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Number System Tutorial I: Integers, Fractions, Prime & Composite


Classification of Real Numbers


It is possible to design a family tree of numbers in the following manner.

Rational Numbers

Any number which can be expressed in the form p/q, where p and q are integers (both + and -) and q≠0q≠0, is called a rational number. Eg: 12,−34,5,0,−712,-34,5,0,-7 etc.
Irrational Numbers

Obviously this is the number which can't be expressed in the form pqpq, where pp and qq are integers (both + and -) and q≠0q≠0. Eg: 2–√,π,e2,π,e etc. For a clear understanding about irrational numbers, it is better to consider the classification of Decimal Numbers.


Terminating and Non-terminating Decimals

If the decimal part of a number consisting a finite number of digits, then it is terminating decimal, otherwise it is a non-terminating decimal.
Eg: 0.2,1.25,123.123450.2,1.25,123.12345 etc are the examples for terminating decimals.
0.333...,0.15427821570.333...,0.1542782157... are the examples for non-terminating decimal.

Recurring Decimal

In a non-terminating decimal if the decimal part is an infinite repetition of a number or a group of numbers, and then it is called a recurring decimal.
Eg: 0.33333... (only 3 repeating), this can be expressed as 0.3¯0.3¯
0.545454... (pair 54 repeating), this can be expressed as 0.54¯¯¯¯0.54¯
0.12366666... (only 6 repeating), this can be expressed as 0.1236¯0.1236¯

Definition for irrational numbers

If a decimal is non-terminating and non-recurring, then it is an irrational number. Eg: 2–√=1.414213562...,3–√=1.7320508...


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