1(one, also calledunit,unity, and(multiplicative) identity) is anumber, and a numerical digit used to represent that number in numerals. It represents a single entity, the unit of counting or measurement. For example, a line segment ofunit lengthis a line segment of length 1. It is also the first of the infinite sequence ofnatural numbers, followed by 2.

The wordonecan be used as a noun, an adjective and a pronoun.^[1]

It comes from the English wordan,^[1]which comes from the Proto-Germanic root *ainaz.^[1]The Proto-Germanic rootainazcomes from the Proto-Indo-European root **oi-no-.^[1]

Compare the Proto-Germanic root *ainazto Old Frisianan, Gothicains,Danishen,Dutcheen,GermaneinsandOld Norseeinn.

Compare the Proto-Indo-European root *oi-no-(which means “one, single”^[1]) toGreekoinos(which means “ace” on dice^[1]),Latinunus(one^[1]), Old Persianaivam, Old Church Slavonic-inuandino-, Lithuanianvienas, Old Irishoinand Bretonun(one^[1]).

One, sometimes referred to asunity,^[2]is the first non-zeronatural number. It is thus theintegerbefore two and after zero, and the first positive odd number.

Any number multiplied by one remains that number, as one is the identity for multiplication. As a result, 1 is its own factorial, its own square andsquare root, its own cube and cube root, 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 nor prime with respect to division, but instead considered a unit (meaning of ring theory).

The glyph used today in the Western world to represent the number 1, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Brahmic script of ancient India, where it was a simple vertical line. It was transmitted to Europe via Arabic during the Middle Ages.

In some countries, the serif at the top is sometimes extended into a long upstroke, sometimes as long as the vertical line, which can lead to confusion with the glyph for seven in other countries. Where the 1 is written with a long upstroke, the number 7 has a horizontal stroke through the vertical line.

While the shape of the 1 character has an ascender in most moderntypefaces, in typefaces with text figures, the character usually is of x-height, as, for example, in [[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/thumb/8/81/TextFigs148.svg/62px-TextFigs148.svg.png|//upload.wikimedia.org/wikipedia/commons/thumb/8/81/TextFigs148.svg/93px-TextFigs148.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/TextFigs148.svg/124px-TextFigs148.svg.png 2x|Horizontal guidelines with a one fitting within lines, a four extending below guideline, and an eight poking above guideline|h40|w62]].

Many older typewriters do not have a separate symbol for1and use the lowercase letterlinstead. It is possible to find cases when the uppercaseJis used, while it may be for decorative purposes.

Mathematically, 1 is:

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 (1x) always equals 1, its inverse does not exist (which would be called thelogarithmbase 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, bute(from the GermanEinheit, “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 characteristicnnot equal to 0, the element called 1 has the property thatn1 = 1n= 0 (where this 0 is the additive identity of the ring). Important examples are finite fields.

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 typicallynormalizedto fall within the unit interval from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters. Likewise,vectorsare often normalized to give unit vectors, that is vectors of magnitude one, because these often have more desirable properties. Functions, too, are often normalized by the condition that they haveintegralone, maximum value one, or square integral one, depending on the application.

Because of the multiplicative identity, iff(x) is a multiplicative function, thenf(1) must equal 1.

It is also the first and second number in the Fibonacci 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,iand −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”.)

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 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 unit matrix (more usually called an identity matrix). Note that the termunit matrixis 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

In category theory, 1 is sometimes used to denote the terminal object of a category.

Innumber 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.

In the philosophy of Plotinus and a number of other neoplatonists, The One is the ultimate reality and source of all existence. Philo of Alexandria (20 BC – AD 50) regarded the number one as God’s number, and the basis for all numbers (“De Allegoriis Legum,” ii.12 [i.66]).