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Terminating and recurring decimals

There are three types of decimal numbers:

  • Terminating decimals: these have a finite number of digits after the decimal point.
  • Recurring decimals: these have one or more repeating numbers or sequences of numbers after the decimal point, which continue infinitely.
  • Decimals which go on for ever, never ending and never forming a repeating pattern. These numbers are called irrational numbers and cannot be expressed as a fraction. We cover these decimals in the number course.

In this section we explore terminating and repeating decimals.

Example: terminating decimal

The fraction \(\frac{3}{5}\) when expressed as a decimal equals 0.6 and the fraction \(\frac{4}{25}\) when expressed as decimal is 0.16.

The fractions \(\frac{3}{5}\) and \(\frac{4}{25}\) can be expressed as terminating decimals. These are numbers with a finite number of decimal places.

Example: recurring decimal

The fraction \(\frac{2}{3}\) when expressed as a decimal produces a decimal which never ends: a recurring decimal. \(\frac{2}{3}\) when expressed as a decimal is 0.666666666666… \(\frac{1}{6}\) when expressed as a decimal is 0.166666666666…

Writing recurring decimals

We write recurring decimal using a dot over the recurring digit like this.

Two thirds expressed as 0.6 recurring (dot over the six). One sixth expressed as 0.16 recurring (dot over the six).

Sometimes not just one digit recurs in the decimal part. In this example of four elevenths, the 3 and the 6 recur. We place a dot over the three and the six to signify that both digits recur.

Four elevenths expressed as 0.36 recurring (dots over three and six)

A whole sequence of digits can recur. In these cases we do not place a dot over each of the recurring digits, just over the first and the last digits in the recurring sequence of digits.

Three sevenths expressed as 0.428571 recurring (dots over four and one, the first and last recurring digits)

Practice: convert and sort

We’ll be using Padlet for this task, so that you can sort your answers into two groups.

  1. Choose a fraction at random.
  2. Convert it into a decimal.
  3. Add your fraction and its decimal equivalent to the Decimal Sort Padlet for this task in the right group: terminating decimal or recurring decimal. Repeat this several times more choosing other fractions at random. This may help you answer the question in the next step.

If you are unsure, add it to the unknown column. Posts can be moved between columns if you think they are in the wrong place. Feel free to post images using your mobile phone, if you prefer.

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This article is from the free online course:

Maths Subject Knowledge: Fractions, Decimals, and Percentages

National STEM Learning Centre