centered heptagonal number is a centered figurate number that represents a heptagon with a dot in the center and all other dots surrounding the center dot in a heptagonal lattice.

Centered octagonal number: -A centered octagonal number is a centered figurate number that represents a octagon with a dot in the center and all other dots surrounding the center dot in a octagonal lattice.

Centered nonagonal number: -A centered nonagonal number is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in a nonagonal lattice.

Centered decagonal number: -A centered decagonal number is a centered figurate number that represents a decagon with a dot in the center and all other dots surrounding the center dot in a decagonal lattice.

Climbing number or Ascending number: -If the digits in a number is in ascending order means then it is called as climbing number.

Complex number:-It is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = −1.

Complement:- Filling up to the required state.

Complementary number: - It is the number complement to the given decimal number with respect to the base 10.

Composite number: - All the real numbers except the prime numbers are called as composite numbers.

Congruence: - Equivalence

Consecutive number or Following number: - It is the following number which present successively in the given number sequence.

Corollary : A concept derived from a main concept.

Cube number: - The product of one number with the same number three times is called as the cube of the number.

Cullen number: -It is a number of the form 2nn + 1.

Cyclic number: - The numbers formed by constructing new numbers from the same digits by moving them cyclically are called as cyclic numbers.

Decimal number: - It is the real number represented as the power of 10 with the radix 10.

Decagonal number: - The numbers whose dots form a lattice work of decagons are called as decagonal numbers.

Decagonal triangular number: -The numbers whose dots form a lattice work of decagonal triangles are called as decagonal triangular numbers.

Decagonal square number: - The numbers whose dots form a lattice work of decagonal squares are called as decagonal square numbers.

Decagonal pentagon number: -The numbers whose dots form a lattice work of decagonal pentagons are called as decagonal pentagon numbers or Pentagonal decagon numbers or Pentadecagonal numbers.

Decagonal hexagon number: -The numbers whose dots form a lattice work of decagonal hexagons are called as decagonal hexagon numbers.

Decagonal heptagon number: -The numbers whose dots form a lattice work of decagonal heptagons are called as decagonal heptagon numbers.

Decagonal octagon number: -The numbers whose dots form a lattice work of decagonal octagons are called as decagonal octagon numbers.

Decagonal nonagon number: -The numbers whose dots form a lattice work of decagonal nonagons are called as decagonal nonagon numbers.

Demlo number: - If we add the first and last digit/s and we get the digit/s in between the numbers then this type of phenomenon is called as demlo number.

Descending number: - If the digits in a number are in descending order means then it is called as descending number.

Divisible number: - If we have divisor/s for a number apart from 1 and the same number itself means it is a divisible number.

Dodecagonal number: - The numbers whose dots form a lattice work of dodecagons are called as dodecagonal numbers.

Dozenal number or Duodecimal number: - It is the real number represented as the power of 12 with the radix 12.

Even number or Paired number: - If a real number is divisible by 2 or multiples of 2 means it is an even number.

Fermat number: - Numbers of the form 2(2 to the power n) + 1, whether prime or composite are called Fermat numbers.

Fibonacci number: - If a number falls in the Fibonacci sequence means it is a Fibonacci number.

Fractional number:-If a number is not divided by another number perfectly means then both the numerator and denominator is collectively called as fractional number.

Harshad number: - It is defined as the number whose summation of the digits is one of the divisors of the number.

Hendecagonal number: - The numbers whose dots form a lattice work of hendecagons are called as hendecagonal numbers.

Heptagonal number: - The numbers whose dots form a lattice work of heptagons are called as heptagonal numbers.

Heptagonal triangular number: -The numbers whose dots form a lattice work of heptagonal triangles are called as heptagonal triangular numbers.

Heptagonal square number: - The numbers whose dots form a lattice work of heptagonal squares are called as heptagonal square numbers.

Heptagonal pentagon number: -The numbers whose dots form a lattice work of heptagonal pentagons are called as heptagonal pentagon numbers.

Heptagonal hexagon number: - The numbers whose dots form a lattice work of heptagonal hexagons are called as heptagonal hexagon numbers.

Heptagonal pyramidal number: -The numbers whose dots form a lattice work of heptagonal pyramids are called as heptagonal pyramidal numbers.

Heptadecagonal number: - The numbers whose dots form a lattice work of heptadecagons are called as heptadecagonal numbers.

Hexadecimal number: - It is the real number represented as the power of 16 with the radix 16.

Hexagonal number or cornered hexagonal number: - The numbers whose dots form a lattice work of hexagons are called as hexagonal numbers.

Hexagonal square number: -The numbers whose dots form a lattice work of hexagonal squares are called as hexagonal square numbers.

Hexagonal pentagon number: - The numbers whose dots form a lattice work of hexagonal pentagons are called as hexagonal pentagon numbers.

Hexagonal pyramidal number: -The numbers whose dots form a lattice work of hexagonal pyramids are called as hexagonal pyramidal numbers.

Hexadecagonal number: -The numbers whose dots form a lattice work of hexadecagons are called as hexadecagonal numbers.

Icosagonal number: - The numbers whose dots form a lattice work of dodecagons are called as dodecagonal numbers.

Icosidigonal number: - The numbers whose dots form a lattice work of icosidigons are called as icosidigonal numbers.

Icositrigonal number: - The numbers whose dots form a lattice work of icositrigons are called as icositrigonal numbers.

Icositetragonal number: -The numbers whose dots form a lattice work of icositetragons are called as icositetragonal numbers.

Icosipentagonal number: -The numbers whose dots form a lattice work of icosipentagons are called as icosipentagonal numbers.

Icosihexagonal number: -The numbers whose dots form a lattice work of icosihexagons are called as icosihexagonal numbers.

Icosihenagonal number: - The numbers whose dots form a lattice work of icosihenagons are called as icosihenagonal numbers.

Icosiheptagonal number: -The numbers whose dots form a lattice work of icosiheptagons are called as icosiheptagonal numbers.

Icosioctagonal number: -The numbers whose dots form a lattice work of icosioctagons are called as icosioctagonal numbers.

Icosinonagonal number: -The numbers whose dots form a lattice work of icosinonagons are called as icosinonagonal numbers.

Indo Arabic numbers or Arabic numbers: - This is the widespread method of numbering system in the world inclusive of zero in it.

Integer: - A real number.

Iteration: - Repetition.

Lucas number: - The Fibonacci rule of adding the latest two to get the next is kept, but here we start from 2 and 1 (in this order) instead of 0 and 1 for the (ordinary) Fibonacci numbers. The numbers obtained are called as Lucas numbers.

Kaprekar number: - When an ‘n’ digit number is squared and in the square number if the right hand ‘n’ digits are added to the left hand ‘n ’ digits and is equal to the original number itself means then it is called as Kaprekar number.

Mersenne number: -A Mersenne number, named after Marin Mersenne, is a positive integer that is one less than a power of two. That is 2n-1.

Mersenne prime number: - If the Mersenne number derived is a prime number means then it is called as Mersenne prime number.

Negative number: - Negative numbers are the reverse of real numbers. It ends in 0.

Nonagonal number: -The numbers whose dots form a lattice work of nonagons are called as nonagonal numbers.

Nonagonal triangular number: -The numbers whose dots form a lattice work of nonagonal triangles are called as nonagonal triangular numbers.

Nonagonal square number: - The numbers whose dots form a lattice work of nonagonal squares are called as nonagonal square numbers.

Nonagonal pentagon number: - The numbers whose dots form a lattice work of nonagonal pentagons are called as nonagonal pentagon numbers.

Nonagonal hexagon number: -The numbers whose dots form a lattice work of nonagonal hexagons are called as nonagonal hexagon numbers.

Nonagonal heptagon number: -The numbers whose dots form a lattice work of nonagonal heptagons are called as nonagonal heptagon numbers.

Nonagonal octagon number: -The numbers whose dots form a lattice work of nonagonal octagons are called as nonagonal octagon numbers.

Nonadecagonal number: - The numbers whose dots form a lattice work of nonadecagons are called as nonadecagonal numbers.

Octal number: - It is the real number represented as the power of 8 and with the radix 8.

Octagonal number or cornered octagonal number: - The numbers whose dots form a lattice work of octagons are called as octagonal numbers.

Octagonal triangular number: -The numbers whose dots form a lattice work of octagonal triangles are called as octagonal triangular numbers.

Octagonal square number: - The numbers whose dots form a lattice work of octagonal squares are called as octagonal square numbers.

Octagonal pentagon number: -The numbers whose dots form a lattice work of octagonal pentagons are called as octagonal pentagon numbers.

Octagonal hexagon number: -The numbers whose dots form a lattice work of octagonal hexagons are called as octagonal hexagon numbers.

Octagonal heptagon number: -The numbers whose dots form a lattice work of octagonal heptagons are called as octagonal heptagon numbers.

Octadecagonal number: - The numbers whose dots form a lattice work of octadecagons are called as octadecagonal numbers.

Octahedral number: - The numbers whose dots form a lattice work of octahedrons are called as octahedral numbers.

Odd number: - If a real number is indivisible by 2 or multiples of 2 means it is an odd number.

Ordinal number or Unpaired number: - Counting of objects in terms of words like first, second, third… is called as ordinal number.

Orthogonal: - perpendicular

Ordinate: - Y-axis or Y-co ordinate.

Palindrome number or symmetrical number: - When a number is written in the reverse order is equal to the original number itself means it is called as a palindrome number.

Pandigital number: - It is an integer that in a given base has among its significant digits each digit used in the base at least once.

Pascal number: - If a number falls in the Pascal sequence or if it is a member of Pascal triangle means it is a Pascal number.

Pentagonal number: - The numbers whose dots form a lattice work of pentagons are called as pentagonal numbers.

Pentagonal triangular number: -The numbers whose dots form a lattice work of pentagonal triangles are called as pentagonal triangular numbers.

Pentagonal square number: -The numbers whose dots form a lattice work of pentagonal squares are called as pentagonal square numbers.

Pentagonal pyramidal number: -The numbers whose dots form a lattice work of pentagonal pyramids are called as pentagonal pyramidal numbers.

Pentagonal bipyramidal number: -The numbers whose dots form a lattice work of pentagonal bipyramids are called as pentagonal bipyramidal numbers.

Pental number: - It is the real number represented as the power of 5 and with the radix 5.

Perfect number: - A perfect number is defined as the number whose summations of the factors (excluding the number itself) give the same number.

Polygonal number: - The numbers whose dots form a lattice work of polygons are called as polygonal numbers. This is the general terminology of all forming numbers from triangular number onwards.

Power number: -It is a general terminology of a number whose product with the same number ‘n’ times.

Preceding number or Precursor number: - It is the preceding number which present prior in the given number sequence.

Prime number or Indivisible number: - Prime number is a number which does not have a factor or divisor apart from 1 and the number itself.

Probable number: -The possible number that can be formed using the given digits is called as probable number.

Pronic number: - It is a number which is the product of two consecutive integers, that is, n (n + 1) where ‘n’ is a real number.

Pythagorean triple number: - If x, y and z are real numbers and obey x^2+y^2=z^2 then x, y and z are called as Pythagorean triple numbers.

Pyramidal number or Triangular pyramidal number: - The numbers whose dots form a lattice work of pyramids are called as pyramidal numbers.

Ramanujan number: - Ramanujan number is defined as the number which can be represented as the summation of two numbers or with the powers in two different ways.

Rational number: - Rational numbers include negative numbers in the whole numbers system.

Real number or Natural number or Positive number: - Real numbers are the positive numbers. They start from 1 onwards.

Reciprocal: - Inverse.

Rectangular number: - The numbers whose dots form a lattice work of rectangles are called as rectangular numbers.

Reflux number or Mirror image number: - This is the number obtained by the mirror image of the original number.

Repunit number: - Numbers formed with repeated digit is called as repunit numbers.

Roman numbers: - Numbers starting from I onwards are called as Roman numbers. Here zero is not present.

Science notation number: -It is the method of representing all the numbers as the powers of 10.

Seesaw number: - If the left hand side of the digits in a number with respect to the middle digit/s is increasing/decreasing and again the right hand side of the number is decreasing/increasing then the number is called as seesaw number.

Square number: - The product of one number with the same number two times is called as the square of the number.

Square pyramidal number: - The numbers whose dots form a lattice work of square pyramids are called as square pyramidal numbers.

Square Pyramidal square Numbers: - The numbers whose dots form a lattice work of square pyramids as well as squares are called as square pyramidal square numbers.

Square Pyramidal tetrahedral Numbers: - The numbers whose dots form a lattice work of square pyramids as well as tetrahedrons are called as square pyramidal tetrahedral numbers.

Square bipyramidal number: - The numbers whose dots form a lattice work of square bipyramids are called as square bipyramidal numbers.

Square Pyramidal triangular Numbers: - The numbers whose dots form a lattice work of square pyramids as well as triangles are called as square pyramidal triangular numbers.

Square triangular number: -The numbers whose dots form a lattice work of square triangles are called as square triangular numbers.

Supplement: - Additional state to acquire the required state.

Surd: - Approximate square root number of the power.

Ternary number: - It is the real number represented as the power of 3 and with the radix 3.

Tetradecagonal number: - The numbers whose dots form a lattice work of tetradecagons are called as tetradecagonal numbers.

Tetrahedral number: - The numbers whose dots form a lattice work of tetrahedrons are called as tetrahedral numbers.

Tetrahedral square Numbers: - The numbers whose dots form a lattice work of tetrahedrons as well as squares are called as tetrahedral square numbers.

Tetrahedral triangular Numbers: - The numbers whose dots form a lattice work of tetrahedrons as well as triangles are called as tetrahedral triangular numbers.

Transcendental number:- It is a number (possibly a complex number) which is not algebraic that is, it is not a root of a non-constant polynomial equation with rational coefficients.

Transpose number: - The numbers derived by interchanging the digits of the original number are called as transpose number.

Triacontagonal number: -The numbers whose dots form a lattice work of triacontagons are called as triacontagonal numbers.

Triangular number: - The numbers whose dots form a lattice work of triangles are called as triangular numbers.

Triangular bipyramidal number: - The numbers whose dots form a lattice work of triangular bipyramids are called as triangular bipyramidal numbers.

Triangular square number: - The numbers whose dots form a lattice work of triangles as well as that of squares are called as triangular square numbers.

Tridecagonal number: - The numbers whose dots form a lattice work of tridecagons are called as tridecagonal numbers.

Whole number: - Whole numbers include zero in the real number system. That is it starts from 0 onwards.

Annexure 2:- Comparison of mathematical properties

Comparison of mathematical properties

S.No

Properties

Arithmetic addition

Arithmetic subtraction

Arithmetic multiplication

Arithmetic division

1.

Additive property

Yes

No

No

No

2.

Subtractive property

No

Yes

No

No

3.

Multiplicative property

No

No

Yes

No

4.

Divisive property

No

No

No

Yes

5.

Determinative property

Yes

Yes

Yes

Yes

6.

Reflux property

No

No

No

No

7.

Palindrome property

No

No

No

No

8.

Symmetric property

Yes

No

Yes

No

9.

Transitive property

Yes

No

Yes

No

10.

Commutative property

Yes

No

Yes

No

11.

Anti commutative property

No

Yes

No

No

12.

Left associative property

Yes

No

Yes

No

13.

Right associative property

Yes

No

Yes

No

14.

Left cancellation property

No

No

Yes

Yes

15.

Right cancellation property

No

No

Yes

Yes

16.

Identity property

Yes

Yes

Yes

Yes

17.

Inverse property

Yes

Yes

No

No

18.

Left distributive property

Yes

Yes

No

No

19.

Right distributive property

Yes

Yes

No

No

20.

Closure property

Yes

Yes

Yes

Yes

21.

Upset property

No

No

Yes

No

22.

Upset reflux property

No

No

No

No

Comparison of mathematical properties (continued)

S.No

Properties

Powers

Mod

Ratio

Percentage

Average

1.

Additive property

Yes

Yes

Yesf

Yes

Yes

2.

Subtractive property

Yes

Yes

Yesg

Yes

No

3.

Multiplicative property

Yes

Yes

Yes

Yes

No

4.

Divisive property

Yes

Yes

Yes

Yes

No

5.

Determinative property

Yes

Yes

Yes

Yes

Yes

6.

Reflux property

No

No

Yes

No

No

7.

Palindrome property

No

No

No

No

No

8.

Symmetric property

No

Yes

Yes

No

No

9.

Transitive property

No

Yes

Yes

No

No

10.

Commutative property

No

Yes

Yes

No

No

11.

Anti commutative property

No

Yesa

No

No

No

12.

Left associative property

No

Yesb

Yesh

No

No

13.

Right associative property

No

Yesc

Yesi

No

No

14.

Left cancellation property

Yes

Yes

Yesj

No

No

15.

Right cancellation property

Yes

Yes

Yesk

No

No

16.

Identity property

Yes(1^2)

Yes

No

No

No

17.

Inverse property

No

No

No

No

No

18.

Left distributive property

No

Yesd

Yesl

Yes

No

19.

Right distributive property

No

Yese

Yesm

Yes

No

20.

Closure property

No

Yes

Yesn

No

No

21.

Upset property

Yes*

No

No

No

No

22.

Upset reflux property

No

No

No

No

No

* Yes when powers are involved in multiplication operation only.

a Yes for mod subtraction only.

b Yes for mod addition and mod multiplication only.

c Yes for mod addition and mod multiplication only.

d Yes for mod addition and subtraction only.

e Yes for mod addition and subtraction only.

f Yes for addition of precedent part of ratio with another precedent part of the ratio.

g Yes for subtraction of precedent part of ratio with another precedent part of the ratio.

h Yes for ratio addition and multiplication only.

i Yes for ratio addition and multiplication only.

j Yes for multiplication and division only.

k Yes for multiplication and division only.

l Yes for multiplication and division only.

m Yes for multiplication and division only.

n Yes for multiplication and division only.