##### Document Text Contents

Page 1

SHAKUNTALA DEVI'S

NUMBEBS

Everything you always wanted to know about

numbers but was difficult to understand.

Page 73

to reduce multiplication to addition with the aid

of tables of quarter squares. The method is very

old. There are many tables of quarter squares and

the most extensive table lists quarter squares of

integers upto 200,000. The method is very useful

and economical.

147

WHAT ARE 'QUARTERNIONS' ?

If you do not use the properties of order and the

commutative law of multiplication, we obtain an

interesting further extension of the complex number

system known as Quarternions q.v. in abbreviation.

Quarternions are of the form a + bi + cj + dk

where a, b, c, d are real.

148

WHAT IS THE 'THEORY OF NUMBERS' ?

The Theory of Numbers is concerned with the

study of the properties of integers or whole num-

bers. The proofs of many of the Theory use the

deepest resources of mathematics. However, some

of the most interesting conjectures are still un-

proved.

Topics of the Theory of the Numbers can be listed

as follows?

71

Page 74

I.Divisibility and Primality:

a. Elementary Definitions; Factorizations into

Primes.

b. Residue Classes; Eulers Theorem

c. Congruences in one unknown

d. Quadratic Residues; The Quadratic Recipro-

city Law

e. Factorization of Numbers; Mersenne Primes

II.Representation by forms:

a. Binary Quadratic forms

b. Genera of Quadratic*. forms; Formulas for

Number of Representations

c. The Numbers Represented by a Quadratic

Form: universal forms Representation of

zero.

d. Automorphs and Reduction of indefinite

Binary Quadratic Forms

e. Diophantine Equations

III.Topics in Analytic Number Theory:

a. Gauss's Class-Number Conjencture

b. Distribution of Primes; Asymptotic Formulas

IV. Additive Theory of Numbers:

a. Partitions

b. The waring problem and related problems

72

Page 145

Also by Shakuntala Devi

Puzzles to Puzzle You

Match Your Wits with the Human Comput r!

The 150 puzzles included in this book are enjoyable

exercises in reason intended to sharpen your wits.

"Engrossing and entertaining book - there is

something of interest for all".

Statesman

"Provides excitement as well as recreation".

Patriot

More Puzzles to Puzzle You

Original, Maddening and...Irresistible!

This book offers over 300 tantalising puzzles, brain-

teasers and riddles. The puzzles include every possible

type of mathematual recreation; time and distance

problems, age and money riddles, puzzles involving

geometry and elementary algebra and just plain

straight thinking.

"She leaves you spellbound with tabulating feats that

could be the envy of the best computers in the world".

l imes of India

Page 146

The Human Computer

explains and simplifies

EVERYTHING YOU ALWAYS WANTED

TO KNOW ABOUT NUMBERS BUT

WAS DIFFICULT TO UNDERSTAND

Little Known Facts

9

99Ninth power of the ninth power of nine is the largest in the world of numbers that can be

expressed with just 3 digits. The answer is mind

staggering!

It will contain 369 mill ion digits. To read it

normally would take more than a year. To write

down would require 1164 miles of paper!

Brain Teasers

A necklace was broken during an amorous

struggle. 1/3rd of the pearls fell to the ground,

1/5th stayed on the couch, 1/6th were found by the

girl, and 1/10th recovered by her lover; six pearls

remained on the string. Say of how many pearls the

necklace was composed.

Numer ica l concepts Simpl i f ied and Explained

Numerals & Digits • Perfect Numbers & Amicab le

Numbers • Algor i thm & Logarithm • Associative &

Commutat ive Laws of Multipl ication • What are

G o o g o l & Googolp lex .

A n d over 20 ready — reference tables to

help you understand and master numbers.

O R I E N T R \ P E R B A C K S

Illustration on Cover: Diabolic Magic Square similar to one inscribed

at Khajuraho around 12th Century.

ISBN A 1 - 2 2 2 - Q D D L - D

SHAKUNTALA DEVI'S

NUMBEBS

Everything you always wanted to know about

numbers but was difficult to understand.

Page 73

to reduce multiplication to addition with the aid

of tables of quarter squares. The method is very

old. There are many tables of quarter squares and

the most extensive table lists quarter squares of

integers upto 200,000. The method is very useful

and economical.

147

WHAT ARE 'QUARTERNIONS' ?

If you do not use the properties of order and the

commutative law of multiplication, we obtain an

interesting further extension of the complex number

system known as Quarternions q.v. in abbreviation.

Quarternions are of the form a + bi + cj + dk

where a, b, c, d are real.

148

WHAT IS THE 'THEORY OF NUMBERS' ?

The Theory of Numbers is concerned with the

study of the properties of integers or whole num-

bers. The proofs of many of the Theory use the

deepest resources of mathematics. However, some

of the most interesting conjectures are still un-

proved.

Topics of the Theory of the Numbers can be listed

as follows?

71

Page 74

I.Divisibility and Primality:

a. Elementary Definitions; Factorizations into

Primes.

b. Residue Classes; Eulers Theorem

c. Congruences in one unknown

d. Quadratic Residues; The Quadratic Recipro-

city Law

e. Factorization of Numbers; Mersenne Primes

II.Representation by forms:

a. Binary Quadratic forms

b. Genera of Quadratic*. forms; Formulas for

Number of Representations

c. The Numbers Represented by a Quadratic

Form: universal forms Representation of

zero.

d. Automorphs and Reduction of indefinite

Binary Quadratic Forms

e. Diophantine Equations

III.Topics in Analytic Number Theory:

a. Gauss's Class-Number Conjencture

b. Distribution of Primes; Asymptotic Formulas

IV. Additive Theory of Numbers:

a. Partitions

b. The waring problem and related problems

72

Page 145

Also by Shakuntala Devi

Puzzles to Puzzle You

Match Your Wits with the Human Comput r!

The 150 puzzles included in this book are enjoyable

exercises in reason intended to sharpen your wits.

"Engrossing and entertaining book - there is

something of interest for all".

Statesman

"Provides excitement as well as recreation".

Patriot

More Puzzles to Puzzle You

Original, Maddening and...Irresistible!

This book offers over 300 tantalising puzzles, brain-

teasers and riddles. The puzzles include every possible

type of mathematual recreation; time and distance

problems, age and money riddles, puzzles involving

geometry and elementary algebra and just plain

straight thinking.

"She leaves you spellbound with tabulating feats that

could be the envy of the best computers in the world".

l imes of India

Page 146

The Human Computer

explains and simplifies

EVERYTHING YOU ALWAYS WANTED

TO KNOW ABOUT NUMBERS BUT

WAS DIFFICULT TO UNDERSTAND

Little Known Facts

9

99Ninth power of the ninth power of nine is the largest in the world of numbers that can be

expressed with just 3 digits. The answer is mind

staggering!

It will contain 369 mill ion digits. To read it

normally would take more than a year. To write

down would require 1164 miles of paper!

Brain Teasers

A necklace was broken during an amorous

struggle. 1/3rd of the pearls fell to the ground,

1/5th stayed on the couch, 1/6th were found by the

girl, and 1/10th recovered by her lover; six pearls

remained on the string. Say of how many pearls the

necklace was composed.

Numer ica l concepts Simpl i f ied and Explained

Numerals & Digits • Perfect Numbers & Amicab le

Numbers • Algor i thm & Logarithm • Associative &

Commutat ive Laws of Multipl ication • What are

G o o g o l & Googolp lex .

A n d over 20 ready — reference tables to

help you understand and master numbers.

O R I E N T R \ P E R B A C K S

Illustration on Cover: Diabolic Magic Square similar to one inscribed

at Khajuraho around 12th Century.

ISBN A 1 - 2 2 2 - Q D D L - D