One, sometimes referred to as unity, is the integer before two and after zero. One is the first non-zero number in the natural numbers as well as the first odd number in the natural numbers.

Any number multiplied by one is that number, as one is the identity for multiplication. As a result, one is its own factorial, its own square, its own cube, and so on. One is also the empty product, as any number multiplied by one is itself.

As a digit[edit]

Evolution1glyph.svg

The glyph used today in the Western world to represent the number 1, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Indians, who wrote 1 as a horizontal line, much like the Chinese character 一. The Gupta wrote it as a curved line, and the Nagari sometimes added a small circle on the left (rotated a quarter turn to the right, this 9-look-alike became the present day numeral 1 in the Gujarati and Punjabi scripts). The Nepali also rotated it to the right but kept the circle small.[1] This eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral \mathrm{I}. In some European (e.g., Germany) and Asian (e.g., Israel) countries, the little serif at the top is sometimes extended into a long upstroke, sometimes as long as the vertical line, which can lead to confusion with the glyph for seven in other countries. Where the 1 is written with a long upstroke, the number 7 has a horizontal stroke through the vertical line.

While the shape of the 1 character has an ascender in most modern typefaces, in typefaces with text figures, the character usually is of x-height, as, for example, in TextFigs148.svg.

The 24-hour tower clock in Venice, using J as a symbol for 1.

Many older typewriters do not have a separate symbol for 1 and use the lowercase l instead. It is possible to find cases when the uppercase J is used, while it may be for decorative purposes.

Mathematics[edit]

Mathematically, 1 is: in arithmetic (algebra) and calculus, the natural number that follows 0 and precedes 2 and the multiplicative identity of the integers, real numbers and complex numbers; more generally, in abstract algebra, the multiplicative identity ("unity"), usually of a ring.

One cannot be used as the base of a positional numeral system; sometimes tallying is referred to as "base 1", since only one mark (the tally) is needed, but this is not a positional notation.

The logarithms base 1 are undefined, since the function 1x always equals 1 and so has no unique inverse.

In the real-number system, 1 can be represented in two ways as a recurring decimal: as 1.000... and as 0.999... (q.v.).

Formalizations of the natural numbers have their own representations of 1: in the Peano axioms, 1 is the successor of 0; in Principia Mathematica, 1 is defined as the set of all singletons (sets with one element); in the Von