Fundamentals Of Number System

Fundamentals of Number System: Number: This is one of the simplest mathematical structure. This theory is one of the oldest part of mathematics and fundamental to study of subjects like mathematics, science , physics etc. India’s contribution to the world in Number theory is ZERO ‘0’. Brahmagupta was the astronomer who introduced the concept of Zero for the first time. Number System: A number system is the set of symbols used to express quantities as the basis for counting, determining order, comparing amounts, performing calculations, and representing value. It is the set of characters and mathematical rules that are used to represent a number. Classification of Numbers: Number Line: A number line is a one-dimensional graph used to represent the numbers with ‘Zero’ as the reference point such that the positive numbers are to the right of ‘0’and negative numbers are to its left.

-3

-2

-1

0

1

2

3

All Numbers can be classified in two categories. Real Numbers: All numbers that can be represented on the number line are called real number. Complex Number: The numbers that cannot be represented on the number line are called complex numbers..

All real numbers can be classified further as follows.

Real Numbers

Rational

Irrational

Integer

Fraction

Proper (p/q, Where q>p)

Whole Number

Algebraic

Transcendental

Improper

Mixed

(p/q, Where p>q)

(a p/q, where q>p and a is an integer)

Natural Numbers

Negative Integers

Important Notes:      

No number can be both rational and irrational. All rational numbers are terminating or recurring Zero is neither negative nor positive All irrational numbers are non-terminating and non-recurring. One is neither a prime nor a composite number. Two is only even prime number.

Complex Numbers: Simplest form in which a complex number is written is a+ib, where a and b are the real number and i is the imaginary unit whose value is √-1. Example: 5i, -2i, √3i Set of complex numbers is denoted by C. For a complex number a+ib, can not be equal to zero. If a=0 and b≠0 then the number becomes a purely imaginary number. Any real number can be written in the form of complex number . since any real number x can be written as x +i(0). Real Numbers: The set of real numbers includes all integers, positive and negative; all fractions; and the irrational numbers, those whose decimal expansions never repeat.

It is very useful to picture the real numbers as points on the real line, as shown here.

PROPERTIES OF REAL NUMBERS The following table lists the defining properties of the real numbers (technically called the field axioms). These laws define how the things we call numbers should behave.

Addition

Multiplication

Commutative

Commutative

For all real a, b

For all real a, b

a+b=b+a

ab = ba

Associative

Associative

For all real a, b, c

For all real a, b, c

a + (b + c) = (a + b) + c

(ab)c = a(bc)

Identity

Identity

There exists a real number 0 such that for every real a

There exists a real number 1 such that for every real a

a+0=a

a×1=a

Additive Inverse (Opposite)

Multiplicative Inverse (Reciprocal)

For every real number a there exist a real number, denoted (−a), such that

For every real number a except 0 there exist a real number,

a + (−a) = 0

denoted

a×

, such that

=1

Distributive Law For all real a, b, c a(b + c) = ab + ac, and (a + b)c = ac + bc

Rational: