1 2 → Ordinal 1st(first) Factorization 1 | Cardinal one Divisors 1 | |

**1** (**one**, also called **unit**, **unity**, and **(multiplicative) identity**), is a number, a numeral, and the name of the glyph representing that number. 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 series of natural numbers, followed by 2.

## Contents

## Etymology

The word **one** can be used as a noun, an adjective and a pronoun.

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

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

Compare the Proto-Indo-European root ***oi-no-** (which means one, single) to Greek **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 **un** (one).

## As a number

One, sometimes referred to as **unity**, is the first non-zero natural number. It is thus the integer before two and after zero, and the first positive odd number.

Any number multiplied by one is that number, as one is the identity 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 nor prime with respect to division, but instead considered a unit.

## As a digit

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 Indians, who wrote 1 as a horizontal line, much like the Chinese character 一. The Gupta wrote it as a curved line, and the Nagari sometimes added a small circle on the left (rotated a quarter turn to the right, this 9-look-alike became the present day numeral 1 in the Gujarati and Punjabi scripts). The Nepali also rotated it to the right but kept the circle small. This eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral I. In some countries, the little 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 modern typefaces, in typefaces with text figures, the character usually is of x-height, 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.

## Mathematics

Mathematically, 1 is:

**multiplicative identity**(also called

*unity*), usually of a group or a ring.

1 cannot be used as the base of a positional numeral system, as the only digit that would be permitted in such a system would be 0. (Sometimes tallying is referred to as "base 1", since only one mark — the tally itself — is needed, but 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).

There are two ways to write the real number 1 as a recurring decimal: as 1.000..., and as 0.999.... There is only one way to represent the real number 1 as a Dedekind cut:

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 *n*1 = 1*n* = 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 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 number 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 1s in its lexicographic ordering. Sloane's later *Encyclopedia of Integer Sequences* and its Web counterpart, the *On-Line Encyclopedia of Integer Sequences*, ignore initial 1s in their lexicographic ordering of sequences, because such initial 1s often correspond to trivial cases.

1 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. (For example, 4 = 2^{2}, but if units are included, is also equal to, say, (−1)^{6} × 1^{23} × 2^{2}, 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.

1 is one of three possible values of the Möbius function: it takes the value 1 for square-free integers with an even number of distinct prime factors.

1 is the only odd number in the range of Euler's totient function φ(*x*), in the cases *x* = 1 and *x* = 2.

1 is the only 1-perfect number (see multiply perfect number).

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 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.

The ancient Egyptians represented all fractions (with the exception of 2/3) in terms of sums of fractions with numerator 1 and distinct denominators. For example, 2/5 = 1/3 + 1/15. Such representations are popularly known as Egyptian Fractions or Unit Fractions.

The generating function that has all coefficients 1 is given by

1/1 −*x*= 1 +

*x*+

*x*

^{2}+

*x*

^{3}+ …

This power series converges and has finite value if and only if, |*x*| < 1.

## In technology

## In science

## In astronomy

## In philosophy

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]).