One

''For the year AD 1, see 1.

One (1) is the natural number following zero and preceding two. It represents a single entity. One is sometimes referred to as unity. The Roman numeral for one is I.

For any number x:

x·1 = 1·x = x (see Multiplication);
x/1 = x (see Division);
x¹ = x and 1^x = 1 (see Exponentiation).

Using ordinary addition, we have 1+1=2; depending on the interpretation of the symbol "+" and the numeral system used, the expression can have many different values, listed at One_plus_one.

One cannot be used as the base of a positional numeral system in the ordinary way. Sometimes tallying is referred to as "base 1", since only one mark (the tally) is needed, but this doesn't work in the same way as other positional numeral systems. Related to this, one cannot take logarithms with base 1 (same as one cannot divide by zero, since log_n x is log_e x / log_e n, and log 1 = 0).

In the Von Neumann representation of natural numbers, 1 is defined as the set {0}. This set has cardinality 1 and hereditary rank 1. Sets like this with a single element are called singletonss.

In a multiplicative group or monoid, the identity element is sometimes called 1, but e (from the German Einigkeit, unity) is more traditional. However, 1 is especially common for the multiplicative identity of a ring.

One is its own factorial, the first triangular number, and its own square. It is also the first and second numbers in the Fibonacci sequence.

One is most often used for representing 'true' as a Boolean datatype in computer science (though any other number than zero would also represent true).

One is not always thought of as a number, although (unlike zero) it has been accepted as such since antiquity. Reflecting this, many languages retain a distinction between singular and plural forms of a noun, the former reserved for the case when only one object is being referred to.

Other meanings of one

One is also:

A song by U2
A song by Metallica
A song by Marvin Hamlisch from the musical A Chorus Line
(In all-caps) A trademark of Nestlé Purina PetCare for a line of pet food products (acronym for the "optimum nutritional effectiveness" they are claimed to possess)
The atomic number of hydrogen.
The denomination of U.S. dollar bill with George Washington's portrait, and the denomination of coin with Sacagawea's portrait.
In cents of a U.S. dollar, the denomination of coin with Abraham Lincoln's portrait.
Part of the designation of Interstate H-1, a freeway in Hawaii.
The designation of the Trans-Canada Highway in the four western provinces of Canada, as well as Prince Edward Island and Newfoundland and Labrador.
The designation of a U. S. Route 1, one of the earliest such designations, which extends from the Florida Keys to Maine and traverses some of the oldest roads in the U.S.
The designation of U.K. trunk road, A1 road, and U.K. motorway M1 motorway.
Air Force One is the callsign of aircraft SAM 28000 (as of 2003).
The code for international direct dial phone calls to countries participating in the North American Numbering Plan, such as the United States and Canada.
An enneagram personality type.
The DVD region of the United States and Canada.

Cultural meaning

Many human cultures have given the concept of one-ness symbolic meanings. Many religions consider God to be a perfect example of one-ness. See monad for a detailed discussion of other types of one-ness.

Something is unique if it is the only one of its kind. More loosely and exaggeratingly (especially in advertising) the term is used for something very special.

One is also an (archaic) expression of the first person singular ("one is not amused") and of the second person singular ("does one take sugar?)".

See also: zero, one, two, integer, list of numbers, unity.