Numbers : Arithmetic and Computation

This book contains a number of elementary ideas on numbers, their representations, interesting arithmetical problems and their analytical solutions, fundamentals of computers and programming plus programming solutions as an alternative to the analytical solutions and much more. Spanning seven chapters, this book, while keeping its lucid storytelling verve, describes integers, real numbers and numerous interesting properties and historical references; followed by a good collection of arithmetic problems and their analytical solutions. Please note: Taylor & Francis does not sell or distribute the Hardback in India, Pakistan, Nepal, Bhutan, Bangladesh and Sri Lanka. This book contains a number of elementary ideas on numbers, their representations, interesting arithmetical problems and their analytical solutions, fundamentals of computers and programming. Please note: T&F does not sell or distribute the Hardback in South Asia. Cover 1 Title Page 2 Copyright Page 3 Preface 4 Table of Contents 8 1 Introduction 12 2 Integers 20 2.1 Representation 20 2.2 Test of divisibility 22 2.3 Curious patterns 24 2.3.1 Multiplication 25 2.3.2 Multiplication, addition and division 25 2.3.3 With consecutive integers 26 2.3.4 Pascal’s triangle 26 2.4 Iterations 28 2.4.1 Number of even, odd and total digits 28 2.4.2 Sum of powers of the digits 29 2.4.3 Magic show with 1089 29 2.4.4 Kaprekar numbers 30 2.4.5 Collatz conjecture and hailstone numbers 31 2.5 Prime numbers 31 2.5.1 Test of primality 32 2.5.2 Types of primes 34 2.6 Composite numbers and their divisors 37 2.7 Taxicab numbers 39 2.8 Pythagorean triples 40 2.9 Sequences 41 2.10 Miscellany 43 2.10.1 Representation of large numbers 43 2.10.2 Fascination of integers 44 3 Real Numbers 45 3.1 Introduction 45 3.2 Rational numbers 45 3.3 Irrational numbers 46 3.4 Transcendental numbers 47 3.5 Decimal and binary representation 47 3.6 Continued fraction representation 50 3.7 Different types of mean of a set of real numbers 52 3.8 Special rational numbers 53 3.8.1 Bernoulli numbers 53 3.8.2 Unit fractions 54 3.9 Special irrational and transcendental numbers 57 3.9.1 Golden sections 58 3.9.2 Oldest universal mathematical constant π 65 3.9.3 Base of natural logarithm e 67 3.9.4 Euler’s constant γ 68 4 Problems 70 5 Solutions 85 6 Computer and programming fundamentals 146 6.1 Advantages of binary representation of numbers 146 6.2 A computer as a black box 149 6.3 Logic operation and logic gates 150 6.3.1 NOT, AND and OR logic 150 6.3.2 The derived logic operations and logic gates 151 6.4 Implementing the half-adder 151 6.4.1 The Half-adder (HA) using simple logic gates 151 6.4.2 Half-adder is not enough for addition 152 6.4.3 Full-adder (FA) 153 6.5 Representing negative numbers 154 6.6 A full-subtractor 155 6.7 Multiplication and division 156 6.8 Fractional number and floating point representation 156 6.8.1 Fractional representation 156 6.8.2 IEEE 754 standard for floating point 157 6.8.3 IEEE 754 notation 157 6.9 Character and other codes 158 6.9.1 ASCII code 158 6.9.2 Other codes 159 6.9.3 UNICODE 159 6.10 Organisation of a typical PC 159 6.11 C programming language: A tutorial Introduction 160 6.11.1 C programming 160 6.11.2 Program development cycle and running your program 161 6.12 The role of the operating system (OS) 162 6.13 A few simple programs 162 6.13.1 Arithmetic operations 162 6.13.2 Temperature conversion 163 6.13.3 Copy input (keyboard) to output (monitor) program 164 6.13.4 Using your own function—other than main() 164 6.14 Recursive function 165 6.15 More programming examples: Counting characters, words and lines 166 6.15.1 Counting characters 166 6.15.2 Word counting 166 6.15.3 Counting alpha-numerals 167 6.16 Data objects, constants and declarations 167 6.17 Arithmetic, relational and logical operators 169 6.18 Bitwise Operation 169 6.19 C Keywords 170 6.20 Control flow 170 6.20.1 Loop statement: do . . . while 170 6.20.2 Branching 171 6.21 The pointer type 172 6.22 Local variables 172 6.23 Type conversion 174 6.24 The task ahead 174 7 Programming Solutions 175 7.1 Some useful functions 175 7.2 Solution to simple problems 183 7.3 Computers are good at repetition 185 7.4 Conjectures 187 7.4.1 Conjecture—got a formal proof 189 7.4.2 Conjecture proved to be wrong 190 7.4.3 Problems to be solved through computer programs 191 7.5 Solutions to some of the problems 192 7.6 Arithmetic with large integers 197 7.6.1 Factorial of n (≥ 21) 198 7.6.2 Solution to a Problems with big numbers 200 7.7 A computer cannot solve all problems 200 7.7.1 Finding counter example for Euler’s conjecture 201 7.7.2 The travelling salesman problem 201 7.7.3 Cryptography—RSA (failure is the pillar of security) 202 7.7.4 Difficult for analytical solution but easy for a computer 205 7.8 Enhancing computing power through distribution 206 7.9 Road ahead 206 A Brahmagupta’s Equation 207 A.1 Solution of Equation (1.7) 207 A.2 Solution of Equation (2.29) 209 B Bernoulli Numbers 210 C Unit Fractions 212 D Pattern in Decimal Representation 214 E Platonic Solids 216 F Conversion table, Boolean Algebra rules, Floating Point Format etc. 218 F.1 Decimal to Binary, Octal and Hexadecimal 218 F.2 Laws of Boolean Algebra and De-Morgan’s Laws 218 F.3 FLOATING POINT: Normalised, De-normalised and NaN 219 F.3.1 Examples of Floating Point Encoding: Normalized 220 F.3.2 Examples of Encoding: De-Normalized and NaN 220 F.3.3 Range 221 F.3.4 Rounding 221 F.4 ASCII Table 222 F.5 ASCII version of hello.c 223 F.6 C keywords 223 G Your own Library, command line parameters and dealing with big numbers 224 G.1 Your own library 224 G.2 Command line parameters 227 G.3 Functions for big numbers 228 Bibliography 232 Index 234 ASCII,code;,Bernoulli,numbers;,Collatz,conjecture,and,hailstone,numbers;,computing,fundamentals;,IEEE,754,notation;,Kaprekar,numbers;,Levant,books;,Pascal’s,triangle;,programming;,Sarat,Book,house;,Taxicab,numbers ASCII code,Bernoulli numbers,Collatz conjecture and hailstone numbers,computing fundamentals,IEEE 754 notation,Kaprekar numbers,Levant books,Pascal’s triangle,programming,Sarat Book house,Taxicab numbers