ch1.3.12-infinite-decimal-fractions

Chapter XII – Of Infinite Decimal Fractions

Summary: Shows how to convert any vulgar fraction to a decimal by long division, explains why some decimals terminate and others repeat indefinitely, and gives a systematic method to convert any repeating decimal back to a vulgar fraction.

Sources: chapter-1.3.12

Last updated: 2026-05-02

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Converting a vulgar fraction to a decimal

To convert $a/b$ to a decimal, perform long division of $a.0000\dots$ by $b$ (§526). Placing the decimal point correctly yields the decimal expansion.

Terminating decimals

Fractions whose denominators have only the prime factors 2 and 5 terminate:

$\frac{1}{2}=0.5,\frac{1}{4}=0.25,\frac{3}{4}=0.75,\frac{1}{8}=0.125,\frac{1}{5}=0.2,\frac{3}{8}=0.375$

Non-terminating repeating decimals

All other fractions produce infinite repeating decimals. The pattern must eventually repeat because at each step of the division the remainder belongs to a finite set of possibilities; once a remainder recurs, the digit pattern cycles.

Fraction	Decimal	Repeating block
$1/3$	$0.\overline{3}$	1 digit
$2/3$	$0.\overline{6}$	1 digit
$1/6$	$0.1\overline{6}$	1 digit
$1/7$	$0.\overline{142857}$	6 digits
$1/9$	$0.\overline{1}$	1 digit
$1/11$	$0.\overline{09}$	2 digits

For denominator 7 the period is exactly 6, because the possible non-zero remainders are $1,2,3,4,5,6$ — six values — so the pattern must recur within six steps (§533).

Converting a repeating decimal to a vulgar fraction

The key tool is the infinite geometric series formula (see ch1.3.11-geometrical-progressions, §520).

General rule

If $k$ figures repeat, multiply $s$ by $10^k$, subtract $s$, and solve (§537–538):

Repeating pattern	Equation	Result
Single digit $a$: $0.\overline{a}$	$10s-s=a$	$s=a/9$
Two digits $ab$: $0.\overline{ab}$	$100s-s=ab$	$s=ab/99$
Three digits $abc$: $0.\overline{abc}$	$1000s-s=abc$	$s=abc/999$

In general, a repeating block of $k$ digits gives denominator $\underset{k}{\underbrace{99\cdots 9}}$.

Examples

• $0.\overline{142857}$: $s=142857/999999=1/7$ (§530–531).
• $0.\overline{296296}$: $s=296/999=8/27$ (§538).
• $0.\overline{09}$ (denominator 11): $99s=9$, so $s=9/99=1/11$ (§536).

Algebraic proof for $1/7$

Setting $s=0.\overline{142857}$ and computing $1000000s-s=142857$ gives $999999s=142857$, so $s=142857/999999=1/7$ (§531).

Connection to geometric series

A repeating decimal is an infinite geometric progression. For example, $0.\overline{142857}$ has first term $142857/10^6$ and ratio $1/10^6$; the infinite-series sum formula gives $142857/(10^6-1)=1/7$ (§530). This is an application of the result in ch1.3.11-geometrical-progressions.

Extended worked example: $1/(10!)$

Euler computes $1/(1\cdot 2\cdot 3\cdots 10)$ to 14 decimal places by successive long divisions, dividing first by 2, then 3, then 4, …, then 10 (§539). The result is approximately $0.00000027557319$.

Related pages

• repeating-decimals
• decimal-fractions
• ch1.3.11-geometrical-progressions
• infinite-series
• fractions