1 (number)

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 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 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 1-automorphic number in any place-based numbering system.

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).

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 singletonss.

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 On-Line 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 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 equal to the sum of its digits in any place-based numbering system, making it an all-Harshad number.

One is the number of n × n 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.

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