Numbers: Arithmetic and Computation

Cover
Title Page
Copyright Page
Preface
Table of Contents
1 Introduction
2 Integers
	2.1 Representation
	2.2 Test of divisibility
	2.3 Curious patterns
		2.3.1 Multiplication
		2.3.2 Multiplication, addition and division
		2.3.3 With consecutive integers
		2.3.4 Pascal’s triangle
	2.4 Iterations
		2.4.1 Number of even, odd and total digits
		2.4.2 Sum of powers of the digits
		2.4.3 Magic show with 1089
		2.4.4 Kaprekar numbers
		2.4.5 Collatz conjecture and hailstone numbers
	2.5 Prime numbers
		2.5.1 Test of primality
		2.5.2 Types of primes
	2.6 Composite numbers and their divisors
	2.7 Taxicab numbers
	2.8 Pythagorean triples
	2.9 Sequences
	2.10 Miscellany
		2.10.1 Representation of large numbers
		2.10.2 Fascination of integers
3 Real Numbers
	3.1 Introduction
	3.2 Rational numbers
	3.3 Irrational numbers
	3.4 Transcendental numbers
	3.5 Decimal and binary representation
	3.6 Continued fraction representation
	3.7 Different types of mean of a set of real numbers
	3.8 Special rational numbers
		3.8.1 Bernoulli numbers
		3.8.2 Unit fractions
	3.9 Special irrational and transcendental numbers
		3.9.1 Golden sections
		3.9.2 Oldest universal mathematical constant π
		3.9.3 Base of natural logarithm e
		3.9.4 Euler’s constant γ
4 Problems
5 Solutions
6 Computer and programming fundamentals
	6.1 Advantages of binary representation of numbers
	6.2 A computer as a black box
	6.3 Logic operation and logic gates
		6.3.1 NOT, AND and OR logic
		6.3.2 The derived logic operations and logic gates
	6.4 Implementing the half-adder
		6.4.1 The Half-adder (HA) using simple logic gates
		6.4.2 Half-adder is not enough for addition
		6.4.3 Full-adder (FA)
	6.5 Representing negative numbers
	6.6 A full-subtractor
	6.7 Multiplication and division
	6.8 Fractional number and floating point representation
		6.8.1 Fractional representation
		6.8.2 IEEE 754 standard for floating point
		6.8.3 IEEE 754 notation
	6.9 Character and other codes
		6.9.1 ASCII code
		6.9.2 Other codes
		6.9.3 UNICODE
	6.10 Organisation of a typical PC
	6.11 C programming language: A tutorial Introduction
		6.11.1 C programming
		6.11.2 Program development cycle and running your program
	6.12 The role of the operating system (OS)
	6.13 A few simple programs
		6.13.1 Arithmetic operations
		6.13.2 Temperature conversion
		6.13.3 Copy input (keyboard) to output (monitor) program
		6.13.4 Using your own function—other than main()
	6.14 Recursive function
	6.15 More programming examples: Counting characters, words and lines
		6.15.1 Counting characters
		6.15.2 Word counting
		6.15.3 Counting alpha-numerals
	6.16 Data objects, constants and declarations
	6.17 Arithmetic, relational and logical operators
	6.18 Bitwise Operation
	6.19 C Keywords
	6.20 Control flow
		6.20.1 Loop statement: do . . . while
		6.20.2 Branching
	6.21 The pointer type
	6.22 Local variables
	6.23 Type conversion
	6.24 The task ahead
7 Programming Solutions
	7.1 Some useful functions
	7.2 Solution to simple problems
	7.3 Computers are good at repetition
	7.4 Conjectures
		7.4.1 Conjecture—got a formal proof
		7.4.2 Conjecture proved to be wrong
		7.4.3 Problems to be solved through computer programs
	7.5 Solutions to some of the problems
	7.6 Arithmetic with large integers
		7.6.1 Factorial of n (≥ 21)
		7.6.2 Solution to a Problems with big numbers
	7.7 A computer cannot solve all problems
		7.7.1 Finding counter example for Euler’s conjecture
		7.7.2 The travelling salesman problem
		7.7.3 Cryptography—RSA (failure is the pillar of security)
		7.7.4 Difficult for analytical solution but easy for a computer
	7.8 Enhancing computing power through distribution
	7.9 Road ahead
A Brahmagupta’s Equation
	A.1 Solution of Equation (1.7)
	A.2 Solution of Equation (2.29)
B Bernoulli Numbers
C Unit Fractions
D Pattern in Decimal Representation
E Platonic Solids
F Conversion table, Boolean Algebra rules, Floating Point Format etc.
	F.1 Decimal to Binary, Octal and Hexadecimal
	F.2 Laws of Boolean Algebra and De-Morgan’s Laws
	F.3 FLOATING POINT: Normalised, De-normalised and NaN
		F.3.1 Examples of Floating Point Encoding: Normalized
		F.3.2 Examples of Encoding: De-Normalized and NaN
		F.3.3 Range
		F.3.4 Rounding
	F.4 ASCII Table
	F.5 ASCII version of hello.c
	F.6 C keywords
G Your own Library, command line parameters and dealing with big numbers
	G.1 Your own library
	G.2 Command line parameters
	G.3 Functions for big numbers
Bibliography
Index