Decimal Representation Of Rational Numbers | Solved Examples

Decimal Representation of Rational Numbers

 A rational number is a number that can be written in the form p/q where p and q are integers, and q ≠ 0. The set of rational numbers is denoted by Q or \(\mathbb{Q}\). Examples:1/4, −2/5, 0.3 (or) 3/10, −0.7(or) −7/10, 0.151515... (or) 15/99. Rational numbers can be represented as decimals. The different types of rational numbers are Integers like -1, 0, 5, etc., fractions like 2/5, 1/3, etc., terminating decimals like 0.12, 0.625, 1.325, etc., and non-terminating decimals with repeating patterns (after the decimal point) such as 0.666..., 1.151515..., etc.

1.	Decimal Representation of Rational Numbers
2.	Decimal Representation of Terminating Rational Number
3.	Decimal Representation of Non-Terminating Rational Number
4.	Solved Examples on Decimal Representation of Rational Numbers
5.	Practice Questions on Decimal Representation of Rational Numbers
6.	Frequently Asked Questions (FAQs)

Decimal Representation of Rational Numbers

The decimal representation of a rational number is converting a rational number into a decimal number that has the same mathematical value as the rational number. A rational number can be represented as a decimal number with the help of the long division method. We divide the given rational number in the long division form and the quotient which we get is the decimal representation of the rational number. A rational number can have two types of decimal representations (expansions):

• Terminating
• Non-terminating but repeating

Note: Any decimal representation that is non-terminating and non-recurring, will be an irrational number. 

Let's try to understand what are terminating and non-terminating terms. While dividing a number by the long division method, if we get zero as the remainder, the decimal expansion of such a number is called terminating.

Example: 1/2

Let us see the long division of 1 by 2 in the following image:

1/2 = 0.5 is a terminating decimal

And while dividing a number, if the decimal expansion continues and the remainder does not become zero, it is called non-terminating.

Example: 1/3

Let us see the long division of 1 by 3 in the following image:

 

1/3 = 0.33333... is a recurring, non-terminating decimal. You can notice that the digits in the quotient keep repeating.

Decimal Representation of Terminating Rational Number

The terminating decimal expansion means that the decimal representation or expansion terminates after a certain number of digits. A rational number is terminating if it can be expressed in the form: p/(2n×5m). The rational number whose denominator is a number that has no other factor than 2 or 5, will terminate the result sooner or later after the decimal point. Consider the rational number 1/16.

Here, the decimal expansion of 1/16 terminates after 4 digits. Here 16 in the denominator is 16 = 24. Note that in terminating decimal expansion, you will find that the prime factorization of the denominator has no other factors other than 2 or 5.

Decimal Representation of Non-Terminating Decimal Number

The non-terminating but repeating decimal expansion means that although the decimal representation has an infinite number of digits, there is a repetitive pattern to it. The rational number whose denominator is having a factor other than 2 or 5, will not have a terminating decimal number as the result.

For example:

Note that in non-terminating but repeating decimal expansion, you will find that the prime factorization of the denominator has factors other than 2 or 5.

Related topics:

• Operations on Rational Numbers
• Rational Numbers
• Irrational Numbers

Important Notes

1. If a number can be expressed in the form p/(2n×5m) where p ∈ Z and m,n ∈ W, then the rational number will be a terminating decimal.
2. Terminating decimal expansion means that the decimal representation or expansion terminates after a certain number of digits.
3. Every non-terminating but repeating decimal representation corresponds to a rational number even if the repetition starts after a certain number of digits.