CBSE Class 11 Maths Notes: Number Systems - Natural, Whole, Integers and Rational and Irrational

NATURAL NUMBERS

The numbers 1, 2, 3, 4…. Are called natural numbers, their set is denoted by N. Thus N = {1, 2, 3, 4, 5….}

WHOLE NUMBERS

The numbers 0, 1, 2, 3, 4…… Are called whole numbers, their set is denoted by W.
Thus W = {0, 1, 2, 3, 4….}

INTEGERS

The numbers …– 3, –2, –1, 0, 1, 2, 3….are called integers and their set is denoted by I & Z

Set of positive integers denoted by I⁺and consists of {1, 2, 3…}, also known as set of natural numbers.
Set of negative integers denoted by I^–and consists of {…– 3, – 2, – 1}
Set of non-negative integers is {0,1,2, 3…} also known as set of whole numbers
Set of non-positive integers is {…– 3, – 2, – 1, 0}

RATIONAL NUMBERS

All numbers of the form p/q where p and q are integer and q ¹ 0, are called rational. Thus

it may be noted that every integer is a rational number it can be written as p/1. Examples are 1/3, – 4/9 and 57

The rational numbers are precisely the real numbers with decimal expansions that are either

Terminating (ending in an infinite string of zeros), for example 3/4 = .75000… = .75
or
Non- Terminating Repeating (ending with a block of digits that repeats over and over).

For example 23/11 = 2.090909… = 2.09. The bar indicates the block of repeating digits.

IRRATIONAL NUMBERS

Real numbers that are not rational are called irrational numbers. They are precisely the real numbers with decimal expansions that are non-terminating non-repeating. Their set is denoted by Q^c (i.e. complementary set of Q)