How to simplify repeating decimal numbers

Finite decimal numbers vs. infinite decimal numbers

Up to now, we’ve been dealing with only finite decimal numbers (numbers with a finite number of decimal places). For example, $32.18476$ is a finite decimal number, because it ends at the $6$ .

In contrast, there are two kinds of decimal numbers that go on forever and ever. Some decimals that go on forever eventually get to a point where a certain digit (or sequence of digits) repeats infinitely, but some decimal number that go on forever don’t repeat.

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A decimal number where a digit or sequence of digits repeats infinitely is called a repeating decimal. An example is

$32.184766666666...$

The ... means that the $6$ repeats forever. We can rewrite a repeating decimal in compact form by writing the repeating digit/sequence just once and putting a bar over it. For example, we can write $32.184766666666...$ as

$32.1847\overline{6}$

An example of the other kind of infinite decimal is $\pi$ , whose decimal representation goes on forever but never repeats. Here are the first $46$ digits of $\pi$ .