Terminating and Non-Terminating Decimals

# Terminating and Non-Terminating Decimals

## Terminating Decimals and Non-Terminating Decimals

When we want to convert a fraction into a decimal form, we generally divide the numerator by the denominator till the remainder becomes zero. In some cases, we obtain zero as remainder but in other cases, the non-zero remainder is obtained.
Let us convert the following fractions into the decimal form.

1. 15/20                                                      2. 1/8
We observe that on division, remainder zero is obtained. Such fractions which gives zero as a remainder on division are called terminating decimals.
Now, let us take some more examples and convert them into decimals.

1. 10/9                                                              2. 13/6
Thus, we observe that on division, we get non-zero remainder infinitely. Such decimal are called non-terminating decimals.
If in a non-terminating decimal, a digit or a group of digits is repeating again and again, then the decimal is called the non-terminating repeating (recurring) decimal.

## Converting a Terminating Decimal to a Vulgar Fraction

To convert a terminating decimal into a vulgar fraction, we put 1 in the denominator and add as many zeroes as the number of digits after decimal.

Example: Convert the following decimals into fractions.
a. 0.7                                        b. 6.75

Solution:
a. 0.7 = 7/10                            b. 6.75 = 675/100

## Converting a Non-terminating Repeating Decimal to a Vulgar Fraction

To convert a non-terminating repeating (recurring) decimal into a vulgar fraction, look at the following example.

Example: Convert the following decimals into vulgar fractions.
a. 0.777777…..                                         b. 0.262626…...

Solution:
a. Let x = 0.777777…..        (1)
Multiply equation (1) by 10, we get
10x = 7.777777……            (2)
Subtracting equation (1) from (2), we get
9x = 7
x = 7/9

b. Let x = 0.262626…...         (1)
Multiply equation (1) by 100, we get
100x = 26.262626……           (2)
Subtracting equation (1) from (2), we get
99x = 26
x = 26/99

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