One Big Happy

1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by  2, although by other definitions 1 is the second natural number, following  0.

The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is the smallest possible difference between two distinct natural numbers.

The unique mathematical properties of the number have led to its unique uses in other fields, ranging from science to sports. It commonly denotes the first, leading, or top thing in a group.

Etymology

The word ''one'' can be used as a noun, an adjective, and a pronoun.

It comes from the English word ''an'', which comes from the Proto-Germanic root . The Proto-Germanic root comes from the Proto-Indo-European root ''*oi-no-''.

Compare the Proto-Germanic root to Old Frisian ''an'', Gothic ''ains'', Danish ''en'', Dutch ''een'', German ''eins'' and Old Norse ''einn''.

Compare the Proto-Indo-European root ''*oi-no-'' (which means "one, single") to Greek ''oinos'' (which means "ace" on dice), Latin ''unus'' (one), Old Persian , Old Church Slavonic ''-inu'' and ''ino-'', Lithuanian ''vienas'', Old Irish ''oin'' and Breton ''un'' (one).

As a number

One, sometimes referred to as unity, is the first non-zero natural number. It is thus the integer after zero.

Any number multiplied by one remains that number, as one is the identity for multiplication. As a result, 1 is its own factorial, its own square and square root, its own cube and cube root, and so on. One is also the result of the empty product, as any number multiplied by one is itself. It is also the only natural number that is neither composite nor prime with respect to division, but is instead considered a unit (meaning of ring theory).

As a digit

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 Brahmic script of ancient India, where it was a simple vertical line. It was transmitted to Europe via the Maghreb and Andalusia during the Middle Ages, through scholarly works written in Arabic.

In some countries, the 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 used for seven in other countries. In styles in which the digit 1 is written with a long upstroke, the digit 7 is often written with a horizontal stroke through the vertical line, to disambiguate them. Styles that do not use the long upstroke on digit 1 usually do not use the horizontal stroke through the vertical of the digit 7 either.

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

Many older typewriters lack a separate key for ''1'', using the lowercase letter ''l'' or uppercase ''I'' instead. It is possible to find cases when the uppercase ''J'' is used, though it may be for decorative purposes. In some typefaces, different glyphs are used for I and 1, but the numeral 1 resembles a small caps version of I, with parallel serifs at top and bottom, with the capital I being full-height.

Mathematics

Definitions

Mathematically, 1 is:
*in arithmetic ( algebra) and calculus, the natural number that follows 0 and the multiplicative identity element of the integers, real numbers and complex numbers;
*more generally, in algebra, the multiplicative identity (also called ''unity''), usually of a group or a ring.

Formalizations of the natural numbers have their own representations of 1. In the Peano axioms, 1 is the successor of 0. In '' Principia Mathematica'', it is defined as the set of all singletons (sets with one element), and in the Von Neumann cardinal assignment of natural numbers, it is defined as the set .

In a multiplicative group or monoid, the identity element is sometimes denoted 1, but ''e'' (from the German ''Einheit'', "unity") is also traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. When such a ring has characteristic ''n'' not equal to 0, the element called 1 has the property that (where this 0 is the additive identity of the ring). Important examples are finite fields.

By definition, 1 is the magnitude, absolute value, or norm of a unit complex number, unit vector, and a unit matrix (more usually called an identity matrix). Note that the term ''unit matrix'' is sometimes used to mean something quite different.

By definition, 1 is the