1 (one) is a number, numeral, and the name of the glyph representing that number. It is the natural number following 0 and preceding 2. It represents a single entity. One is sometimes referred to as unity or unit as an adjective. For example, a line segment of "unit length" is a line segment of length 1.

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 positional numeral system.
x/1 = x (see division) x1 = x, 1x = 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. 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 real number system, 1 can be represented in two ways as a recurring decimal: as 1.000... and as 0.999... This identity is not immediately obvious to many people, and a full understanding of why it is true requires an understanding of the properties of the real numbers. See the article 0.999... for more details. 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 Principia Mathematica, 1 is defined as the set of all singletons.

or monoid, the identity element is sometimes denoted "1", but "e" (from the German Einheit, 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). One is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number to name just a few. Because of the multiplicative identity, if f(x) is a multiplicative function, then f(1) must equal 1. 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 the smallest positive odd integer. One is a harmonic divisor number. One is most often used for representing 'true' as a Boolean datatype in computer science. One is currently considered neither a prime number, nor a composite number — although it used to be considered prime. Defining a prime as a number that is only divisible by one and itself, one is a prime. 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 positive divisors, one and itself. The last professional mathematician to publicly label 1 a prime number was Henri Lebesgue in 1899. (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). By definition, 1 is the magnitude or absolute value of a unit vector and a unit matrix (more usually called an identity matrix). Note that the term unit matrix is usually used to mean something quite different. One is the value of the sine and cosine at π/2 and 0 radians, respectively. One is the most common leading digit in many sets of data, a consequence of Benford's law. Sequence of natural numbers always ends with the number 1 (Collatz conjecture). See also −1.