Recurring Decimals as Rational Numbers

From the previous concept of rational numbers, we are clear about the meaning of rational number. A rational number is a number in \(\frac{p}{q}\) form where ‘p’ and q’ are the integers and ‘q’ is not equal to zero. Both ‘p’ and ‘q’ could be negative as well as positive. We have also seen as to how rational numbers can be converted to both terminating and non-terminating decimal numbers. Now, non- terminating decimal numbers can be further classified into two types which are recurring and non- recurring decimal numbers.

Recurring numbers: Recurring numbers are those numbers which keep on repeating the same value after decimal point. These numbers are also known as repeating decimals.

For example:

\(\frac{1}{3}\) = 0.333...... (3 repeats forever)

\(\frac{1}{7}\) = 0.142857142857....... (14285714 repeats forever)

\(\frac{77}{600}\)= 0.128333...... (3 repeats forever)

To show a repeating digits in a decimal number, often we put a dot or a line above the repeating digit as given below:                                                                                            

For Example:

\(\frac{1}{3}\) = 0.333..…  = 0.\(\dot{3}\) = 0.\(\overline{3}\)

Non- recurring numbers: Non- recurring numbers are those, which do not repeat their values after decimal point. They are also known as non- terminating and non- repeating decimal numbers.

For Example: 

√2 = 1.4142135623730950488016887242097…...

√3 = 1.7320508075688772935274463415059…...

π = 3.1415926535897932384626433832795….....

e = 2.7182818284590452353602874713527….....

In previous topic, we have already seen how to convert rational numbers into decimal fractions (may it be terminating or non- terminating decimal number). In this topic we’ll try to understand the steps involved in conversion of recurring (or repeating) decimal numbers into rational fractions. The steps involved are as follows:-

Step I: Let us assume ‘x’ to be the repeating decimal number we are trying to convert into rational number.

Step II: Carefully examine the repeating decimal to find the repeating digits.

Step III: Place the repeating digits to the left of decimal point.

Step IV: After step 3 place the repeating digits to the right of decimal point.

Step V: Now subtract left sides of the two equations. Then, subtract the right sides of the two equations. As we subtract, just make sure the differences of both the sides are positive. 

For having a better understanding let’s have look at some of the examples as shown below:

1. Convert 0.7777… into rational fraction.

Solution:

Step I: x = 0.7777

Step II: After examining we find that repeating digit is 7.

Step III: Place the repeating digit (7) to the left of decimal point. To do so, we need to move the decimal point 1 place to the right. This can also be done by multiplying the given no. by 10.

So, 10x = 7.777

Step IV: After step 3 place the repeating digits to the right of decimal point. In this case if we place the repeating digits to the right of decimal point it becomes the original number.

x = 0.7777

Step V: The two equations are-

 x = 0.7777, 

⟹ 10x = 7.777

Now we have to subtract the right and left hand sides-

10x - x = 7.777- 0.7777

⟹ 9x = 7.0

⟹ x = \(\frac{7}{9}\)

Hence, x= \(\frac{7}{9}\) is the required rational number.

2. Convert 4.567878….. into rational fraction.

Solution:

The conversion of the given decimal number into rational fraction can be carried out by using following conversion steps:

Step I: Let x = 4.567878…

Step II: After examining we find that the repeating digits are ‘78’.

Step III: Now we place the repeating digits ‘78’ to the left of decimal point. To do so we need to shift the decimal point to the right by 4 places. This can be done by multiplying the given number by’10,000’.

10,000x = 45678.787878

Step IV: 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 = 456.787878

Step V: Now the two equations become:

10,000x = 45678.787878, and

100x = 456.787878

Step VI: Now we have two subtract both the left and right hand sides of the two equations and equate them so that the equality remains the same.

10,000x - 100x = 45678.787878 - 456.787878

⟹ 9,900x = 45,222

⟹ x = \(\frac{45222}{9900}\)

This rational fraction can further reduced to

x = \(\frac{7537}{1650}\) (divide both numerator and denominator by 6)

So, the rational conversion of the given decimal number is \(\frac{7537}{1650}\).

All the conversion of this type can be carried out by using the above mentioned steps carefully.

Short-cut method of Conversion of recurring decimal to rational numbers

The method of conversion of recurring decimals in the form p/q is as follows.

Recurring decimal = 

\(\frac{\textrm{The whole number obtained by writing the digits in their order - The whole number made by the nonrecurring digits in order}}{10^{\textrm{The number of digits after the decimal point}} - 10^{\textrm{The number of digits after the decimal point that do not recur}}}\)

For example:

Express 15.0\(\dot{2}\) as a rational number.

Solution:

Here, the whole number obtained by writing the digits in their order = 1502,

The whole number made by the nonrecurring digits in order = 150

The number of digits after the decimal point = 2 (two)

The number of digits after the decimal point that do not recur = 1 (one).

Therefore, 

15.0\(\dot{2}\) = \(\frac{1502 - 150}{10^{2} - 10^{1}} = \frac{1352}{100 - 10} = \frac{1352}{90}\)

Rational Numbers

Rational Numbers

Decimal Representation of Rational Numbers

Rational Numbers in Terminating and Non-Terminating Decimals

Recurring Decimals as Rational Numbers

Laws of Algebra for Rational Numbers

Comparison between Two Rational Numbers

Rational Numbers Between Two Unequal Rational Numbers

Representation of Rational Numbers on Number Line

Problems on Rational numbers as Decimal Numbers

Problems Based On Recurring Decimals as Rational Numbers

Problems on Comparison Between Rational Numbers

Problems on Representation of Rational Numbers on Number Line

Worksheet on Comparison between Rational Numbers

Worksheet on Representation of Rational Numbers on the Number Line

9th Grade Math

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