Recurring Decimal

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Repeating Decimal

Recurring decimal numbers are actually those numbers that keep on repeating the same value after a decimal point. These numbers are also called as repeating decimals. For example:

1/3 = 0.33333..... (3 repeats forever)

1/7 = 0.142857142857142857....... (14285714 repeat forever)

77/600= 0.128333333...... (3 repeat forever)

To display a repeating digit in a decimal number, often we put a dot or a line over the repeating digit as shown below:

For Example:

\[1/3 = 0.33333. . . =  0.\dot{3} = 0.\bar{3}\]


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Non - Recurring Numbers

Non - recurring numbers are those in mathematics that do not repeat their values after a decimal point. They are also called as non- terminating decimals and non- repeating decimal numbers. For Example:

√2 = 1.41421356237309504…...

√7 = 2.647568759…...

π = 3.1415926535897932384626….....

e = 2.7182818284590452353602….....


Terminating and Recurring Decimals

When performing the conversion from fractions to decimals, you can formulate whether the decimal will terminate or recur.


Types of Decimal Numbers

Decimal numbers are classified into three types i.e.:

  1. Terminating Decimals: these decimals have a finite number of digits followed by the decimal point.

  2. Recurring Decimals: These consist of one or more repeating numbers or sequences of numbers followed by the decimal point, which keeps on infinitely.

  3. Irrational: These Decimals go on forever, are never ending and also never form a repeating pattern. These numbers are known as irrational numbers and cannot be written in the form of a fraction.

Converting Recurring Decimals to Fractions

Lets understand and perform an example to understand how we a recurring decimal to a fraction.

Let’s check the steps involved in converting repeating decimals to fractions recurring (rational). The steps involved are as given:-

Step I: Let ‘x’ be the repeating decimal number that we want to convert into a rational number.

Step II: observe the repeating decimal in order to identify the repeating digits.

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

Step IV: Place the repeating digits to the right of the decimal point.

Step V: Now deduct the left sides of the two equations. Then, do the subtraction in the right side of the two equations. As we subtract, just ensure the differences of both the sides will be positive.


Solved Examples

Example: Convert 0.7 (one recurring digit) into a fraction.

Solution: Follow the below steps to convert recurring decimals to fractions:

Let x= recurring decimal

Let n = the number of recurring digits

Multiply the recurring decimal by 10

Subtracting (1) from (3) in order to remove the recurring part

Solve for x, expressing the answer as the fraction in the simplest form

Now, to solve for the given example

x = 0.777777....

10x = 7.77777

10x –x = 7

9x = 7

X = 7/9

Therefore, x = 7/9 is the required rational number.


Example: Convert 1.256 (two recurring digit) into a fraction

Solution:

x = 0.125656....

10x = 125.6565....

100x – x = 125.6565... -1.256565

99x = 124.4

x = 124.4/99 = 1244/990 = 622/495


Example of Conversion Repeating Decimal to Fraction

Convert the numerical digit 4.567878….. into a rational fraction.

Solution:

Converting the given decimal number into rational fraction can be performed by undertaking the following conversion steps:

Step I: Let x = 4.56787878…

Step II: After analyzing the expression, we identified that the repeating digits are ‘78’.

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

10,000x = 45678.787878

Step IV: Now we would require moving the repeating digits to the left of the decimal point in the original decimal number. For this purpose, we will have to multiply the original number by ‘100’.

100x = 456.787878

Step V: here, the two equations become:

10,000x = 45678.787878, and

100x = 456.787878

Step VI: Now we need to subtract both the left and right hand sides of the two equations and equate them in such a manner that the equality remains the same.

10,000x - 100x = 45678.787878 - 456.787878

⟹ 9,900x = 45,222

⟹ x = 45222/9900

This rational fraction can further be reduced to

x = 75371650 (dividing both denominator and numerator by 6)

Thus, the rational conversion of the provided decimal number comes out to be 75371650.

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

FAQ (Frequently Asked Questions)

Q1. What is Meant by a Rational Number?

Answer: A rational number is a number in the form of p q where ‘p’ and q’ are the integers and ‘q’ is not equal to zero (0). Both ‘p’ and ‘q’ could be positive as well as negative. A rational number can also be converted to both terminating and non-terminating decimal numbers. Remember that, non- terminating decimal numbers can be further categorized into two types that are recurring decimal and non- recurring decimal numbers.

Q2. How to Simplify, Solve and Express 15.02˙ as a Rational Number?

Answer: Firstly, we would require writing down the whole number in an order. For example, the whole number we got by writing the digits in its order = 1502. The whole number formed by the non recurring digits in order = 150. The number of digits after the decimal point = 2. The number of digits after the decimal point that do not repeat = 1. Hence, 15.02˙ = 1502−150102 − 101=1352100 − 10 = 135290.

Q3. What is a Repeating Decimal?

Answer: A repeating decimal, also known as the recurring decimal, is a number whose decimal display eventually becomes periodic (i.e., the same sequence of digits repeats indefinitely, infinite number of times).