Math CentralQuandaries & Queries


Question from Grace:

What is non repeating decimal

Hi Grace,

A repeating decimal number is one that has a string of digits that repeats indefinitely. For example

\[89.3333\cdot\cdot\cdot \]

where the digit $3$ repeats indefinitely. The repeating pattern may contain more than 1 digit as in

\[587.67313131\cdot\cdot\cdot \]

where the pair of digits $31$ repeats or

\[0.543254325432\cdot\cdot\cdot \]

where the pattern $5432$ repeats.

A non-repeating decimal is one that is not a repeating decimal number. The examples that are usually given are $\sqrt{2}$ and $\pi,$ but it is reasonably easy to construct one yourself. For example

\[0.101001000100001\cdot\cdot\cdot \]

where the number of zeros between two ones continues to increase.

You might wonder why anyone cares if a decimal is repeating or not. The reason is that when you convert a common fraction $\large \frac{a}{b}$ where $a$ and $b$ are integers, to decimal form by dividing the denominator into the numerator, the decimal is repeating. Try it with for example $\large \frac{5}{7}.$

The converse is also true, any repeating decimal can be expressed as a fraction. For example see our response to an earlier question.

Hence a non-repeating decimal number cannot be expressed as a common fraction.

I hope this helps,

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