Repeating Decimal

Repeating decimals are numbers having digits reoccurring after the decimal point. These are numbers that have digits repeating infinitely but in periods after the decimal point.

Types of Repeating Decimals

Some decimals can have only 1 digit repeating. Other decimals can have 2 or more digits repeating together periodically. The diagram below shows some more examples of repeating digits with repetitive patterns according to periods.

Let us see how to write a repeating decimal.

How to use Symbol for Repeating Decimals

Repeating decimals can be represented with bars or dots on the top according to the pattern of the repetition. The diagram below shows how to write repeating decimals based on the periods of their repetitions.

For example; 0.3333… can be written as ${ 0.\overline{3}}$ with the bar on 3 as it is repeating at period 1.

We also use dots over the digits that are recurring as shown in the diagram below.

Are Repeating Decimals Rational or Irrational

YES! Some rational numbers are repeating decimals.

Repeating decimals are rational numbers because when we write them in p/q form, the numerator ‘p’and the denominator ‘q’ are whole numbers.

For example, if we divide 1 by 3 by long division method, we get the quotient as 0.33333….. However, its fractional form is ${\dfrac{1}{3}}$, where both 1 and 3 are whole numbers. Hence, repeating decimals are rational numbers.

Let us learn how to write a repeating decimal as a fraction.

Converting Repeating Decimal to Fraction

Suppose we have a decimal 0.5555… We will convert it to fraction as shown in the figure below.

Convert the fraction 1/11 into decimal.

Solution:

1/11 = 0.09090909…
It is a repeating decimal.

Which fraction has a repeating decimal as its decimal expansion, 9/12 or 3/11?

Solution:

Case 1: 9/12 = 0.75, remainder = 0, the decimal expansion is a terminating decimal, as the division ends at 5.
Case 2: 3/11 = 0.2727…, remainder is never 0, it comes as 8 and 3 in repetitions, so the quotient is a repeating decimal where 2 and 7 are repeating in periods of 2.
∴ 0.2727…. = ${0.\overline{27}}$