Decimal Expansion

Before going into a representation of the decimal expansion of rational numbers, let us understand what rational numbers are. A Rational Number is a number that can be written in form of p/q  where p, q are Integers and q != 0. Example: 2/3, 1/4, 4/5, etc. Rational numbers are denoted by Q. As every Integer can be represented in p/q form so all Integers are rational numbers.

Example: -1, -2, -6, 4, 5 can be represented as -1/1, -2/1, 4/1, 5/1

There are generally 3 types of decimal expansion:

1. Terminating
2. Non-terminating Repeating
3. Non-terminating Non Repeating

Terminating Decimals

Terminating decimals are those decimal numbers that have a finite number of digits. That means the number comes to end after a decimal point after a certain number of repetitions.

Example: 0.5, 0.678, 14.123445, 1.23, 0.00024, etc. 

Non-Terminating Decimals

Non-Terminating decimals are those decimal numbers that have an infinite number of digits. Here the number doesn’t comes to end.

Example: 1.33333….., 52.36363636…, 2.343537684904…, 3.1415926535897…, etc.

Repeating Decimals: Repeating decimals are those numbers in which a specific number repeats uniformly after a decimal point.

Example: 0.5555…, 13.262626…, 1.8769876…, etc.

Note: Non terminating and repeating decimals are rational numbers. They can be expressed in p/q form, where q != 0.

Non Repeating Decimals: In Non Repeating decimals, there is no uniform repetition of a number.

Example: 4.345627238…, 1.61803398…, 2.718281828459.., 1342.352567545…, etc.

Note: Non terminating and nonrepeating decimals are irrational numbers. They cannot be expressed in p/q form.

The Combination of a set of rational and irrational numbers is called real numbers. All the real numbers can be expressed on the number line. The numbers other than real numbers that cannot be represented on the number line are called imaginary numbers (unreal numbers). They are used to represent complex numbers. Below is a classification diagram of real numbers.