One is the multiplicative identity, not a prime
The number one is the smallest positive integer and the multiplicative identity: anything times one is itself. By modern convention it counts as neither prime nor composite. Every natural number after zero is built by adding one to its predecessor, and in real-world data sets, one leads roughly 30% of all figures.
One passes the naive test for a prime, dividing evenly only by one and itself, yet mathematicians now exclude it. In algebra it stands for the identity element of any unital ring or field, and any element with a multiplicative inverse is called a unit in its honour. Every power of one is one. Giuseppe Peano's original axioms started the natural numbers at one, though he later revised them to start at zero. Tally marks are a base-1 system: simple, but hard to read in bulk.
Engineers and scientists scale values into the interval from zero to one, so one is the probability of a certain event and the length of a unit vector. Benford's law explains why real-world figures that grow exponentially begin with one about 30% of the time. Adrien-Marie Legendre's constant for counting primes, introduced in 1808, turned out to equal one. Physicists set constants such as the speed of light to one in Planck units, and quantum wavefunctions must be normalised so their squared magnitude integrates to one.
The English word descends from Old English an, the Germanic root ainaz and a Proto-Indo-European root meaning one or unique. It hides inside alone, none, once, atone and lonely, while Latin uni- and sol- and Greek mono- give unicycle, solo and monorail. Sumerian tablets from the early third millennium BCE used the same symbol for one and sixty, a habit Babylonian cuneiform kept.
The vertical stroke used in the West goes back to the Brahmi script in Ashoka's edicts around 250 BCE and reached Europe through the Maghreb and Al-Andalus. German writers stretch the top serif into a long upstroke, so they cross their sevens to keep the two apart. In computing, one means a switch that is on, and true equals one in many programming languages. The Pythagoreans did not call one a number at all, but the source of all numbers.
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