Recurring Decimals

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Transum, Wednesday, August 30, 2017 "Here's something to think about, discuss with your friends and share with your teacher: What is the difference between $0.\dot 9$ and one? " Ann Roberts, London Sunday, September 27, 2020 "Most fractions convert to recurring (repeating) decimals. Think of ⅓ converting to 0.333... There are some fractions that convert to terminating decimals. Think of ½ converting to 0.5. Which fractions convert to terminating decimals? I'll tell you. Write your fraction in its lowest terms n/d. Calculate the prime factorisation of the denominator d. Your fraction will only convert to a terminating decimal if: the prime factorisation of the denominator d contains ONLY 2s and/or 5s. In other words, the prime factorisation might include only 2s, or only 5s or a combinations of 2s and 5s. No other number is allowed. Let me show you. 1/20 converts to a terminating decimal because the prime factorisation of 20 is 2x2x5 (so only 2s and 5s). 1/9 converts to a recurring decimal because the prime factorisation of 9 is 3x3. (3s are not allowed). Use this rule to see if you can find some more fractions that convert to terminating decimals." Sudeep, Twitter Monday, May 6, 2024 " All doable without tedious calculations! pic.twitter.com/fpjikr0w5W — Sudeep (@boss_maths) August 11, 2023 " Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.

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A decimal with a repeating digit (or set of digits) is called a recurring decimal.

For example $0.77777777...$ is a recurring decimal and is called "nought point seven recurring"

$9.247347347...$ is also a recurring decimal and is called "nine point two four seven recurring"

The period of a recurring decimal is the number of digits in the repeating section so for the second example above the period is three.

A more efficient way of writing out a recurring decimal is by only writing the repeating digit once but putting a dot over the first and last number in the repeating sequence. Another method is drawing a line over the repeating digit or digits.Here are some examples.

$0.333333333... = 0.\dot 3 = 0.\overline 3$

$0.76531531531... = 0.76\dot 53\dot 1 = 0.76\overline{531}$

A fraction can be converted to a decimal using long division; dividing the numerator by the denominator. If the decimal is recurring the repeating pattern of numbers will be spotted in the long division working. The following example shows the repeating patterns when converting $ \frac{7}{11} $ to a decimal:

$Using long division to convert a fraction into a recurring decimal$

There are two common methods for converting a recurring decimal to a fraction:

Method 1

1 repeating digit

Let the recurring decimal be represented by $x$

$$x = 0.8888888...$$

Multiply both sides by 10 (as there is one repeating digit)

$$10x = 8.8888888...$$

Subtract the first equation from the second

$$9x = 8$$ $$x = \frac{8}{9}$$

2 repeating digits

Let the recurring decimal be represented by $x$

$$x = 1.36363636...$$

Multiply both sides by 100 (as there are two repeating digits)

$$100x = 136.36363636...$$

Subtract the first equation from the second

$$99x = 135$$ $$x = \frac{135}{99}$$ $$x = \frac{15}{11}$$

3 repeating digits

The method is the same but multiply both sides by 1000.

Method 2

1 repeating digit

Example: convert $0.8888888...$ to a fraction.

This method requires you to know that $\frac19 = 0.1111111...$

$0.8888888...$ is exactly eight times $0.1111111...$

$$\therefore 0.8888888... = \frac{8}{9}$$

2 repeating digits

Example: convert $0.45454545$ to a fraction in its lowest terms.

This method requires you to know that $\frac{1}{99} = 0.01010101...$

$0.45454545...$ is exactly forty five times $0.01010101...$

$$\therefore 0.45454545... = \frac{45}{99}$$ $$0.45454545... = \frac{5}{11}$$

3 repeating digits

Example: convert $0.\dot 61\dot 2$ to a fraction in its lowest terms.

This method requires you to know that $\frac{1}{999} = 0.\dot 00\dot 1$

$0.\dot 61\dot 2$ is exactly six hundred and twelve times $0.\dot 00\dot 1$

$$\therefore 0.\dot 61\dot 2 = \frac{612}{999}$$ $$0.\dot 61\dot 2 = \frac{68}{111}$$

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Recurring Decimals

Change recurring decimals into their corresponding fractions and vica versa.

Level 1 Level 2 Exam-Style Help More Decimals

Instructions

More Activities:

Featured Activity

Tower of Hanoi

Answers

Go Maths

Maths Map

Teachers

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Method 1

1 repeating digit

2 repeating digits

3 repeating digits

Method 2

1 repeating digit

2 repeating digits

3 repeating digits