In the earlier articles, we have discussed on what is meant by Rational Number and its representation. Here we will step ahead and discuss the classification of non-terminating decimals i.e. recurring and non-recurring decimals. We will elaborate and explain the definitions of recurring and non-recurring decimals as well as the steps involved in how to turn a recurring decimal into a rational number with enough examples. Go through the article completely and be well versed with the concept.
Recurring Decimals – Definition
Recurring Decimals are the numbers that keep on repeating the same value after the decimal point. These are also called repeating decimals. For Example: 1/3 = 0.3333….
To denote a repeating digit in the recurring decimal we often place a bar over the repeating digit.
Non-Recurring Decimals – Definition
In the case of Non-Recurring Decimals, the numbers don’t repeat after the decimal point. These are also known as Non-Terminating or Non-Repeating Decimals. For Example: √3 = 1.7320508075688772935274463415059……
How to Write Recurring Decimals as Rational Numbers?
Go through the simple steps listed below to convert between recurring decimals to rational numbers. The steps involved are as under
- Firstly, let us assume x to be the recurring decimal we need to change as a rational number.
- Examine the repeating digits carefully in the recurring decimal.
- Place the repeating digit to the left side of the decimal point.
- Next to the above step place the repeating digits on the right of the decimal point.
- Now, subtract both the left-hand sides and right-hand sides accordingly to obtain the rational number.
Shortcut Method for turning a Recurring Decimal into a Rational Number
(The whole number obtained by writing the digits in their order – The whole number made by the nonrecurring digits in order )/(10^The number of digits after the decimal point−10^The number of digits after the decimal point that does not recur)
Let us better understand the method by having a glance at the example.
Example:
Convert 13.288888…. to a rational number?
Solution:
Given Repeating Decimal Number = 13.288888…….
As per the formula, (The whole number obtained by writing the digits in their order – The whole number made by the nonrecurring digits in order )/(10^The number of digits after the decimal point−10^The number of digits after the decimal point that does not recur)
Here the whole number obtained by writing the digits in order is 1328
The whole number obtained by non-recurring digits in order is 132
The number of digits after the decimal point = 2 (two)
The number of digits after the decimal point that do not recur = 1 (one)
13.288888……. = \(\frac { (1328-132) }{ 10^2 -10^1 } \)
=\(\frac { 1196 }{ 90 } \)
=\(\frac { 598 }{ 45 } \)
See More: Rational Numbers in Terminating and Non-Terminating Decimals
Solved Examples on Writing Repeating Decimals as Rational Numbers
Example 1.
Convert 0.3333…. to a rational number?
Solution:
Let us assume the recurring decimal as x i.e. x = 0.3333……. (1)
After examining we found the repeating digit is 3
Now let us place the repeating digit on the left side of the decimal point. To do so we need to move the decimal point 1 place to the right. We can do so by multiplying with 10 i.e. 10x =3.3333…..(2)
Subtracting left-hand sides and right-hand sides equations we have
10x-x = 3.3333…..0.333333
9x = 3
x=3/9
Therefore, recurring decimal 0.3333…. converted to a rational number is 3/9
Example 2.
Convert 2.567878….. to a rational number?
Solution:
Let us assume the recurring decimal as x i.e. x = 2.567878……
After examining we found the repeating digit is 78
Now let us place the repeating digit on the left side of the decimal point. To do so we need to move the decimal point 4 place to the right. We can do so by multiplying with 10000 i.e. 10000x =25678.7878…..(1)
Now we need to shift the repeating digits to the left of the decimal point in the original decimal number. To do so we need to multiply the original number by ‘100’.
100x = 256.7878…..(2)
Subtracting left-hand sides and right-hand sides equations we have
10000x-100x = 25678.7878…..-256.7878…..
9900x = 25422
x=25422/9900 = 4237/1650
Therefore, recurring decimal 2.567878….. converted to a rational number is 4237/1650