 One redirects here. For alternate uses, see One (disambiguation).
1 (one) is the natural number following 0 and preceding 2. It represents a single entity. One is sometimes referred to as unity, and unit is sometimes used as an adjective in this sense. (For example, a line segment of unit length is a line segment of length 1.)
Evolution of the glyph
The glyph used today in the Western world to represent the number 1, a vertical line, often with a little serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Brahmin Indians, who wrote 1 as a horizontal line (in Chinese today this is the way it is written). 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 9lookalike 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. 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 I.
History
Some Ancient Greeks did not consider one as a number: they considered it to be the unit, two being the first proper number as it represented a multiplicity.
In mathematics
For any number x:

x·1 = 1·x = x (This expresses the fact that 1 is the multiplicative identity.) As a consequence of this, 1 is a 1automorphic number in any placebased numbering system.

x/1 = x (see division)

x^{1} = x, 1^{x} = 1, and for nonzero x, x^{0} = 1 (see exponentiation)

x↑↑1 = x and 1↑↑x = 1 (see tetration).
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 meanings, 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, since the "exponential function" with base 1 is the constant function 1.
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 singletons.
In a multiplicative group or monoid, the identity element is sometimes denoted "1", but "e" (from the German Einigkeit, unity) is more traditional. However, "1" is especially common for the multiplicative identity of a ring. (Note that this multiplicative identity is also often called "unity".)
One is its own factorial, and its own square and cube (and so on, as 1 × 1 × ... × 1 = 1). As a consequence of its being its own square, one is also a Kaprekar number. One is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number to name just a few.
It is also the first and second numbers in the Fibonacci sequence, and is the first number in a lot of mathematical sequences. As a matter of convention, Sloane's early Handbook of Integer Sequences added an initial 1 to any sequence that didn't already have it, and considered these initial 1's in its lexicographic ordering. Sloane's later Encyclopedia of Integer Sequences and its Web counterpart, the OnLine Encyclopedia of Integer Sequences, ignore initial ones in their lexicographic ordering of sequences, because such initial ones often correspond to trivial cases.
One is the empty product.
One is a harmonic divisor number.
One is most often used for representing 'true' as a Boolean datatype in computer science.
One is currently not considered as a prime number, although it used to be, and it would be under a simple definition for primality: that the number be only divisible by one and itself  one is certainly itself. However, for purposes of factorization and especially the fundamental theorem of arithmetic, it is more convenient to not think of one as a prime factor, or to think of it as an implicit factor that's always there but need not be written down. To exclude the number one from the list of prime numbers, primality is defined as a number having exactly two distinct divisors, one and itself, itself being a number other than one. The last professional mathematician to publicly label 1 a prime number was Henri Lebesgue in 1899, although Carl Sagan included one in a list of prime numbers in his book Contact in 1985.
One is one of three possible return values of the Möbius function. Passed an integer that is squarefree with an even number of distinct prime factors, the Möbius function returns one.
One is the only odd number that is in the range of Euler's totient function φ(x), in the cases x = 1 and x = 2.
One is the only 1perfect number (see multiply perfect number).
One is equal to the sum of its digits in any placebased numbering system, making it an allHarshad number.
One is the number of n × n magic squares for n = 1, 3.
One is the number of nqueens problem solutions for n = 1.
One is a meandric number, a semimeandric number, and an open meandric number.
By definition, 1 is the magnitude or absolute value of a unit vector and a unit matrix.
One is the value of the sine and cosine at π/2 and 0 radians, respectively.
See also 1.
In science
One is:
 set equal to celerity (c), the speed of light, in Heaviside notation to simplify calculations.
 the factor in ratios for unit conversions.
 the total density ratio for a flat universe.
 The atomic number of hydrogen
In astronomy,

Messier object M1, a magnitude 7.0 supernova remnant in the constellation Taurus, also known as the Crab Nebula.

The New General Catalogue object NGC 1, a 13th magnitude 13 spiral galaxy in the constellation Pegasus

The Saros number of the solar eclipse series which began on 2872 June 4 and ended on 1592 July 11 . The duration of Saros series 1 was 1280.1 years, and it contained 73 solar eclipses.

The Saros number of the lunar eclipse series which began on 2588 March 2 and ended on 1272 April 30. The duration of Saros series 1 was 1316.2 years, and it contained 74 lunar eclipses.
In human society
Many human cultures have given the concept of oneness symbolic meanings. Many religions consider God to be a perfect example of oneness. See monad for a detailed discussion of other types of oneness.
One represents unity, togetherness, and absence of separation or discrimination, e.g. "We are all one."
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?)".
In Western culture, it is believed by many that the maximum number of girlfriends or boyfriends one may have at one time is 1.
In other fields
One is:
 the name of a train operating company in East Anglia, England. Anglia Railways
 "... the loneliest number that you'll ever do"—from the first line of "One" by Three Dog Night.
 the title of songs by Metallica, U2, Creed, Marvin Hamlisch (in the musical A Chorus Line), Alanis Morissette, Harry Nilsson, Three Dog Night, and the Bee Gees. Also the title of a bestselling compilation album of all Beatles songs that reached number 1 in the UK or US charts.
 the denomination of U.S. dollar bill with George Washington's portrait, and the denomination of coin with Sacagawea's portrait.
 the denomination of Canadian dollar coin with a swimming loon on the reverse, hence its universally employed nickname, the "loonie"
 in cents of a U.S. dollar, the denomination of coin with Abraham Lincoln's portrait.
 in cents of a Canadian dollar, the denomination of coin with two maple leaves on the reverse, commonly known as a penny.
 the designation of many roads, listed at Route 1.
 a subway service in New York City. See 1 (New York City Subway service).
 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.
 the universal highest position on most lists and charts. (see List of Number 1 Hits (USA))
 part of a nickname for the U. S. Army's First Infantry Division, "The Big Red One".
 in the phrase "number 1," a euphemism for urination (deriving from traditional U. S. elementary school practice of holding up one or two fingers to indicate the approximate time of a requested absence).
 as the capitalized word One, a euphemism for "God". (See also Oneness).
 as a word in all capitals, ONE, a trademark of Nestlé Purina PetCare for a line of pet food products (acronym for the "optimum nutritional effectiveness" they are claimed to possess).
 as ONE, the selfreferential acronym for ONE North East, the regional development agency in North East England.
 the house number of Number One Observatory Circle, the US VicePresident's residence.
 as Air Force One is the callsign of any United States Air Force aircraft carrying the President of the United States.
 an indication of a tonic chord in music (expressed as the Roman numeral I).
 in rugby union, the number of the loosehead prop .
 in baseball, the number representing the pitcher's position.
 in football, the number of the goalkeeper.
 in the DOS Shell and many Windows programs, the F1 key calling up online help.
 in Astrology, Aries is the 1st astrological sign of the Zodiac.
 the number of surat alFatiha in the Qur'an.
 the number of gods that exist according to monotheism (see also: tawhid)
 the number of syllables in a monosyllabic word.
 the number of humps on a dromedary.
 the number of wheels on a unicycle.
 the number of players in solitaire.
 the years: A.D. 1, immediately followed by 1 B.C., or the year 1 in a century.
 the address of Apsley House, known simply as Number 1, London.
Last updated: 05082005 05:07:15