Recurring decimals and fractions

Why the ×10ⁿ-and-subtract trick turns any recurring decimal into a fraction, how to choose the right multiplier, and which fractions terminate (and why the denominator's primes decide it).

Number · Fractions & decimals

Recurring decimals and fractions

Why the ×10ⁿ-and-subtract trick turns any recurring decimal into a fraction, how to choose the right multiplier, and which fractions terminate (and why the denominator's primes decide it).

Why it works

A recurring decimal like

0.\dot{4} = 0.4444\ldots

never ends — so you can't "just write it over a power of ten". The trick: shift it so the tails line up, then subtract — the infinite tail cancels itself.

x = 0.4444\ldots, \qquad 10x = 4.4444\ldots

10x - x = 9x = 4 \qquad\Rightarrow\qquad x = \tfrac{4}{9}.

The two tails are identical — both are

.4444\ldots

forever — so the subtraction wipes them out exactly, leaving ordinary whole numbers. This is why

0.\dot{4}

is exactly

\frac{4}{9}

, not "about" it (and why it is NOT

0.4 = \frac{2}{5}

— the tail matters).

The multiplier matches the repeating block. Shift by one full period so the tails align:

one repeating digit → $\times 10$ : $0.\dot{4}$ as above;
two repeating digits → $\times 100$ :

x = 0.\dot{4}\dot{5}

100x = 45.\dot{4}\dot{5}

99x = 45

x = \frac{45}{99} = \frac{5}{11}

.

Using

10x

on a two-digit block misaligns the tails (

4.5454\ldots

0.4545\ldots

) and nothing cancels.

Non-repeating digits in front: subtract two shifted copies. For

x = 0.2\dot{7}

(only the 7 repeats): the tails align between

10x = 2.777\ldots

and

100x = 27.77\ldots

, so

100x - 10x = 90x = 25 \qquad\Rightarrow\qquad x = \tfrac{25}{90} = \tfrac{5}{18}.

Choose the two multipliers so BOTH copies end just before an identical tail.

Read the dots precisely. Dots mark the first and last digits of the repeating block:

0.\dot{4}\dot{5} = 0.454545\ldots

but

0.4\dot{5} = 0.45555\ldots

— different numbers, different multipliers.

Which fractions terminate? Decimals are built from tenths, and

10 = 2 \times 5

. A fraction in lowest terms terminates exactly when its denominator's primes are only 2s and 5s:

\frac{7}{40}

terminates (

40 = 2^3 \times 5

, in fact

0.175

), but

\frac{5}{18}

recurs (

18 = 2 \times 3^2

— the 3 can never divide a power of 10). No long division needed: factorise the denominator and read the verdict.