Repeating Decimal

🏆Practice periodic decimal fraction

What is a repeating decimal?
A repeating decimal is a number with a fractional part that, after the decimal point, the digits repeat infinitely, in a periodic manner.

Conversion from Fraction to Repeating Decimal

First step: We will write the fraction as a long division exercise
Second step: We will add the decimal point to the dividend and then 55 zeros (the value of the number is not affected)
Third step: We will solve the division and copy the decimal point to the result exactly in the same place it was.
Fourth step: we will put ellipses at the result to signal that the number continues.

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Test yourself on periodic decimal fraction!

einstein

Write the decimal fraction as a simple fraction:

\( 0.\overline{5}= \)

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Repeating Decimal

In this article, we will introduce you to the repeating decimal and even teach you how to get there without needing to use a calculator. Are we ready?


What is a repeating decimal?

To understand what a repeating decimal looks like, we must know the meaning of the word "repeating".
Something repeating is something that repeats and reappears continuously, periodically.
Even a TikTok video that repeats over and over again does so periodically.
Generally, repeating decimals are also infinite, they do not end and their numbers repeat over and over again...
Example of a repeating decimal 0.4545454545..........0.4545454545..........
Notice that the digits 44 and 55 repeat continuously.


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Where will we see repeating decimals?

Every fraction whose denominator we cannot change to the power of 1010100100, 10001000, 1000010000 etc. by amplification.


How do you convert a fraction to a repeating decimal?

With a few simple steps

First step: We will write the fraction as we write a division horizontally.

Second step: We will add the decimal point to the dividend and then several zeros, about 55 (the value of the number will change)
Third step: We will solve the division and copy the decimal point to the result exactly in the same place it was.
Fourth step: We will write ellipses at the result to signal that the number continues.


Do you know what the answer is?

Let's see it in practice

Convert the fraction 292 \over 9 to a repeating decimal

Solution:
We will proceed step by step:
First, we will write the exercise as a division problem.

1a - Convert the fraction

In the second step, we will add a decimal point and some zeros to the dividend.

2a - In the second step, we will add a decimal point and some zeros to the dividend

In the third step, we will solve the exercise according to the division rules throughout and remember to copy the decimal point to the result.

3a -In the third step, we will solve the exercise according to the division rules throughout

Notice what we have obtained - remainder 22 that repeats successively
The result obtained is, without a doubt, a repeating decimal.
(Let's add ellipses after the last digit to represent continuity)

4a - The result obtained is, without a doubt, a repeating decimal


Another exercise

Convert the fraction 2112 \over 11 to a repeating decimal

Solution:
First, let's write the exercise as a long division and do not forget to add the decimal point and then the zeros to the dividend.

5a - a long division and do not forget to add the decimal point

Great. Now let's solve the exercise according to the division rules and do not overlook copying the decimal point to the result.

6a - a decimal whose digits 1 and 8 repeat successively in a periodic manner

Notice what we obtained: a decimal whose digits 11 and 88 repeat successively in a periodic manner, that is, a repeating decimal number.


Check your understanding

Important note

It may be that in the exam they do not mention that the fraction is, in fact,
a repeating decimal and that they simply ask you to convert it to a decimal number.
Also in this case the way to solve it and the solution will be the same.


Examples and exercises with solutions of repeating decimal numbers

Exercise #1

Write the decimal fraction as a simple fraction:

0.5= 0.\overline{5}=

Video Solution

Answer

59 \frac{5}{9}

Exercise #2

Write the decimal fraction as a simple fraction:

0.333= 0.333=

Video Solution

Answer

13 \frac{1}{3}

Exercise #3

Write the decimal fraction as a simple fraction:

0.123= 0.\overline{123}=

Video Solution

Answer

41333 \frac{41}{333}

Exercise #4

Write the decimal fraction as a simple fraction:

0.67= 0.\overline{67}=

Video Solution

Answer

6799 \frac{67}{99}

Exercise #5

Write the decimal fraction as a simple fraction:

0.81= 0.\overline{81}=

Video Solution

Answer

911 \frac{9}{11}

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