Recurring decimals

Type 1 ÷ 7 into a calculator and the screen fills up. Does it ever stop? Long division can tell you, and it does not need a bigger screen.

A fraction is a division:

\frac{3}{8} = 3 \div 8

. Doing the division gives a decimal that does one of two things:

Long division shows why. Each step has a remainder, and the remainder is always less than the denominator. If a remainder is 0, the decimal stops. If not, a remainder must come back sooner or later, and from then on the same digits repeat.

The repeating block is marked with a bar over it, or dots over its first and last digits:

0.\overline{142857}

0.\dot{1}4285\dot{7}

;

0.1\overline{6}

0.1\dot{6}

means

0.1666\ldots

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

One third

Try dividing 1 by 3 on paper for a few steps. What remainder do you keep getting?

Does the decimal for 1/3 ever stop?No: the 3s go on foreverYes, after about 10 digitsYes: 0.33

TryTry it on the board and compare the result with your prediction.

Divide it out

Change the numerator a and the denominator b. Watch the remainder column.

Try it yourselfYour edits are saved in this browser.

Try 1/3, 1/8 and 1/7. Find a fraction whose repeating block is 6 digits long. Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Why it repeats

Here is the routine.

Worked exampleBefore each step, say what you think comes next.

$1 \div 6$ Bring down a 0: 10 ÷ 6 = 1 remainder 4.
$40 \div 6 = 6 \text{ r } 4$ Remainder 4 again.
$\tfrac{1}{6} = 0.1\overline{6}$ The 6 repeats forever; the 1 does not.
$\tfrac{1}{8} = 0.125$ A remainder of 0: it stops.

Read Zoe's working. Does 1/7's decimal repeat?

How should 0.272727… be written with a bar?A bar over 2 onlyA bar over 27: 0.27 with the bar over both digitsA bar over the last 7 only0.27 with no bar

Zoe decides whether 1/7 is a rational number (a fraction). Click the line where her reasoning goes wrong.

Divide 1 by 11 (on paper, or on the Run board above). How many digits are in its repeating block?

ModifyChange the problem on purpose and see what happens.

Stop, repeat, delay

Need a hint?

Try denominators 2, 4, 8.

Another hint

1/8 = 0.125.

Need a hint?

Try sixths and twelfths.

Try it yourselfYour edits are saved in this browser.

Find a fraction whose decimal stops after exactly 3 decimal places.

Use the board. This ticks when you get there.

Find a fraction with some digits before the repeating block starts.

Use the board. This ticks when you get there.

Divide 1 by 13. How long is its repeating block?

Use the board. This ticks when you get there.

1/3 = 0.333…, so 3 × 1/3 = 0.999… What does that tell you?1/3 is not really 0.333…0.999… is exactly 1Multiplying recurring decimals does not work0.999… is a little less than 1

PractiseA few on your own, to check you have it.