Halving sequence

Halving is the oldest division there is: Egyptian arithmetic multiplied and divided by doubling and halving alone. But this particular run became famous as a paradox. Zeno of Elea, in the fifth century BCE, argued that motion is impossible — to cross a room you must first cross half of it, then half of what is left, then half of that, and since the halves never run out, you can never arrive. Aristotle recorded the argument in the Physics in order to reject it, and it kept its grip for two thousand years, because answering it properly means making sense of adding infinitely many things.

Archimedes had the mathematics long before the vocabulary existed. The Quadrature of the Parabola, written around 250 BCE, measures a parabolic segment by filling it with triangles, each generation a quarter of the area of the one before, and establishes that the total is 4/3 of the first triangle — the series 1 + 1/4 + 1/16 + … summed rigorously by a double contradiction argument rather than by a limit. Fourteenth-century scholars returned to such sums as questions about "proportional parts": Nicole Oresme summed several geometric series of this kind, in the same work in which he proved the harmonic series diverges.

The modern footing came with Augustin-Louis Cauchy, whose Cours d'analyse of 1821 defined the sum of an infinite series as the limit of its partial sums. That is the exact sense in which 1 + 1/2 + 1/4 + … equals 2: the partial sums are 2 − 1/2ⁿ, each one short of 2 by precisely the term just added, so no finite stage ever reaches the total while nothing smaller than 2 can bound them all. A popular story ties the hieroglyphic parts of the Eye of Horus to the fractions 1/2 down to 1/64, which total 63/64; the reading is a modern one and Egyptologists have long disputed it.

Key properties

• Each term is half the one before: the nth term is 1/2ⁿ⁻¹. Every term is positive, the limit is zero, and no term is ever zero.
• The first n terms add to 2 − 1/2ⁿ⁻¹ — short of 2 by exactly the last term added. The infinite sum is 2, the textbook example of a convergent series.
• Dividing by m each step instead gives a total of m/(m − 1): halves reach 2, thirds 3/2, quarters 4/3 — the value Archimedes needed — and tenths 10/9 = 1.(1).
• Each term equals the sum of everything after it: 1/2 = 1/4 + 1/8 + 1/16 + …. In binary that is the identity 0.1 = 0.0111…, two names for the same number.
• 1/2ⁿ has exactly n decimal places and its digits are the digits of 5ⁿ: 1/16 = 0.0625 and 5⁴ = 625. The last digit is always 5.
• In binary the terms are 0.1, 0.01, 0.001, … — the place values to the right of the point, which is why every binary fraction is a sum of them.
• These are the dyadic rationals, the fractions with a power of two underneath. They are exactly the values binary floating point holds without error, which is why 0.5 and 0.25 are exact in every language and 0.1 is not.

Where they turn up

• Radioactive decay: what is left of a sample after each successive half-life runs 1, 1/2, 1/4, 1/8. Ernest Rutherford measured the first of them in 1900 and called it the half-value period; carbon-14’s 5,730 years is what dates archaeological material.
• Music halves twice over. Note values run whole, half, quarter, eighth, sixteenth, and halving a string’s sounding length raises its pitch by exactly one octave — the same sequence organising rhythm and tuning at once.
• ISO 216 paper: A4 is half of A3 and twice A5, and because the sheets keep a √2 ratio the halving never changes their shape. Georg Christoph Lichtenberg described the system in a letter of 1786.
• Serial dilution in chemistry and biology steps down a concentration by a constant factor at a time, and first-order drug elimination follows the same curve, which is why dosing intervals are quoted in half-lives.
• Halving an interval is the fastest way to search a sorted list: binary search, merge sort and the knockout bracket all cut the field in two each round, which is where log₂ n comes from.
• Floating-point arithmetic is built from these fractions. IEEE 754 significands are sums of them, so 0.5, 0.25 and 0.125 are stored exactly while 0.1 is not — the reason 0.1 + 0.2 does not print as 0.3.