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Terminating and Repeating Decimals

Any rational number (that is, a fraction in lowest terms) can be written as either a terminating decimal or a repeating decimal . Just divide the numerator by the denominator . If you end up with a remainder of $0$ , then you have a terminating decimal. Otherwise, the remainders will begin to repeat after some point, and you have a repeating decimal.

The division is as follows:

$\begin{array}{l}\begin{array}{c}0.625\\ 8\overline{)5.000}\end{array}\hfill \\ \underset{\_}{48}\hfill \\ 20\hfill \\ \underset{\_}{16}\hfill \\ 40\hfill \\ \underset{\_}{40}\hfill \\ 0\hfill \end{array}$

So, $\frac{5}{8}=0.625$ . This is a terminating decimal.

The division is as follows:

$\begin{array}{l}\begin{array}{c}0.5833\\ 12\overline{)7.0000}\end{array}\hfill \\ \underset{\_}{60}\hfill \\ 100\hfill \\ \underset{\_}{96}\hfill \\ 40\hfill \\ \underset{\_}{36}\hfill \\ 40\hfill \\ \underset{\_}{36}\hfill \\ 4\hfill \end{array}$

So:

$\frac{7}{12}=0.58\bar{3}$

This is a repeating decimal.

The bar over the number, in this case $3$ , indicates the number or block of numbers that repeat unendingly.

See also Converting Repeating Decimals to Fractions .