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1 (one, also called unit, and unity) 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. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and 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.

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 . The Proto-Germanic root comes from the Proto-Indo-European root ''*oi-no-''.

Compare the Proto-Germanic root 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 , 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 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 is instead considered a unit (meaning of ring theory).

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. Whereas the digit 1 is written with a long upstroke, the digit 7 has a horizontal stroke through the vertical line.

While the shape of the character for the digit 1 has an ascender in most modern

typeface

s, in typefaces with

text figures

, the glyph 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

Definitions

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 number

s;
*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 .

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 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 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.

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 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.

It 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 1 is given by

$\frac = 1+x+x^2+x^3+ \ldots$

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

Primality

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, , but if units are included, is also equal to, say, 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.

Table of basic calculations

In technology

* The

resin identification code

used in recycling to identify polyethylene terephthalate.
*The International Telecommunication Union, ITU country code for the North American Numbering Plan area, which includes the United States, Canada, and parts of the Caribbean.
*A binary code is a sequence of 1 and

0

that is used in computers for representing any kind of data.
*In many physical devices, 1 represents the value for "on", which means that electricity is flowing.
*The numerical value of Boolean data type, true in many programming languages.
*1 is the ASCII code of "Start-of-Header, Start of Header".

In science

*Dimensionless quantities are also known as quantities of dimension one.
*1 is the atomic number of hydrogen.
*+1 is the electric charge of positrons and protons.
*Group 1 of the periodic table consists of the alkali metals.
*Period 1 of the periodic table consists of just two elements, hydrogen and helium.
*The dwarf planet Ceres (dwarf planet), Ceres has the minor-planet designation 1 Ceres because it was the first asteroid to be discovered.
*The Roman numeral I often stands for the first-discovered satellite of a planet or minor planet (such as Neptune I, a.k.a. Triton (moon), Triton). For some earlier discoveries, the Roman numerals originally reflected the increasing distance from the primary instead.

In philosophy

In the philosophy of Plotinus (and that of other neoplatonists), Plotinus#One, The One is the ultimate reality and source of all existence. Philo#Numbers, 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]).

In literature

*Number One is a character in the book series ''Lorien Legacies'' by Pittacus Lore.
*Number 1 is also a character in the series ''Artemis Fowl (series), Artemis Fowl'' by Eoin Colfer.

In music

*In a One (Harry Nilsson song), 1968 song by Harry Nilsson and recorded by Three Dog Night, the number one is identified as "the loneliest number".
*''We Are Number One'' is a 2014 song from the children's TV show ''LazyTown'', which gained popularity as a Internet meme, meme.
* 1 (Beatles album), ''1'' (Beatles album), a compilation album by the Beatles.
* One (U2 song), One, a 1991 song by Irish rock band U2.

In comics

*A character in the Italian comic book Alan Ford (comics), Alan Ford (authors Max Bunker and Magnus), very old disabled man, the supreme leader of the group TNT.
*A character in the Italian comic series PKNA and its sequels, an artificial intelligence as an ally of the protagonist Donald Duck in comics#Paperinik (Duck_Avenger), Paperinik.

In sports

*In baseball scoring, the number 1 is assigned to the pitcher.
*In association football (soccer) the number 1 is often given to the goalkeeper (association football), goalkeeper.
*In most competitions of rugby league (though not the Super League, which uses static squad numbering), the starting Fullback (rugby league), fullback wears jersey number 1.
*In rugby union, the starting Rugby union positions#Prop, loosehead prop wears the jersey number 1.
*1 is the lowest number permitted for use by players of the National Hockey League (NHL); the league prohibited the use of 00 and 0 in the late 1990s (the highest number permitted being 98 (number), 98).
*1 is the lowest Uniform number (American football), number permitted for use at most levels of American football. Under National Football League policy, it can only be used by a quarterback or kicking specialist, kicking player (during NFL preseason, preseason play, restrictions are looser, and players of other positions can wear the number and can also, if no other options exist, wear 0).
*In Formula One, the previous year's world champion is allowed to use the number 1.

In film

*''One A.M. (1916 film), One A.M.'' (1916), starring Charlie Chaplin.
*''One More Time (1970 film), One More Time'' (1970), directed by Jerry Lewis and starring Sammy Davis Jr. and Peter Lawford.
*''One Day (2011 film), One Day'' (2011), starring Anne Hathaway and Jim Sturgess.

In other fields

*''Number One'' is Royal Navy informal usage for the chief executive officer of a ship, the captain's deputy responsible for discipline and all normal operation of a ship and its crew.
*1 is the value of an ace in many playing card games, such as cribbage.
*List of highways numbered 1
*List of public transport routes numbered 1
*1 is often used to denote the Gregorian calendar month of January.
*1 CE, the first year of the Common Era
*01, the former dialing code for Greater London
*PRS One, a German paraglider design
*+1 is the code for international telephone calls to countries in the North American Numbering Plan.

See also

*−1
*+1 (disambiguation)
*List of mathematical constants
*One (word)
*Root of unity
*List of highways numbered 1

References

External links

The Number 1The Positive Integer 1

{{DEFAULTSORT:1 (Number)
1 (number),
Integers