Fractions and decimals

Fractions are older than place value. The Rhind Mathematical Papyrus, copied around 1550 BCE by the scribe Ahmose from a text perhaps two centuries older, works almost entirely in unit fractions — sums of reciprocals such as 1/2 + 1/7 + 1/14 — and opens with a table expressing 2/n in that form for odd n up to 101. Babylonian scribes had something closer to our decimals much earlier: their base-sixty place value system extended to the right of the units, which is why we still cut hours and degrees into sixtieths and sixtieths of sixtieths.

Decimal fractions took far longer to settle. Al-Uqlidisi, writing in Damascus around 952, used them in his arithmetic of the Hindu numerals, though historians disagree about how deliberately; al-Kashi applied them systematically in The Key to Arithmetic (1427) while working at Ulugh Beg's observatory in Samarkand. In Europe the decisive popularizer was Simon Stevin, whose short pamphlet De Thiende ("The Tenth", Leiden, 1585) argued that decimal fractions should replace common fractions everywhere, and that weights, measures and coinage should be decimalized too — a proposal that waited for the French Revolution. Stevin's own notation was cumbersome, circling the position of each digit. The modern point arrived with John Napier's Rabdologiae (1617); an earlier printed use by Bartholomaeus Pitiscus in 1608 is sometimes claimed, but that attribution is disputed.

Repeating decimals got their theory from number theory rather than from commerce. Gauss treated the conversion of ordinary fractions into decimals in Section VI of the Disquisitiones Arithmeticae (1801), tying the length of the repeating block to the order of 10 modulo the denominator — the result that explains why 1/7 needs six repeating digits while 1/11 needs only two. The notation never standardized: British schools mark the repetend with dots over its first and last digits, American texts draw an overline, and continental European ones often use brackets, which is why this page lets you pick.

Key properties

• A fraction in lowest terms has a terminating decimal expansion in base ten exactly when its denominator has no prime factors other than 2 and 5.
• Otherwise the expansion repeats forever, and the length of the repeating block is the multiplicative order of 10 modulo the part of the denominator coprime to 10.
• If the reduced denominator is 2^a · 5^b · d with d coprime to 10, the expansion settles into its repeating block after max(a, b) digits.
• Every repeating decimal is rational: 0.(abc) = abc/999, and that identity — multiply by a power of ten, subtract, divide — is the method this page uses.
• The repetend of 1/n is at most n − 1 digits long. Primes achieving that maximum are called full reptend primes, and begin 7, 17, 19, 23, 29, 47.
• 1/7 = 0.(142857), and multiplying 142857 by 2, 3, 4, 5 or 6 gives a cyclic rotation of the same six digits.
• Every terminating decimal has a second exact representation ending in repeating nines: 0.2 = 0.1(9), and 1 = 0.(9).
• Reducing by the greatest common divisor and keeping the denominator positive gives each rational number exactly one canonical form.

Where they turn up

• Machine shops and US customary measurement work in binary fractions — 1/16, 1/32, 1/64 of an inch — because repeated halving stays exact in both fractions and decimals (1/64 = 0.015625).
• US stock markets quoted prices in eighths and sixteenths of a dollar until decimalization was completed in 2001, a change that shrank the minimum tick from 6.25 cents to one cent.
• Binary floating point cannot represent 1/10 exactly, which is why 0.1 + 0.2 does not equal 0.3 in most programming languages and why financial code stores integer cents or uses a decimal type.
• Music notation is a fraction system: a time signature is a fraction of a whole note, and tuplets are the places where the arithmetic stops dividing evenly.
• Gear and pulley ratios are quoted as fractions because the exact integer tooth counts matter — a 7:1 reduction behaves differently from the decimal 0.142857 it rounds to.