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# Classification of Numbers

All numbers can be classified as follows: Natural numbers
All countable numbers are called natural numbers. They are denoted by N.
N = {1, 2, 3, 4…………………. ∞}

Whole numbers
When zero is included in the set of natural numbers, they are called whole numbers. They are denoted by W.
W = {0, 1, 2, 3, 4……………… ∞}

Integers
The integers are natural numbers including 0 (0, 1, 2, 3, ...) and their negatives (0, −1, −2, −3, ...). They are denoted by I.

Rational numbers
Any number which can be expressed in the form of p/q, where p and q are integers and q ≠ 0, is called a rational number. The set of rational numbers is denoted by Q.
Example: 4, 4/5, -5/8, 1/3, 2.74 etc.

Fractions
A fraction denotes a part or parts of a unit. The following are different types of fractions.
(a) Proper fractions – Fractions whose numerators are less than its denominator,
Example: 2/3, 4/5

(b) Improper fractions – Fractions whose numerators are greater than its denominator,
Example: 7/3

(c) Mixed fractions – These fractions have 2 parts – an integer part and a fractional part,
Example: 4½

(d) Decimal fractions
Fractions in which the denominators are powers of ten are called decimal fractions.

Recurring decimals
If in a decimal fraction, a figure or a set of figures is repeated continuously, then such a number is called a recurring decimal.

Example:
a) 1/3 = 0.333333…. = 0.33
b) 22/7 = 3.142857142857 = 3.142857

If all the figures after the decimal point repeat themselves then it is called a pure recurring decimal. If only some figures repeat and others don’t, then it is called a mixed recurring decimal.

Irrational numbers
Any number which cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0, is called an irrational number. Set of irrational numbers is denoted by Q'. They are generally the non-terminating and non-recurring decimal fractions.
Example: √3, π, e etc.

Real numbers
Set of numbers which include both rational and irrational numbers is called real numbers. The set of real numbers is denoted by R. R = Q U Q'

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