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Mathematicsfd
Mathematically, 1 is
in arithmetic (algebra) and calculus, the natural number that follows 0 and precedes 2 and the multiplicative identity of the integers, real numbers and complex numbers;
more generally, in abstract algebra, the multiplicative identity ("unity"), usually of a ring.
One cannot be used as the base of a positional numeral system; sometimes tallying is referred to as "base 1", since only one mark (the tally) is needed, but this is not a positional notation.
The logarithms base 1 are undefined, since the function 1x always equals 1 and so has no unique inverse.
In the real-number system, 1 can be represented in two ways as a recurring decimal: as 1.000... and as 0.999... (q.v.).
Formalizations of the natural numbers have their own representations of 1:
in the Peano axioms, 1 is the successor of 0;
in Principia Mathematica, 1 is defined as the set of all singletons (sets with one element);
in the Von Neumann cardinal assignment of natural numbers, 1 is defined as the set {0}.
In a multiplicative group or monoid, the identity element is sometimes denoted 1, especially in abelian groups, but e (from the German Einheit, "unity") is more traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. When such a ring has characteristic n not equal to 0, the element called 1 has the property that n1 = 1n = 0 (where this 0 is the additive identity of the ring). Important examples are general fields.
One is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few.
In many mathematical and engineering equations, numeric values are typically normalized to fall within the unit interval from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters.
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 (0 is the zeroth) and is the first number in many other mathematical sequences. As a matter of convention, Sloane's early Handbook of Integer Sequences added an initial 1 to any sequence that did not 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 neither a prime number nor a composite number, but a unit, like -1 and, in the Gaussian integers, i and -i. The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units (e.g., 4 = 22 = (-1)6×123×22).
The definition of a field requires that 1 must not be equal to 0. Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the field with one element, which is not a singleton and is not a set at all.
One is the only positive integer divisible by exactly one positive integer (whereas prime numbers are divisible by exactly two positive integers, composite numbers are divisible by more than two positive integers, and zero is divisible by all positive integers). One was formerly considered prime by some mathematicians, using the definition that a prime is divisible only by one and itself. However, this complicates the fundamental theorem of arithmetic, so modern definitions exclude units.
One is one of three possible values of the Möbius function: it takes the value one for square-free integers with an even number of distinct prime factors.
One is the only odd number 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 sometimes used to mean something quite different.
By definition, 1 is the probability of an event that is almost certain to occur.
One is the most common leading digit in many sets of data, a consequence of Benford's law.
