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1 (one, also called unit, and unity) is a
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

number
and a
numerical digit A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent number A number is a mathematical object A mathematical object is an abstract concept arising in mat ...
used to represent that number in
numeral A numeral is a figure, symbol, or group of figures or symbols denoting a number. It may refer to: * Numeral system used in mathematics * Numeral (linguistics), a part of speech denoting numbers (e.g. ''one'' and ''first'' in English) * Numerical di ...
s. It represents a single entity, the
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
of
counting Counting is the process of determining the number of Element (mathematics), elements of a finite set of objects, i.e., determining the size (mathematics), size of a set. The traditional way of counting consists of continually increasing a (mental ...
or
measurement Measurement is the quantification (science), quantification of variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measurement are dependen ...

measurement
. For example, a
line segment In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

line segment
of ''unit length'' is a line segment of
length Length is a measure of distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...

length
 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest
positive integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. It is also sometimes considered the first of the
infinite sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s, followed by 
2
2
, although by other definitions 1 is the second natural number, following 
0
0
. The fundamental mathematical property of 1 is to be a
multiplicative identity In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...
, 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 A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
; 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 Old Frisian was a West Germanic language spoken between the 8th and 16th centuries in the area between the Rhine ), Surselva Surselva Region is one of the eleven administrative districts Administrative division, administrative unitArticl ...
''an'',
Gothic Gothic or Gothics may refer to: People and languages *Goths or Gothic people, the ethnonym of a group of East Germanic tribes **Gothic language, an extinct East Germanic language spoken by the Goths **Crimean Gothic, the Gothic language spoken by ...
''ains'',
Danish Danish may refer to: * Something of, from, or related to the country of Denmark * A national or citizen of Denmark, also called a "Dane", see Demographics of Denmark * Danish people or Danes, people with a Danish ancestral or ethnic identity * Danis ...
''en'',
Dutch Dutch commonly refers to: * Something of, from, or related to the Netherlands * Dutch people () * Dutch language () *Dutch language , spoken in Belgium (also referred as ''flemish'') Dutch may also refer to:" Castle * Dutch Castle Places * ...
''een'',
German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of Germany, see also German nationality law * German language The German la ...

German
''eins'' and
Old Norse Old Norse, Old Nordic, or Old Scandinavian is a stage of development of North Germanic dialects before their final divergence into separate Nordic languages. Old Norse was spoken by inhabitants of Scandinavia Scandinavia; : ''Skades ...
''einn''. Compare the Proto-Indo-European root ''*oi-no-'' (which means "one, single") to
Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ...
''oinos'' (which means "ace" on dice),
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became ...

Latin
''unus'' (one),
Old Persian Old Persian is one of the two directly attested Old Iranian languages The Iranian or Iranic languages are a branch of the Indo-Iranian languagesIndo-Iranian may refer to: * Indo-Iranian languages * Indo-Iranians, the various peoples speaking ...
,
Old Church Slavonic Old Church Slavonic or Old Slavonic () was the first Slavic literary language A literary language is the form of a language A language is a structured system of communication used by humans, including speech (spoken language), gestures (S ...
''-inu'' and ''ino-'',
Lithuanian Lithuanian may refer to: * Lithuanians Lithuanians ( lt, lietuviai, singular ''lietuvis/lietuvė'') are a Balts, Baltic ethnic group. They are native to Lithuania, where they number around 2,561,300 people. Another million or more make up the Lith ...
''vienas'',
Old Irish Old Irish (''Goídelc''; ga, Sean-Ghaeilge; gd, Seann Ghàidhlig; gv, Shenn Yernish or ; Old Irish: ᚌᚑᚔᚇᚓᚂᚉ), sometimes called Old Gaelic, is the oldest form of the Goidelic The Goidelic or Gaelic languages ( ga, teangacha ...
''oin'' and
Breton Breton most often refers to: *anything associated with Brittany Brittany (; french: link=no, Bretagne ; br, Breizh, or ; Gallo language, Gallo: ''Bertaèyn'' ) is a peninsula and cultural region in the west of France, covering the western part ...
''un'' (one).


As a number

One, sometimes referred to as unity, is the first non-zero
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
. It is thus the
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
after
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

zero
. Any number multiplied by one remains that number, as one is the
identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
for
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
. As a result, 1 is its own
factorial In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, its own
square In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...
and
square root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

square root
, its own
cube In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
and
cube root In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

cube root
, and so on. One is also the result of the
empty product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, as any number multiplied by one is itself. It is also the only natural number that is neither
composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials * ...
nor
prime A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
with respect to
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
, but is instead considered a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
(meaning of
ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
).


As a digit

The glyph used today in the Western world to represent the number 1, a vertical line, often with a
serif In typography, a serif () is a small line or stroke regularly attached to the end of a larger stroke in a letter or symbol within a particular font or family of fonts. A typeface or "font family" making use of serifs is called a serif typeface ( ...

serif
at the top and sometimes a short horizontal line at the bottom, traces its roots back to the
Brahmic The Brahmic scripts, also known as Indic scripts, are a family of abugida writing systems. They are used throughout the Indian subcontinent, Southeast Asia and parts of East Asia, including Japan in the form of Siddhaṃ script, Siddhaṃ. T ...
script of ancient India, where it was a simple vertical line. It was transmitted to Europe via
Arabic Arabic (, ' or , ' or ) is a Semitic language The Semitic languages are a branch of the Afroasiatic language family originating in the Middle East The Middle East is a list of transcontinental countries, transcontinental region ...

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 7 is a number, numeral, and glyph. 7 or seven may also refer to: * AD 7, the seventh year of the AD era * 7 BC, the seventh year before the AD era * The month of July Music Artists * Seven (Swiss singer) (born 1978), a Swiss recording artist * Se ...

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
ascenderAscender may refer to: *Ascender (climbing), a rope-climbing device *Ascender Corporation, a font company *Ascender (typography), a font feature *XP-55 Ascender, a prototype aircraft *Isuzu Ascender, a sports utility vehicle *JP Aerospace#Ascender, ...
in most modern
typeface A typeface is the design of lettering Lettering is an umbrella term In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying ...

typeface
s, in typefaces with
text figures Text figures (also known as non-lining, lowercase, old style, ranging, hanging, medieval, billing, or antique figures or numerals) are numeral A numeral is a figure, symbol, or group of figures or symbols denoting a number. It may refer to: * N ...

text figures
, the glyph usually is of
x-height upright 2.0, alt=A diagram showing the line terms used in typography In typography File:metal movable type.jpg, 225px, Movable type being assembled on a composing stick using pieces that are stored in the type case shown below it Typography ...

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 Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
(
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
) and
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
, the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
that follows and the multiplicative
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of the
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s,
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s and
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s; *more generally, in
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
, the multiplicative identity (also called ''unity''), usually of a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
or a ring. Formalizations of the natural numbers have their own representations of 1. In the
Peano axioms In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...
, 1 is the
successor Successor is someone who, or something which succeeds or comes after (see success (disambiguation), success and Succession (disambiguation), succession) Film and TV * The Successor (film), ''The Successor'' (film), a 1996 film including Laura Girli ...
of 0. In ''
Principia Mathematica Image:Principia Mathematica 54-43.png, 500px, ✸54.43: "From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." – Volume I, 1st editionp. 379(p. 362 in 2nd edition; p. 360 in abridged v ...
'', it is defined as the set of all singletons (sets with one element), and in the
Von Neumann cardinal assignment The von Neumann cardinal assignment is a cardinal assignment which uses ordinal number In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly ...
of natural numbers, it is defined as the set . In a multiplicative
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
or
monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
, the
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 Characteristic (from the Greek word for a property, attribute or trait Trait may refer to: * Phenotypic trait in biology, which involve genes and characteristics of organisms * Trait (computer programming), a model for structuring object-oriented ...
''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 field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. By definition, 1 is the
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
,
absolute value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

absolute value
, or
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
of a
unit complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
,
unit vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, and a
unit matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by ''I'n'', or simply by ''I'' if the size is immaterial or can be trivially determined b ...

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 Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

probability
of an event that is absolutely or
almost certain In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
to occur. In
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, 1 is sometimes used to denote the
terminal object In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
of a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
. In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

number theory
, 1 is the value of
Legendre's constant File:Legendre's constant 10 000 000.svg, 250px, Later elements up to 10,000,000 of the same sequence ''an'' = ln(''n'') − ''n''/''π''(''n'') (red line) appear to be consistently less than 1.08366 (blue line). Legendre's cons ...

Legendre's constant
, which was introduced in 1808 by
Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials In physical science and mathematic ...
in expressing the
asymptotic behavior In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing Limit (mathematics), limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very large ...
of the
prime-counting function In mathematics, the prime-counting function is the Function (mathematics), function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the pi, number ). History Of great int ...
. 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 The unary numeral system is the simplest numeral system to represent natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this ...
. Unlike
base 2 Base or BASE may refer to: Brands and enterprises *Base (mobile telephony provider), a Belgian mobile telecommunications operator *Base CRM, an enterprise software company founded in 2009 with offices in Mountain View and Kraków, Poland *Base De ...
or
base 10 The decimal numeral system (also called the base-ten positional numeral system, and occasionally called denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hi ...
, this is not a
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications ope ...
. Since the base 1 exponential function (1''x'') always equals 1, its
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
does not exist (which would be called the
logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

logarithm
base 1 if it did exist). There are two ways to write the real number 1 as a
recurring decimal A repeating decimal or recurring decimal is decimal representation A decimal representation of a non-negative real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be C ...
: as 1.000..., and as 0.999.... 1 is the first
figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are ...
of every kind, such as
triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and Cube (algebra)#In integers, cube numbers. The th triangular number ...
,
pentagonal number 181px, A visual representation of the first six pentagonal numbers A pentagonal number is a figurate number that extends the concept of triangular number, triangular and square numbers to the pentagon, but, unlike the first two, the patterns involv ...

pentagonal number
and
centered hexagonal number A centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") ...
, to name just a few. In many mathematical and engineering problems, numeric values are typically ''normalized'' to fall within the
unit interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters. Likewise,
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
are often normalized into
unit vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s (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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

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 :''Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.'' In number theory, a multiplicative function is an arithmetic ...
, then ''f''(1) must be equal to 1. It is also the first and second number in the
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of I ...

Fibonacci
sequence (0 being the zeroth) and is the first number in many other . The definition of a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
requires that 1 must not be equal to . Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the
field with one element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, 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 numberIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
for a simply connected algebraic group over a number field. The
generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers (''a'n'') by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinar ...
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 In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
, x, < 1 .


Primality

1 is by convention neither a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
nor a
composite number A composite number is a positive integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calcul ...
, but a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
(meaning of
ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
) like −1 and, in the
Gaussian integers In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number t ...
, '' i'' and −''i''. The
fundamental theorem of arithmetic In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...
guarantees
unique factorization In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

zero
is divisible by all positive integers.


Table of basic calculations


In technology

* The
resin identification code The ASTM International Resin Identification Coding System, often abbreviated RIC, is a set of symbols appearing on plastic Plastics are a wide range of synthetic polymers, synthetic or semi-synthetic materials that use polymers as a main ing ...

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 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