1

One comes from the English word an,[4] which comes from the Proto-Germanic root *ainaz.[4] The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-.[4]

Compare the Proto-Germanic root *ainaz to Old Frisian an, Gothic ains, Danish en, Dutch een, German eins and Old Norse einn.

Compare the Proto-Indo-European root *oi-no- (which means "one, single")[4] to Greek oinos (which means "ace" on dice),[4] Latin unus (one),[4] Old Persian aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin and Breton un (one).[4]

Mathematically, 1 is:

• in arithmetic (algebra) and calculus, the natural number that follows 0 and the multiplicative identity element of the integers, real numbers and complex numbers;
• more generally, in algebra, the multiplicative identity (also called unity), usually of a group or a ring.

Formalizations of the natural numbers have their own representations of 1. In the Peano axioms, 1 is the successor of 0. In Principia Mathematica, it is defined as the set of all singletons (sets with one element), and in the Von Neumann cardinal assignment of natural numbers, it is defined as the set {0}.

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 n1 = 1n = 0 (where this 0 is the additive identity of the ring). Important examples are finite fields.

By definition, 1 is the magnitude, absolute value, or norm of a unit complex number, unit vector, and a unit matrix (more usually called an identity matrix). The term unit matrix is sometimes used to mean a matrix composed entirely of 1s.

By definition, 1 is the probability of an event that is absolutely or almost certain to occur.

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 Adrien-Marie Legendre in expressing the asymptotic behavior of the prime-counting function. Legendre's constant was originally conjectured to be approximately 1.08366, but was proven to equal exactly 1 in 1899.

Tallying 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 does not exist (which would be called the logarithm base 1 if it did exist).

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, vectors are often normalized into unit vectors (i.e., 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 be equal to 1.

There are two ways to write the real number 1 as a recurring decimal: as 1.000..., and as 0.999.... 1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few.

1 is also the first and second number in the Fibonacci sequence (0 being the zeroth) and is the first number in many other mathematical sequences.

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.

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 equal to 1 is a geometric series, given by

${\displaystyle {\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\ldots }$

The zeroth metallic mean is 1, with the golden section equal to the continued fraction [1;1,1,...], and the infinitely nested square root ${\displaystyle \scriptstyle {\sqrt {1+{\sqrt {{\text{ }}1+\cdots {\text{ }}}}}}.}$

The series of unit fractions that most rapidly converge to 1 are the reciprocals of Sylvester's sequence, which generate the infinite Egyptian fraction

${\displaystyle 1={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{43}}+\cdots }$

1 is by convention neither a prime number nor a composite number, but a unit (meaning of ring theory) 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. For example, 4 = 2², but if units are included, is also equal to, say, (−1)⁶ × 1²³ × 2², among infinitely many similar "factorizations".

1 appears to meet the naïve definition of a prime number, being evenly divisible only by 1 and itself (also 1). As such, some mathematicians considered it a prime number as late as the middle of the 20th century, but mathematical consensus has generally and since then universally been to exclude it for a variety of reasons (such as complicating the fundamental theorem of arithmetic and other theorems related to prime numbers).

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.