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4Recurring decimal numbers

If you want to write 13\frac13 as a decimal number, you can simply use the written method for division:

−1:3=0.3333…−10−19‾−1101−19‾−101011−19‾−10010111−19‾−100010−10000⋮\def\arraystretch{1.25} \begin{array}{l}\hphantom{-}1 :3 = 0.3333\dots\\\hphantom{-}10\\\underline{-\hphantom{1}9}\\\hphantom{-1}10\\\hphantom{1}\underline{-\hphantom{1}9}\\\hphantom{-10}10\\\hphantom{11}\underline{-\hphantom{1}9}\\\hphantom{-100}10\\\hphantom{111}\underline{-\hphantom{1}9}\\\hphantom{-1000}10\\\hphantom{-10000}\vdots\end{array}

There is a problem: You could divide again and again without reaching a final result, the division remainder will always be 1. Therefore, your decimals will be 3.

Recurring decimal numbers

Decimal numbers like 0.3333333…0.3333333\dots are called recurring decimal numbers, because the same numbers keep recurring infinitely often. There is another way of notating 0.3333333…0.3333333\dots You can just write 0.3‾0.\overline{3}. Another recurring decimal number is 1.12‾=1.121212…1.\overline{12}=1.121212\dots In this case, two numbers keep recurring.


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