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.

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.

**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 $5$ 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.

Write the decimal fraction as a simple fraction:

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

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?

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 $0.4545454545..........$

Notice that the digits $4$ and $5$ repeat continuously.

Test your knowledge

Question 1

Write the decimal fraction as a simple fraction:

\( 0.\overline{81}= \)

Question 2

Write the decimal fraction as a simple fraction:

\( 0.\overline{123}= \)

Question 3

Write the decimal fraction as a simple fraction:

\( 0.333= \)

Every fraction whose denominator we cannot change to the power of $10$ – $100$, $1000$, $10000$ etc. by amplification.

**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 $5$ (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?

Question 1

Write the decimal fraction as a simple fraction:

\( 0.\overline{67}= \)

Question 2

Write the decimal fraction as a simple fraction:

\( 0.\overline{352}= \)

Question 3

Write the decimal fraction as a simple fraction:

\( 0.1666= \)

Convert the fraction $2 \over 9$ to a repeating decimal

**Solution:**

We will proceed step by step:

First, we will write the exercise as a division problem.

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.

Notice what we have obtained - remainder $2$ that repeats successively

The result obtained is, without a doubt, a repeating decimal.

(Let's add ellipses after the last digit to represent continuity)

Convert the fraction $2 \over 11$ 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.

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

Notice what we obtained: a decimal whose digits $1$ and $8$ repeat successively in a periodic manner, that is, a repeating decimal number.

Check your understanding

Question 1

Write the decimal fraction as a simple fraction:

\( 10.\overline{67}= \)

Question 2

Question 3

Write the decimal fraction as a simple fraction:

\( 0.\overline{142857}= \)

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.

Write the decimal fraction as a simple fraction:

$0.\overline{5}=$

$\frac{5}{9}$

Write the decimal fraction as a simple fraction:

$0.\overline{67}=$

$\frac{67}{99}$

Write the decimal fraction as a simple fraction:

$0.333=$

$\frac{1}{3}$

Write the decimal fraction as a simple fraction:

$0.\overline{123}=$

$\frac{41}{333}$

Write the decimal fraction as a simple fraction:

$0.\overline{81}=$

$\frac{9}{11}$

Do you think you will be able to solve it?

Question 1

Write the decimal fraction as a simple fraction:

\( 0.\overline{63}= \)

Question 2

Question 3

Related Subjects

- The Order of Basic Operations: Addition, Subtraction, and Multiplication
- Order of Operations: Exponents
- Order of Operations: Roots
- Division and Fraction Bars (Vinculum)
- The Numbers 0 and 1 in Operations
- Neutral Element (Identiy Element)
- Order of Operations with Parentheses
- Order or Hierarchy of Operations with Fractions
- Opposite numbers
- Elimination of Parentheses in Real Numbers
- Addition and Subtraction of Real Numbers
- Multiplication and Division of Real Numbers
- Multiplicative Inverse
- Integer powering
- Positive and negative numbers and zero
- Real line or Numerical line
- Mixed Numbers and Fractions Greater Than 1
- Addition and Subtraction of Mixed Numbers
- Multiplication of Integers by a Fraction and a Mixed Number
- Multiplication and Division of Decimal Numbers by 10, 100, etc.
- Multiplication of Decimal Numbers
- Division of Decimal Numbers
- Decimal Measurements
- Density