## ⓘ 1

1 is a number, 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 of unit length is a line segment of length 1. 1 is the smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0.

The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 returns that number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; although universal today, this was a matter of some controversy until the mid-20th century.

## 1. 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 en, 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.

## 2. As a number

One, sometimes referred to as unity, is the first non-zero natural number. It is thus the integer after zero.

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 and square 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.

## 3. 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 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 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.

### * 4.1. Mathematics * Definitions

Mathematically, 1 is:

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

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 n 1 = 1 n = 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. 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.

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

In number theory, 1 is the value of Legendres constant, which was introduced in 1808 by Adrien-Marie Legendre in expressing the asymptotic behavior of the prime-counting function. Legendres constant was originally conjectured to be approximately 1.08366, but was proven to equal exactly 1 in 1899.

### * 4.2. Mathematics * Properties

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.

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.

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 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 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 sequence 0 is 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 Benfords 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

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

This power series converges and has finite value if and only if, | x | < 1 {\displaystyle |x|