Math Core

Every rational number can be written as a fraction, but calculators, price tags and measuring tools usually show decimals. In this lesson you'll use long division to turn any fraction into a decimal, and you'll discover that the decimal always does one of two things: it stops, or it repeats forever.

A fraction is a division

The fraction bar means divide: $\dfrac{3}{8} = 3 \div 8$ . To find the decimal, divide the numerator by the denominator with long division. Write $3$ as $3.000\ldots$ so you can keep bringing down zeros.

\begin{aligned} 30 \div 8 &= 3 \text{ remainder } 6 \\ 60 \div 8 &= 7 \text{ remainder } 4 \\ 40 \div 8 &= 5 \text{ remainder } 0 \end{aligned}

The digits of the answer are $3, 7, 5$ , so $\dfrac{3}{8} = 0.375$ . Once the remainder is $0$ , the division is finished.

Definition

Terminating and repeating decimals

A terminating decimal stops after a finite number of digits, like $0.375$ .

A repeating decimal has a digit or block of digits that repeats forever, like $0.272727\ldots$ We write it with a bar over the repeating block: $0.\overline{27}$ .

When the division never ends

Try $\dfrac{5}{11} = 5 \div 11$ :

\begin{aligned} 50 \div 11 &= 4 \text{ remainder } 6 \\ 60 \div 11 &= 5 \text{ remainder } 5 \\ 50 \div 11 &= 4 \text{ remainder } 6 \\ 60 \div 11 &= 5 \text{ remainder } 5 \end{aligned}

The remainder $5$ came back, which is where we started. From here the same steps happen again and again, so the digits $4, 5$ repeat forever: $\dfrac{5}{11} = 0.454545\ldots = 0.\overline{45}$ .

Why it must stop or repeat

When you divide by $11$ , every remainder is one of $0, 1, 2, \ldots, 10$ . If you ever get $0$ , the decimal terminates. If you never get $0$ , you only have $10$ possible remainders, so sooner or later one must come back. The moment a remainder repeats, the digits start repeating too.

Decimal expansions of rational numbers

The decimal form of every rational number either terminates or eventually repeats. You find it by dividing the numerator by the denominator.

Sometimes a few digits come before the repeating part. For $\dfrac{1}{6}$ : $10 \div 6 = 1$ remainder $4$ , then $40 \div 6 = 6$ remainder $4$ , and the remainder $4$ keeps coming back. So $\dfrac{1}{6} = 0.1666\ldots = 0.1\overline{6}$ . The bar goes over the $6$ only.

Negative numbers and mixed numbers

A negative sign just comes along for the ride: $-\dfrac{3}{8} = -0.375$ .

For a mixed number, keep the whole-number part and convert only the fraction: $2\tfrac{4}{9} = 2 + 0.\overline{4} = 2.\overline{4}$ .

Worked example: A terminating decimal

Write $\dfrac{7}{16}$ as a decimal.

Divide $7.0000$ by $16$ :

\begin{aligned} 70 \div 16 &= 4 \text{ remainder } 6 \\ 60 \div 16 &= 3 \text{ remainder } 12 \\ 120 \div 16 &= 7 \text{ remainder } 8 \\ 80 \div 16 &= 5 \text{ remainder } 0 \end{aligned}

So $\dfrac{7}{16} = 0.4375$ .

Worked example: A repeating decimal

Write $-\dfrac{2}{3}$ as a decimal.

First find $\dfrac{2}{3}$ . $20 \div 3 = 6$ remainder $2$ , and the remainder $2$ comes back every time. So $\dfrac{2}{3} = 0.666\ldots = 0.\overline{6}$ , and

-\frac{2}{3} = -0.\overline{6}

Worked example: A longer repeating block

Write $\dfrac{4}{7}$ as a decimal.

step	digit	remainder
$40 \div 7$	5	5
$50 \div 7$	7	1
$10 \div 7$	1	3
$30 \div 7$	4	2
$20 \div 7$	2	6
$60 \div 7$	8	4

The remainder $4$ matches the numerator we started with, so the block $571428$ repeats: $\dfrac{4}{7} = 0.\overline{571428}$ .

Common mistake

Put the bar over exactly the digits that repeat. $0.1\overline{6}$ means $0.1666\ldots$ , but $0.\overline{16}$ means $0.161616\ldots$ These are different numbers. And don't stop early: writing $\dfrac{2}{3} = 0.6$ or $0.67$ is only an approximation.

Tip

You can predict the type of decimal before dividing. Write the fraction in simplest form and look at the denominator. If its only prime factors are $2$ s and $5$ s (like $8$ , $20$ or $25$ ), the decimal terminates. If it has any other prime factor (like $3$ , $7$ or $11$ ), the decimal repeats.