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One

One redirects here. For alternate uses, see One (disambiguation).

0 1 2 3 4 5 6 7 8 9 >> List of numbers -- Integers 0 10 20 30 40 50 60 70 80 90 >>
Cardinal	1 one
Ordinal	1st first
Numeral system	unary
Factorization	<math> 1 <math>
Divisors	1
Roman numeral	I
Binary	1
Octal	1
Duodecimal	1
Hexadecimal	1

1 (one) is the natural number following 0 and preceding 2. It represents a single entity. One is sometimes referred to as unity. As "one", it is the smallest positive integer requiring one syllable in English. Prefixes for 1 are mono- (Greek) and uni- (Latin).

Evolution of the glyph

The glyph we use 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 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. This eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from simile with the Roman numeral 1.

In mathematics

For any number x:

x·1 = 1·x = x (This is called the multiplicative identity.)

x/1 = x (see Division)

x¹ = x, 1^x = 1, and for nonzero x, x⁰ = 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 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 singletons.

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, and its own square and cube. 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 1s in its lexicographic ordering. Sloane's later Encyclopedia of Integer Sequences and its Web counterpart, the On-Line Encyclopedia of Integer Sequences, ignore initial ones in their lexicographic ordering of sequences, because such initial ones often correspond to trivial cases, and thus it can be debatable precisely where a sequence should start.

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 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. Claiming to have done something "a number of times" may be true but misleading if you have only done it once.

One is currently not usually thought of as a prime number, but it is considered by some to be such, 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.

In mathematics, one is perhaps the number that most often appears as a literal constant in axiomatic equations, but in a way, it appears in all equations as an implicit multiplicand, due to multiplicative identity. As a consequence of the multiplicative identity, 1 is a 1-automorphic number in any place-based numbering system. Also, 1 is equal to the sum of its digits in any place-based numbering system, making it an all-Harshad number.

One is one of three possible return values of the Möbius function. Passed an integer that is square-free 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 1-perfect number (see multiply perfect number).

One is the number of nxn magic squares for n = 1,3.

One is the number of n-Queens Problem solutions for n = 1.

One is a meandric number, a semi-meandric number, and an open meandric number.

See also -1.

In human society

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.

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, the maximum number of spouses a person may have is 1.

In American culture, it is believed by many that the maximum number of girlfriends or boyfriends one may have at one time is 1.

In science

One is:

• The atomic number of hydrogen.
• The number of large natural satellites that Earth has.

In computer science

Like other negative numbers, computers usually represent negative one in two's complement internally. If a programmer is not careful, negative one held in a signed integer in two's complement could be mistaken for a number of the form 2^{sizeof(unsigned int)} - 1 if inadvertently cast to an unsigned integer.

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 U2, Metallica, Marvin Hamlisch (in the musical A Chorus Line), Alanis Morissette, Harry Nilsson, Three Dog Night, and the Bee Gees.
• 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.
• The designation of the following roads:
  • Interstate H-1, a freeway in Hawaii.
  • Trans-Canada Highway in the four western provinces of Canada, as well as Prince Edward Island and Newfoundland and Labrador.
  • U.S. Highway 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. For others, see Highway 1.
  • The U.K. trunk road, A1 road
  • The U.K. motorway M1 motorway.
  • The Pacific Coast Highway, a state road in California.
• 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).
• the name of:
  • As the capitalized word One, a euphemism for "God"
  • 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 self-referential acronym for ONE North East, the regional development agency in North East England
• The house number of Number One Observatory Circle, the US Vice-President's residence.
• 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 1 A.D., 1 B.C., or 1901