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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 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 ...
of
counting Counting is the process of determining the 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) ...
or
measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measurement are dependent on the context and discipline. In natural science Natur ...

measurement
. For example, a
line segment 250px, The geometric definition of a closed line segment: the intersection of all points at or to the right of ''A'' with all points at or to the left of ''B'' In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' ...

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 A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an appa ...

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. Historians credit the 9th-century Byzantine missionaries Saints Cyril and Methodius with standardizing the language and using it in translating the Bible ...
''-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 belonging to the 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 Re ...
after
zero 0 (zero) 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 languag ...

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 Juxtaposition is an act or instance of placing two elements close together or side by side. This is often done in order to compare/contr ...

multiplication
. As a result, 1 is its own
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n ...
, its own
square In Euclidean geometry, a square is a regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger ...
and
square root 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) ...

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 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). ...

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 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 ...
(meaning of
ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure ...
).


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 Typography is the art and technique of arranging type to make written language A written language is the representation of a spoken or gestural language A language is a structured system of communication used by ...

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 , a typeface designed in 1991, uses text figures. . Text figures (also known as non-lining, lowercase, old style, ranging, hanging, medieval, billing, or antique figures or numerals) are numeral system, numerals designed with varying heights in a ...

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 'cra ...
(
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 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 ...
of the
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the 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 Re ...
s,
real 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). ...
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 pr ...
, 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 (mathematic ...
, the
identity 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 ...
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 , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ...

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 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 ...
, and a
unit matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structur ...

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 and Legendre transformation are named a ...
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 ). Image:PrimePi.svg, 400p ...
. 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 numbers: to represent a number ''N'', a symbol representing 1 is repeated ''N'' times. In the unary system, the number 0 (zero) is represented by the empty string, that ...
. 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 of the (or ). More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number ...
. 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 numberThe term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygon ...
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 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 ...
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 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 (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 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). ...

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 mathematician A mathematician is someone who uses an extensive knowledge of mathem ...

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 Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...
, 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 calculu ...
, but a
unit Unit may refer to: Arts and entertainment * 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 ...
(meaning of
ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure ...
) like −1 and, in the
Gaussian integers In number theory, a Gaussian integer is a complex number In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisf ...
, '' 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 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 languag ...

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 Polyethylene terephthalate (or poly(ethylene terephthalate), PET, PETE, or the obsolete PETP or PET-P), is the most common thermoplastic A thermoplastic, or thermosoft plastic, is a plastic polymer A polymer (; Greek ''wikt:poly-, poly-' ...

polyethylene terephthalate
. *The
ITU 260px, ITU Monument, Bern The International Telecommunication Union is a specialized agency of the United Nations responsible for all matters related to information and communication technologies Information and communications technology ...

ITU
country code for the
North American Numbering Plan The North American Numbering Plan (NANP) is a telephone numbering plan A telephone numbering plan is a type of numbering schemeThere are many different numbering schemes for assigning nominal numbers to entities. These generally require an agr ...
area, which includes the United States, Canada, and parts of the Caribbean. *A
binary code A binary code represents text Text may refer to: Written word * Text (literary theory) Text may refer to: Written word * Text (literary theory), any object that can be read, including: **Religious text, a writing that a religious tradition con ...

binary code
is a sequence of 1 and that is used in
computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These ...

computer
s for representing any kind of
data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sense, data are a set of values of qualitative property, qualitative or quantity, quantitative variable (research), variables about one or ...

data
. *In many physical devices, 1 represents the value for "on", which means that electricity is flowing. *The numerical value of
true True most commonly refers to truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherw ...
in many programming languages. *1 is the
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the stu ...
code of " Start of Header".


In science

*
Dimensionless quantities In dimensional analysis In engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline ...
are also known as quantities of dimension one. *1 is the atomic number of
hydrogen Hydrogen is the chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all have the same ...

hydrogen
. *+1 is the
electric charge Electric charge is the physical property A physical property is any property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of th ...
of
positron The positron or antielectron is the antiparticle s (left) and antiparticles (right). From top to bottom; electron The electron is a subatomic particle, symbol or , whose electric charge Electric charge is the physical property of mat ...

positron
s and protons. *Group 1 of the
periodic table The periodic table, also known as the periodic table of (the) chemical elements, is a tabular display of the chemical element upright=1.0, 500px, The chemical elements ordered by link=Periodic table In chemistry Chemistry is ...

periodic table
consists of the
alkali metals The alkali metals consist of the chemical element upright=1.0, 500px, The chemical elements ordered by link=Periodic table In chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a ...
. *Period 1 of the periodic table consists of just two elements,
hydrogen Hydrogen is the chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all have the same ...

hydrogen
and
helium Helium (from el, ἥλιος, helios Helios; Homeric Greek: ), Latinized as Helius; Hyperion and Phaethon are also the names of his father and son respectively. often given the epithets Hyperion ("the one above") and Phaethon ("the shining") ...

helium
. *The dwarf planet
Ceres Ceres most commonly refers to: * Ceres (dwarf planet) Ceres (; minor-planet designation: 1 Ceres) is the smallest recognized dwarf planet, the closest dwarf planet to the Sun, and the List of notable asteroids, largest object in the main astero ...
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 A planet is an astronomical body orbiting a star or Stellar evolution#Stellar remnants, stellar remnant that is massive enough to be Hydrostatic equilibrium, rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and ...

planet
or
minor planet A minor planet is an astronomical object in direct orbit around the Sun (or more broadly, any star with a planetary system) that is neither a planet nor exclusively classified as a comet. Before 2006, the International Astronomical Union (IAU) o ...
(such as Neptune I, a.k.a.
Triton Triton commonly refers to: * Triton (mythology), a Greek god * Triton (moon), a satellite of Neptune Triton may also refer to: Biology * Triton cockatoo, a parrot * Triton (gastropod), a group of sea snails * ''Triton'', a synonym of ''Triturus'', ...

Triton
). For some earlier discoveries, the Roman numerals originally reflected the increasing distance from the primary instead.


In philosophy

In the philosophy of
Plotinus Plotinus (; grc-gre, Πλωτῖνος, ''Plōtînos'';  – 270 CE) was a major Hellenistic The Hellenistic period spans the period of Mediterranean history The Mediterranean Sea is a sea connected to the Atlantic Ocean, surround ...

Plotinus
(and that of other
neoplatonist Neoplatonism is a strand of Platonic Plato's influence on Western culture was so profound that several different concepts are linked by being called Platonic or Platonist, for accepting some assumptions of Platonism, but which do not imply accept ...
s), is the ultimate reality and source of all existence.
Philo of Alexandria Philo of Alexandria (; grc, Φίλων, Phílōn; he, , Yedidia (Jedediah) HaCohen; ), also called Philo Judaeus, was a Hellenistic Jewish philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from ...

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 .66.


In literature

*Number One is a character in the book series ''
Lorien Legacies ''Lorien Legacies'' is a series of young adult fiction, young adult science fiction books, written by James Frey, Jobie Hughes, and formerly, Greg Boose, under the collective pseudonym Pittacus Lore. Lorien Legacies ''I am Number Four'' "I ...
'' by Pittacus Lore. *Number 1 is also a character in the series ''
Artemis Fowl ''The Fowl Adventures'' is a series of ten dark fantasy novels written by Irish people, Irish author Eoin Colfer revolving around various members of the Fowl family. The first cycle, ''Artemis Fowl'', follows Elf (Artemis Fowl), elf LEP Reconn ...
'' by
Eoin Colfer Eoin Colfer (; born 14 May 1965) is an Irish author of children's books. He worked as a primary school teacher before he became a full-time writer. He is best known for being the author of the ''Artemis Fowl'' series. In September 2008, Colf ...

Eoin Colfer
.


In music

*In a 1968 song by
Harry Nilsson Harry Edward Nilsson III (June 15, 1941 – January 15, 1994), known professionally as Nilsson, was an American singer-songwriter who achieved the peak of his commercial success in the early 1970s. His work is characterized by pioneering vocal o ...

Harry Nilsson
and recorded by
Three Dog Night Three Dog Night is an American rock band formed in 1967, with founding members consisting of vocalists Danny Hutton, Cory Wells, and Chuck Negron. This lineup was soon augmented by Jimmy Greenspoon (keyboards), Joe Schermie (bass), Michael Al ...
, the number one is identified as "the loneliest number". *''
We Are Number One "We Are Number One" is a song from the English-language Iceland Iceland ( is, Ísland; ) is a Nordic countries, Nordic island country in the Atlantic Ocean, North Atlantic Ocean, with a population of 356,991 and an area of , making it the m ...
'' is a 2014 song from the children's TV show ''
LazyTown ''LazyTown'' (Icelandic: ''Latibær'') is an Icelandic-American children's television musical series. It was created by Magnús Scheving Magnús Örn Eyjólfsson Scheving (; born 10 November 1964) is an Icelandic writer, entrepreneur, produce ...
'', which gained popularity as a
meme A meme ( ) is an idea, behavior, or style that spreads by means of imitation from person to person within a culture and often carries symbolic meaning representing a particular phenomenon or theme. A meme acts as a unit for carrying culture, c ...
. * ''1'' (Beatles album), a compilation album by the Beatles. *
One 1 (one, also called unit, and unity) 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 ...
, a 1991 song by Irish rock band .


In comics

*A character in the Italian comic book Alan Ford (authors
Max Bunker Max Bunker, pen name of Luciano Massimiliano Secchi (born 24 August 1939), is an Italian comics, Italian comic book writer, and publisher, best known as the co-author of ''Alan Ford (comics), Alan Ford''. Bunker's career started in 1960 when he co- ...
and
Magnus Magnus, meaning "Great" in Latin, was used as of in the first century BCE. The best-known use of the name during the Roman Empire is for the fourth-century Western Roman Emperor Flavius Magnus the assassin, often just called . The name gained ...

Magnus
), very old disabled man, the supreme leader of the group TNT. *A character in the Italian comic series
PKNA ''PK - Paperinik New Adventures'' (''PKNA'') is an Italian comic, published by Disney Italy from 14 March 1996 to 20 December 2000, about the new adventures of Paperinik, the superhero A superhero or superheroine is a stock character that p ...
and its sequels, an
artificial intelligence Artificial intelligence (AI) is intelligence Intelligence has been defined in many ways: the capacity for abstraction Abstraction in its main sense is a conceptual process where general rules and concept Concepts are defined as abstra ...

artificial intelligence
as an ally of the protagonist
PaperinikDonald Duck Donald Fauntleroy Duck is a cartoon character created in 1934 at Walt Disney Animation Studios. Donald is an Anthropomorphism, anthropomorphic American Pekin, white duck with a yellow-orange bill, legs, and feet. He typically wears a ...
.


In sports

*In
baseball Baseball is a bat-and-ball games, bat-and-ball game played between two opposing teams who take turns batting (baseball), batting and fielding. The game proceeds when a player on the fielding team (baseball), fielding team, called the pi ...

baseball
scoring, the number 1 is assigned to the
pitcher In baseball, the pitcher is the player who pitches the Baseball (ball), baseball from the pitcher's mound toward the catcher to begin each play, with the goal of out (baseball), retiring a batter (baseball), batter, who attempts to either make ...

pitcher
. *In
association football Association football, more commonly known as simply football or soccer, is a team sport played with a sphere, spherical Ball (association football), ball between two teams of 11 football player, players. It is played by approximately 250&nbs ...
(soccer) the number 1 is often given to the
goalkeeper In many team sports which involve scoring goal (sport), goals, the goalkeeper (sometimes termed goaltender, netminder, goalie or keeper) is a designated player charged with directly preventing the opposing team from scoring by blocking or inte ...
. *In most competitions of
rugby league Rugby league football, commonly known as just rugby league or simply league, rugby, football, or footy, is a full-contact sport played by two teams of thirteen players on a rectangular field Field may refer to: Expanses of open ground * Fi ...
(though not the
Super League The Super League (SL), sponsored as the Betfred Super League and officially known as Super League Europe, is the top-level of the British rugby league system. At present the league consists of twelve teams, of which ten are from Northern Engl ...
, which uses static squad numbering), the starting fullback wears jersey number 1. *In
rugby union Rugby union, commonly known simply as rugby, is a Contact sport#Terminology, close-contact team sport that originated in England in the first half of the 19th century. One of the Comparison of rugby league and rugby union, two codes of rugby f ...
, the starting wears the jersey number 1. *1 is the lowest number permitted for use by players of the
National Hockey League The National Hockey League (NHL; french: Ligue nationale de hockey—LNH) is a professional ice hockey sports league, league in North America comprising 32 teams—25 in the United States and 7 in Canada. It is considered to be the premier pro ...
(NHL); the league prohibited the use of 00 and 0 in the late 1990s (the highest number permitted being 98). *1 is the lowest
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 ...
permitted for use at most levels of
American football American football, referred to simply as football in the United States and Canada and also known as gridiron, is a team sport A team sport includes any sport Sport pertains to any form of Competition, competitive physical acti ...

American football
. Under
National Football League The National Football League (NFL) is a professional American football American football, referred to simply as football in the United States and also known as gridiron, is a team sport played by two teams of eleven players on a rect ...
policy, it can only be used by a
quarterback The quarterback (commonly abbreviated "QB"), colloquially known as the "signal caller", is a position in gridiron football Gridiron football,
quarterback
or kicking player (during 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 Formula One (also known as Formula 1 or F1) is the highest class of international racing for Open wheel car, single-seater Formula racing, formula racing cars sanctioned by the Fédération Internationale de l'Automobile (FIA). The World Dri ...
, the previous year's world champion is allowed to use the number 1.


In film

*'' One A.M.'' (1916), starring
Charlie Chaplin Sir Charles Spencer Chaplin Jr. (16 April 188925 December 1977) was an English comic actor, filmmaker, and composer who rose to fame in the era of . He became a worldwide icon through his screen persona, , and is considered one of the most i ...

Charlie Chaplin
. *'' One More Time'' (1970), directed by
Jerry Lewis Jerry Lewis (born Joseph Levitch; March 16, 1926 – August 20, 2017) was an American comedian, actor, filmmaker, humanitarian and singer. Nicknamed "The King of Comedy", he is regarded as one of the most significant American cultural figur ...

Jerry Lewis
and starring
Sammy Davis Jr. Samuel George Davis Jr. (December 8, 1925 – May 16, 1990) was an American singer, dancer, actor, vaudevillian and comedian whom critic Randy Blaser called "the greatest entertainer ever to grace a stage in these United States". At age th ...
and
Peter Lawford Peter Sydney Ernest Lawford (born Peter Sydney Ernest Aylen; 7 September 1923 – 24 December 1984) was an English-born American actor, producer, and socialite, who resided in the United States throughout his adult life.Obituary ''Variety Obitua ...
. *'' One Day'' (2011), starring
Anne Hathaway Anne Jacqueline Hathaway (born November 12, 1982) is an American actress. She is the recipient of many awards, including an Academy Award The Academy Awards, popularly known as the Oscars, are awards for artistic and technical merit in ...

Anne Hathaway
and
Jim Sturgess James Anthony Sturgess''Births, Marriages & Deaths Index of England & Wales, 1916–2005.''; at ancestry.com (born 16 May 1978) is an English actor and singer-songwriter. His first major role was as Jude in the musical romance drama film '' Acr ...

Jim Sturgess
.


In other fields

*''Number One'' is
Royal Navy The Royal Navy (RN) is the United Kingdom's naval warfare Naval warfare is combat Combat ( French for ''fight'') is a purposeful violent conflict meant to physically harm or kill the opposition. Combat may be armed (using weapon A ...
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 An ace is a playing card A playing card is a piece of specially prepared , heavy paper, thin cardboard, , cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a to ...

ace
in many playing card games, such as
cribbage Cribbage, or crib, is a card game, traditionally for two players, that involves playing and grouping playing cards, cards in combinations which gain points. It can be adapted for three or four players. Cribbage has several distinctive features ...

cribbage
. *
List of highways numbered 1 The following highways are numbered 1. For roads numbered A1, see list of A1 roads. For roads numbered B1, see list of B1 roads. For roads numbered M1, see List of M1 roads. For roads numbered N1, see list of N1 roads. For roads numbered S1 ...
* List of public transport routes numbered 1 *1 is often used to denote the
Gregorian calendar The Gregorian calendar is the used in most of the world. It was introduced in October 1582 by as a modification of the , reducing the average year from 365.25 days to 365.2425 days, and adjusting for the drift in the that the inaccuracy ha ...
month of
January January is the first month of the year in the Julian and Gregorian calendar The Gregorian calendar is the used in most of the world. It was introduced in October 1582 by as a modification of the , reducing the average year from 365. ...
. *
1 CE __NOTOC__ AD 1 (I), 1 AD or 1 CE is the epoch year for the Anno Domini The terms (AD) and before Christ (BC) are used to label or number years in the Julian and Gregorian calendar The Gregorian calendar is the calendar used in ...
, the first year of the
Common Era Common Era (CE) is one of the year notations used for the Gregorian calendar The Gregorian calendar is the used in most of the world. It was introduced in October 1582 by as a modification of the , reducing the average year from 365.2 ...
*01, the former dialing code for
Greater London Greater London is an administrative area Administrative division, administrative unitArticle 3(1). , country subdivision, administrative region, subnational entity, first-level subdivision, as well as many similar terms, are generic names ...

Greater London
* PRS One, a German paraglider design *+1 is the code for international telephone calls to countries in the
North American Numbering Plan The North American Numbering Plan (NANP) is a telephone numbering plan A telephone numbering plan is a type of numbering schemeThere are many different numbering schemes for assigning nominal numbers to entities. These generally require an agr ...
.


See also

*
−1 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 ...
* +1 (disambiguation) *
List of mathematical constantsA mathematical constant A mathematical constant is a key 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. ...
* One (word) *
Root of unity The 5th roots of unity (blue points) in the complex plane 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 containe ...
*
List of highways numbered 1 The following highways are numbered 1. For roads numbered A1, see list of A1 roads. For roads numbered B1, see list of B1 roads. For roads numbered M1, see List of M1 roads. For roads numbered N1, see list of N1 roads. For roads numbered S1 ...


References


External links


The Number 1The Positive Integer 1
{{DEFAULTSORT:1 (Number) Integers