Fractions unitaires

Unit fractions are the oldest way of writing a fraction, and for a long time they were the only way. Egyptian arithmetic had no numerator: a quantity was written as a sum of distinct reciprocals, marked by placing the ro sign — a mouth, read as "part" — above a numeral, with separate symbols reserved for 2/3 and 3/4. The Rhind Mathematical Papyrus, copied by the scribe Ahmose around 1550 BCE from an original perhaps three centuries older, opens with the table every Egyptian calculation needed: 2/n as a sum of unit fractions for each odd n from 5 to 101, starting 2/5 = 1/3 + 1/15. The Egyptian Mathematical Leather Roll, from roughly the same period, records 26 further decompositions. Doubling is the one operation the notation makes awkward, and those tables are the workaround.

The method outlived the civilisation that needed it. Fibonacci's Liber Abaci of 1202 sets out a greedy rule — subtract the largest unit fraction that still fits, then repeat on what is left — and James Joseph Sylvester proved in 1880 that it always terminates, so every positive fraction has such a form.

What drew later mathematicians was the sum. Nicole Oresme, working in Paris around 1350, showed that 1 + 1/2 + 1/3 + … exceeds any bound you care to name: group the terms after the first as 1/2, then 1/3 + 1/4, then the next four, then the next eight, and every group totals at least 1/2. The proof was lost and found again — Pietro Mengoli in 1650, Jacob and Johann Bernoulli around 1689 — and it is still the standard answer to the assumption that terms shrinking to nothing must add to something finite. Euler supplied the rate in the 1730s: the first n terms come to roughly ln n + γ, where γ = 0.5772156649… is the constant named for him and Mascheroni, and which nobody has yet proved irrational.

Propriétés principales

• 1/n is the reciprocal of n: the two multiply to 1. The terms decrease forever, approach zero and never arrive — there is no smallest unit fraction.
• 1/n has a terminating decimal exactly when n has no prime factor other than 2 or 5. Otherwise it repeats, with a period equal to the multiplicative order of 10 modulo whatever is left of n once its 2s and 5s are divided out.
• Consecutive unit fractions telescope: 1/n − 1/(n+1) = 1/(n(n+1)). Adding that identity up gives 1/2 + 1/6 + 1/12 + 1/20 + … = 1 exactly.
• Every unit fraction splits in two: 1/n = 1/(n+1) + 1/(n(n+1)). So 1/2 = 1/3 + 1/6 = 1/3 + 1/7 + 1/42 = …, and no fraction has only one Egyptian representation.
• The partial sums Hₙ = 1 + 1/2 + … + 1/n grow without limit but very slowly, like ln n + γ. Passing 10 takes 12,367 terms; passing 100 takes about 1.5 × 10⁴³.
• No harmonic number after the first is a whole number. H₁ = 1, and a result from 1915 shows every later Hₙ has a denominator the numerator cannot cancel.
• 1/9 = 0.(1), 1/99 = 0.(01) and 1/999 = 0.(001). The repetend of 1/7 is the cyclic number 142857: multiplying it by 2, 3, 4, 5 or 6 only rotates its digits.

Où on les rencontre

• Reciprocals are what add in physics and engineering. Resistors in parallel satisfy 1/R = 1/R₁ + 1/R₂ + …, the thin-lens equation is 1/f = 1/u + 1/v, and springs in series combine the same way — each one a sum of unit fractions in disguise.
• A string stopped at 1/2, 1/3 or 1/4 of its length sounds the octave, the twelfth and the double octave, and the nth harmonic has 1/n of the fundamental wavelength. The overtone series is this sequence, heard rather than written.
• Zipf’s law, set out by George Kingsley Zipf in the 1930s and 1940s, says the nth most common word in a long text appears with frequency roughly proportional to 1/n — which puts the harmonic numbers at the centre of natural-language statistics.
• The coupon collector problem: the expected number of random draws needed to complete a set of n is n·Hₙ, about 225 stickers for an album of 50 and about 1,176 for one of 200.
• Identical blocks stacked with offsets 1/2, 1/4, 1/6, … overhang the table edge by half the harmonic number, so with enough blocks the overhang can be made as large as you like without the stack toppling.
• Fraction walls, fraction strips and fraction number lines — the standard primary-school models for comparing sizes — are drawings of exactly this run, which is why it is worth being able to print one at any length.