1 ( one, also called unit, unity, and (multiplicative) identity) is a number, numeral, and glyph. It represents a single entity, the unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. It is also the first of the infinite sequence of , followed by 2.
It comes from the English word an, which comes from the ProtoGermanic root *ainaz. The ProtoGermanic root *ainaz comes from the ProtoIndoEuropean root *oino.
Compare the ProtoGermanic root *ainaz to Old Frisian an, Gothic language ains, Danish language en, Dutch language een, German language eins and Old Norse einn.
Compare the ProtoIndoEuropean root *oino (which means "one, single") to Greek language oinos (which means "ace" on dice), Latin unus (one), Old Persian aivam, Old Church Slavonic inu and ino, Lithuanian vienas, Old Irish oin and Breton language un (one).
Any number multiplied by one remains that number, as one is the Identity element for multiplication. As a result, 1 is its own factorial, its own square, its own cube, and so on. One is also the result of the empty product, as any number multiplied by one is itself. It is also the only natural number that is neither composite number nor prime number with respect to division, but instead considered a unit (meaning of ring theory).
While the shape of the 1 character has an ascender in most modern , in typefaces with text figures, the character usually is of xheight, as, for example, in .
Many older typewriters do not have a separate symbol for 1 and use the lowercase letter l instead. It is possible to find cases when the uppercase J is used, while it may be for decorative purposes.
Tally mark is often referred to as "base 1", since only one mark – the tally itself – is needed. This is more formally referred to as a unary numeral system. Unlike base 2 or base 10, this is not a positional notation.
Since the base 1 exponential function (1^{ x}) always equals 1, its inverse function does not exist (which would be called the logarithm base 1 if it did exist).
There are two ways to write the real number 1 as a recurring decimal: as 1.000..., and as 0.999....
Formalizations of the natural numbers have their own representations of 1:
In a multiplicative group or monoid, the identity element is sometimes denoted 1, but e (from the German Einheit, "unity") is also 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 (where this 0 is the additive identity of the ring). Important examples are .
1 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 problems, 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. Likewise, vector space are often normalized to give , that is vectors of magnitude one, because these often have more desirable properties. Functions, too, are often normalized by the condition that they have integral one, maximum value one, or square integral one, depending on the application.
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 number in the Fibonacci number sequence (0 is the zeroth) and is the first number in many other mathematical sequences.
1 is neither a prime number nor a composite number, but a unit (meaning of ring theory), like −1 and, in the Gaussian integers, imaginary unit and − i. The fundamental theorem of arithmetic guarantees factorization over the integers only up to units. (For example, , but if units are included, is also equal to, say, among infinitely many similar "factorizations".)
The definition of a field requires that 1 must not be equal to zero. 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.
1 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). 1 was formerly considered prime by some mathematicians, using the definition that a prime is divisible only by 1 and itself. However, this complicates the fundamental theorem of arithmetic, so modern definitions exclude units.
By definition, 1 is the magnitude, absolute value, or norm of a unit complex number, unit vector, and a identity 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.
1 is the most common leading digit in many sets of data, a consequence of Benford's law.
1 is the only known Tamagawa number for a simply connected algebraic group over a number field.
The generating function that has all coefficients 1 is given by
$\backslash frac\{1\}\{1x\}\; =\; 1+x+x^2+x^3+\; \backslash ldots$
This power series converges and has finite value if and only if, $x\; <\; 1$.
In category theory, 1 is sometimes used to denote the terminal object of a category.
In Number theory, 1 is the value of Legendre's constant, which was introduced in 1808 by AdrienMarie Legendre in expressing the asymptotic behavior of the primecounting function. Legendre's constant was originally conjectured to be approximately 1.08366, but was proven to equal exactly 1 in 1899.
1 × x  1  2  3  4  5  6  7  8  9  10 !  11  12  13  14  15  16  17  18  19  20 !  21  22  23  24  25 !  50  100  1000 
1 ÷ x  1  0.5  0.  0.25  0.2  0.1  0.  0.125  0.  0.1 !  0.  0.08  0.  0.0  0.0 
x ÷ 1  1  2  3  4  5  6  7  8  9  10 !  11  12  13  14  15 
1  1  1  1  1  1  1  1  1  1  1 !  1  1  1  1  1  1  1  1  1  1 
x  1  2  3  4  5  6  7  8  9  10 !  11  12  13  14  15  16  17  18  19  20 

