View source for 1

{{Short description|Number}}
{{Hatnote|This article is about the number. For the year AD 1, and other uses, see One (disambiguation) and Number One (disambiguation).}}
{{Infobox number
|number=1
|numeral=[[Unary numeral system|unary]]
|factorization=[[Empty product|∅]]
|divisor=1
|roman =I, i
|greek prefix=mono-/haplo-
|latin prefix=uni-
|lang1=Greek numeral
|lang1 symbol=α'
|lang2=[[Eastern Arabic numerals|Arabic]], Kurdish, [[Social:Persian language|Persian]], [[Social:Sindhi language|Sindhi]], Urdu
|lang2 symbol={{resize|150%|١}}
|lang3=[[Social:Assamese language|Assamese]] & [[Social:Bengali language|Bengali]]
|lang3 symbol={{resize|150%|১}}
|lang4=Chinese numeral
|lang4 symbol=一/弌/壹
|lang5=Devanāgarī
|lang5 symbol={{resize|150%|१}}
|lang6=Ge'ez
|lang6 symbol={{resize|150%|፩}}
|lang7=[[Georgian numerals|Georgian]]	
|lang7 symbol={{resize|130%|Ⴀ/ⴀ/ა}}(Ani)

|lang8=[[Hebrew numerals|Hebrew]]
|lang8 symbol={{resize|150%|א}}
|lang9=Japanese numeral
|lang9 symbol=一/壱
|lang10=Kannada
|lang10 symbol={{resize|150%|೧}}
|lang11=[[Khmer numerals|Khmer]]
|lang11 symbol={{resize|150%|១}}
|lang13=[[Social:Malayalam|Malayalam]]
|lang13 symbol=൧
|lang14=[[Social:Meitei language|Meitei]]
|lang14 symbol={{resize|150%|꯱}}
|lang15=[[Thai numerals|Thai]]
|lang15 symbol={{resize|150%|๑}}
|lang16=[[Social:Tamil language|Tamil]]
|lang16 symbol={{resize|150%|௧}}
|lang17=[[Social:Telugu language|Telugu]]
|lang17 symbol={{resize|150%|೧}}
|lang18=[[Counting rods|Counting rod]]
|lang18 symbol=𝍠}}

'''1''' ('''one''', '''unit''', '''unity''') is a [[Number|number]] representing a single or the only [[Philosophy:Entity|entity]]. 1 is also a [[Numerical digit|numerical digit]] and represents a single unit of [[Counting|counting]] or [[Measurement|measurement]]. For example, a [[Line segment|line segment]] of ''unit length'' is a line segment of [[Physics:Length|length]]&nbsp;1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of [[Natural number|natural number]]s, followed by&nbsp;[[2|2]], although by other definitions 1 is the second natural number, following&nbsp;[[0|0]].

The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same 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|prime number]]; this was not universally accepted until the mid-20th century. Additionally, 1 is the smallest possible difference between two distinct [[Natural number|natural number]]s.

The unique mathematical properties of the number have led to its unique uses in other fields, ranging from science to sports. It commonly denotes the first, leading, or top thing in a group.

== As a word ==
=== Etymology ===
''One'' originates from the [[Social:Old English|Old English]] word ''an'', derived from the Germanic root {{not a typo|''*ainaz''}}, from the Proto-Indo-European root ''*oi-no-'' (meaning "one, unique").<ref name="etymonline">{{cite web |title=Online Etymology Dictionary |url=http://www.etymonline.com/index.php?term=one |website=etymonline.com |publisher=Douglas Harper |access-date=2013-12-30 |archive-date=2013-12-30 |archive-url=https://web.archive.org/web/20131230234708/http://www.etymonline.com/index.php?term=one |url-status=live }}</ref>

=== Modern usage ===
Linguistically, ''one'' is a [[Cardinal number|cardinal number]] used for counting and expressing the number of items in a collection of things.{{sfn|Hurford|1994|loc=pp. 23-24}} ''One'' is commonly used as a determiner for singular countable nouns, as in ''one day at a time''.{{sfn|Huddleston|Pullum|Reynolds|2022|loc=p. 117}} ''One'' is also a gender-neutral pronoun used to refer to an unspecified person or to people in general as in ''one should take care of oneself''.{{sfn|Huddleston|Pullum|Reynolds|2022|loc=p. 140}} Words that derive their meaning from ''one'' include ''alone'', which signifies ''all one'' in the sense of being by oneself, ''none'' meaning ''not one'', ''once'' denoting ''one time'', and ''atone'' meaning to become ''at one'' with the someone. Combining ''alone'' with ''only'' (implying ''one-like'') leads to ''lonely'', conveying a sense of solitude.{{sfn|Conway|Guy|1996|loc=pp.3-4}} Other common [[Numeral prefix|numeral prefix]]es for the number 1 include uni- (e.g., unicycle, [[Astronomy:Universe|universe]], unicorn), sol- (e.g., solo dance), derived from latin, or mono- (e.g., [[Engineering:Monorail|monorail]], [[Social:Monogamy|monogamy]], [[Monopoly|monopoly]]) derived from greek.<ref>{{cite web |last1=Chrisomalis |first1=Stephen |url=https://phrontistery.info/numbers.html |title=Numerical Adjectives, Greek and Latin Number Prefixes |work=The Phrontistery |access-date=2022-02-24 |archive-date=2022-01-29 |archive-url=https://web.archive.org/web/20220129005307/https://phrontistery.info/numbers.html |url-status=live }}</ref>{{sfn|Conway|Guy|1996|loc=pp.4}}

== Symbols and representation ==
[[File:Clock 24 J.jpg|thumb|alt=Decorative clay/stone circular off-white sundial with bright gold stylized sunburst in center of the 24-hour clock face, one through twelve clockwise on right, and one through twelve again clockwise on left, with J shapes where ones' digits would be expected when numbering the clock hours. Shadow suggests 3 PM toward the lower left.|The 24-hour tower clock in Venice, using ''J'' as a symbol for 1]]thumb|This Woodstock typewriter from the 1940s lacks a separate key for the numeral 1.[[File:Mediaevalziffern.svg|thumb|Hoefler Text, a typeface designed in 1991, uses text figures and represents the numeral 1 as similar to a small-caps I.]]
Among the earliest known record of a numeral system, is the Sumerian [[Decimal|decimal]]-[[Sexagesimal|sexagesimal]] system on clay tablets dating from the first half of the third millennium BCE.{{sfn|Conway|Guy|1996|loc= pp. 17}} The Archaic Sumerian numerals for 1 and 60 both consisted of horizontal semi-circular symbols. {{sfn|Chrisomalis|2010|loc = pp.241}} By c. 2350 BCE, the older Sumerian curviform numerals were replaced with [[Social:Cuneiform|cuneiform]] symbols, with 1 and 60 both represented by the same symbol [[File:Babylonian 1.svg|20px]]. The Sumerian cuneiform system is a direct ancestor to the Eblaite and Assyro-Babylonian Semitic cuneiform decimal systems. {{sfn|Chrisomalis|2010|loc = pp.244}} Surviving Babylonian documents date mostly from Old Babylonian (c. 1500 BCE) and the Seleucid (c. 300 BCE) eras.{{sfn|Conway|Guy|1996|loc = pp.17}} The Babylonian cuneiform script notation for numbers used the same symbol for 1 and 60 as in the Sumerian system.{{sfn|Chrisomalis|2010|loc= pp.249}}

The most commonly used glyph in the modern Western world to represent the number 1 is the [[Arabic numerals|Arabic numeral]], a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom. It can be traced back to the [[Brahmi numerals|Brahmic]] script of ancient India, as represented by [[History:Ashoka|Ashoka]] as a simple vertical line in his Edicts of Ashoka in c. 250 BCE.<ref>{{cite journal|doi=10.3126/jie.v14i1.20077 |title=Evidences of Hierarchy of Brahmi Numeral System |date=2018 |last1=Acharya |first1=Eka Ratna |journal=Journal of the Institute of Engineering |volume=14 |pages=136–142 |doi-access=free }}</ref> This script's numeral shapes were transmitted to Europe via the [[Place:Maghreb|Maghreb]] and Al-Andalus during the Middle Ages, through scholarly works written in [[Social:Arabic|Arabic]].{{citation needed|date=November 2023}} In some countries, the serif at the top may be extended into a long upstroke as long as the vertical line. This variation can lead to confusion with the glyph used for [[7|seven]] in other countries and so to provide a visual distinction between the two the digit 7 may be written with a horizontal stroke through the vertical line.{{citation needed|date=November 2023}}

In modern [[Typeface|typeface]]s, the shape of the character for the digit 1 is typically typeset as a ''lining figure'' with an ascender, such that the digit is the same height and width as a capital letter. However, in typefaces with [[Text figures|text figures]] (also known as ''Old style numerals'' or ''non-lining figures''), the glyph usually is of x-height and designed to follow the rhythm of the lowercase, as, for example, in [[File:TextFigs148.svg|50px|alt=Horizontal guidelines with a one fitting within lines, a four extending below guideline, and an eight poking above guideline]].{{sfn|Cullen|2007|loc=p. 93}} In ''old-style'' typefaces (e.g., Hoefler Text), the typeface for numeral 1 resembles a small caps version of {{rn|I}}, featuring parallel serifs at the top and bottom, while the capital {{rn|I}} retains a full-height form. This is a relic from the [[Roman numerals]] system where {{rn|I}} represents 1.<ref>{{Cite web|url=https://www.typography.com/|title=Fonts by Hoefler&Co.|website=www.typography.com|access-date=2023-11-21}}</ref><ref name="medium-typewriters">{{Cite web|url=https://medium.com/@PostHasteCo/why-old-typewriters-lack-a-1-key-83d777f1e9d0|title=Why Old Typewriters Lack A "1" Key|first=Post Haste Telegraph|last=Company|date=April 2, 2017}}</ref> The modern digit '1' did not become widespread until the mid-1950s. As such, many older typewriters do not have dedicated key for the numeral 1 might be absent, requiring the use of the lowercase letter ''l'' or uppercase ''I'' as substitutes.<ref name="medium-typewriters" /> The lower case "{{rn|j}}" can be considered a swash variant of a lower-case Roman numeral "{{rn|i}}", often employed for the final {{rn|i}} of a "lower-case" Roman numeral. It is also possible to find historic examples of the use of ''j'' or ''J'' as a substitute for the Arabic numeral 1.<ref>{{Cite web|url=https://books.google.com/books?id=QO5UAAAAcAAJ&dq=%22JO+JJ+J2+J3%22&pg=PA70|title=Der allzeitfertige Rechenmeister|first=Christian|last=Köhler|date=November 23, 1693|via=Google Books}}</ref><ref>{{Cite web|url=https://books.google.com/books?id=MIW8-UrpEwIC&dq=%22JO+JJ+J2+J3%22&pg=PA341|title=Naeuw-keurig reys-boek: bysonderlijk dienstig voor kooplieden, en reysende persoonen, sijnde een trysoor voor den koophandel, in sigh begrijpende alle maate, en gewighte, Boekhouden, Wissel, Asseurantie ... : vorders hoe men ... kan reysen ... door Neederlandt, Duytschlandt, Vrankryk, Spanjen, Portugael en Italiën ...|date=November 23, 1679|publisher=by Jan ten Hoorn|via=Google Books}}</ref><ref>{{Cite web|url=https://books.google.com/books?id=UJ-VoRZUhaYC&dq=JO+JJ&pg=PA3|title=Articvli Defensionales Peremptoriales & Elisivi, Bvrgermaister vnd Raths zu Nürmberg, Contra Brandenburg, In causa die Fraiszlich Obrigkait [et]c: Produ. 7. Feb. Anno [et]c. 33|date=November 23, 1586|publisher=Heußler|via=Google Books}}</ref><ref>{{Cite web|url=https://books.google.com/books?id=gc9TAAAAcAAJ&dq=j0+jj+jz+j3&pg=PA285|title=Gustavi Seleni Cryptomenytices Et Cryptographiae Libri IX.: In quibus & planißima Steganographiae a Johanne Trithemio ... magice & aenigmatice olim conscriptae, Enodatio traditur; Inspersis ubique Authoris ac Aliorum, non contemnendis inventis|first=Braunschweig-Lüneburg|last=August (Herzog)|date=November 23, 1624|publisher=Johann & Heinrich Stern|via=Google Books}}</ref>

== In mathematics ==
Mathematically, the number 1 has unique properties and significance. In normal [[Arithmetic|arithmetic]] ([[Algebra|algebra]]), the number 1 is the first [[Natural number|natural number]] after [[0|0]] (zero) and can be used to make up all other integers (e.g., <math>1=1</math>; <math>2=1+1</math>; <math>3=1+1+1</math> etc).
The product of 0 numbers (the ''[[Empty product|empty product]]'') is 1 and the [[Factorial|factorial]] 0! evaluates to 1, as a special case of the empty product.{{sfn|Graham|Knuth|Patashnik|1988|loc = pp.111}} Any number <math>n</math> multiplied or divided by 1 remains unchanged (<math>n \times 1 = n/1 = n</math>). This makes it a mathematical [[Unit (ring theory)|unit]], and for this reason, 1 is often called ''unity''. Consequently, if <math>f(x)</math> is a multiplicative function, then <math>f(1)</math> must be equal to 1. This distinctive feature leads to 1 being is its own [[Factorial|factorial]] (<math>1!=1</math>), its own [[Square (algebra)|square]] (<math>1^2=1</math>) and [[Square root|square root]] (<math>\sqrt{1} = 1</math>), its own [[Cube (algebra)|cube]] (<math>1^3=1</math>) and [[Cube root|cube root]] (<math>\sqrt[3]{1} = 1</math>), and so forth. By definition, 1 is the [[Magnitude (mathematics)|magnitude]], [[Absolute value|absolute value]], or [[Norm (mathematics)|norm]] of a unit complex number, [[Unit vector|unit vector]], and a unit matrix (more usually called an [[Identity matrix|identity matrix]]). It is the multiplicative identity of the [[Integer|integer]]s, [[Real number|real number]]s, and [[Complex number|complex number]]s. 1 is the only natural number that is neither [[Composite number|composite]] (a number with more than two distinct positive divisors) nor [[Prime number|prime]] (a number with exactly two distinct positive divisors) with respect to [[Division (mathematics)|division]].<ref name=":0">{{Cite web|last=Weisstein|first=Eric W.|title=1|url=https://mathworld.wolfram.com/1.html|access-date=2020-09-22|website=mathworld.wolfram.com|language=en|archive-date=2020-07-26|archive-url=https://web.archive.org/web/20200726111421/https://mathworld.wolfram.com/1.html|url-status=live}}</ref>

In algebraic structures such as multiplicative [[Group (mathematics)|group]]s and [[Monoid|monoid]]s the identity element is often 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. Moreover, if a ring has [[Characteristic (algebra)|characteristic]] ''n'' not equal to 0, the element represented by 1 has the property that {{nowrap|''n''1 {{=}} 1''n'' {{=}} 0}} (where this 0 denotes the additive identity of the ring). Important examples that involve this concept are [[Finite field|finite field]]s.{{citation needed|date=November 2023}} A ''[[Matrix of ones|matrix of ones]]'' or ''all-ones matrix'' is defined as a [[Matrix (mathematics)|matrix]] composed entirely of 1s.{{sfn|Horn|Johnson|2012|loc= pp. 8}}

Formalizations of the natural numbers have their own representations of 1. For example, in the original formulation of the [[Peano axioms]], 1 serves as the starting point in the sequence of natural numbers.{{sfn|Peano|1889|loc=p. 1}} [[Biography:Giuseppe Peano|Peano]] later revised his axioms to state 0 as the "first" natural number such that 1 is the [[Successor function|successor]] of 0.{{sfn|Peano|1908|loc=p. 27}}  In the [[Von Neumann cardinal assignment]] of natural numbers, numbers are defined as the [[Set (mathematics)|set]] containing all preceding numbers, with 1 represented as the singleton {0}.{{sfn|Halmos|1974|loc=p. 32}} In [[Lambda calculus|lambda calculus]] and [[Computability theory|computability theory]], natural numbers are represented by [[Church encoding]] as functions, where the Church numeral for 1 is represented by the function <math>f</math> applied to an argument <math>x</math> once (1<math>fx=fx</math>).{{sfn|Hindley|Seldin|2008|loc=p. 48}} 1 is both the first and second number in the [[Fibonacci number|Fibonacci]] sequence (0 being the zeroth) and is the first number in many other [[Sequence|mathematical sequences]]. As a pan-polygonal number, 1 is present in every polygonal number sequence as the first [[Figurate number|figurate number]] of every kind (e.g., [[Triangular number|triangular number]], pentagonal number, [[Centered hexagonal number|centered hexagonal number]]).{{citation needed|date=December 2023}}

The simplest way to represent the natural numbers is by the [[Unary numeral system|unary numeral system]], as used in tallying.{{sfn|Hodges|2009|loc = p.14}} This is often referred to as "base 1", since only one mark&nbsp;– the tally itself&nbsp;– is needed. Unlike base&nbsp;2 or base&nbsp;10, this is not a [[Positional notation|positional notation]]. Since the base 1 exponential function (1<sup>''x''</sup>) always equals 1, its [[Inverse function|inverse]] (i.e., the [[Logarithm|logarithm]] base&nbsp;1) does not exist.{{citation needed|date=November 2023}}

The number 1 can be represented in decimal form by two recurring notations: 1.000..., where the digit 0 repeats infinitely after the decimal point, and [[0.999...|0.999...]], which contains an infinite repetition of the digit 9 after the decimal point. The latter arises from the definition of decimal numbers as the limits of their summed components, such that "0.999..." and "1" represent {{em|exactly}} the same number.{{sfn|Stillwell|1994|loc = p.42}}

===Primality===
Although 1 appears to meet the naïve definition of a prime number, being evenly divisible only by 1 and itself (also 1), by convention 1 is neither a [[Prime number|prime number]] nor a [[Composite number|composite number]]. This is because 1 is the only positive integer divisible by exactly one positive integer, whereas prime numbers are divisible by exactly two positive integers and composite numbers by more than two positive integers. As late as the beginnings of the 20th century, some mathematicians considered 1 a prime number.{{sfn|Caldwell|Xiong|2012|loc=pp. 8,9 }} However, the prevailing and enduring mathematical consensus has been to exclude due to its impact upon the [[Fundamental theorem of arithmetic|fundamental theorem of arithmetic]] and other theorems related to prime numbers. For example, the [[Fundamental theorem of arithmetic|fundamental theorem of arithmetic]] guarantees [[Factorization|unique factorization]] over the integers only up to units, i.e., {{nowrap|4 {{=}} 2{{sup|2}}}} represents a unique factorization. However, if units are included, 4 can also be expressed as {{nowrap|(−1){{sup|6}} × 1{{sup|23}} × 2{{sup|2}},}} among infinitely many similar "factorizations".{{sfn|Caldwell|Xiong|2012|loc=pp. 2,7}} Furthermore, [[Euler's totient function]] and the sum of divisors function are different for prime numbers than they are for 1.{{sfn|Sierpiński|1988|loc=p. 245}}{{sfn|Sandifer|2007|loc=p. 59}}

===Other mathematical attributes and uses===

In many mathematical and engineering problems, numeric values are typically ''normalized'' to fall within the [[Unit interval|unit interval]] from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters. For example, by definition, 1 is the [[Probability|probability]] of an event that is absolutely or almost certain to occur.{{sfn|Graham|Knuth|Patashnik|1988|loc = pp.381}} Likewise, [[Vector space|vectors]] are often normalized into [[Unit vector|unit vector]]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|integral]] one, maximum value one, or square integral one, depending on the application.{{sfn|Blokhintsev|2012|loc=pp. 35}}{{sfn|Sung|Smith|2019}}

In [[Category theory|category theory]], 1 is the terminal object of a [[Category (mathematics)|category]] if there is a unique [[Morphism|morphism]].{{sfn|Awodey|2010|loc=p. 33}} In [[Number theory|number theory]], 1 is the value of [[Legendre's constant]], which was introduced in 1808 by [[Biography:Adrien-Marie Legendre|Adrien-Marie Legendre]] in expressing the [[Asymptotic analysis|asymptotic behavior]] of the [[Prime-counting function|prime-counting function]]. The value was originally conjectured by Legendre to be approximately 1.08366, but was proven in 1899 to equal exactly 1 by [[Biography:Charles Jean de la Vallée Poussin|Charles Jean de la Vallée Poussin]].<ref>La Vallée Poussin, C. Mém. Couronnés Acad. Roy. Belgique 59, 1–74, 1899</ref><ref>{{Cite journal |last=Pintz |first=Janos |date=1980 |title=On Legendre's Prime Number Formula |url=https://www.jstor.org/stable/2321863 |journal=[[The American Mathematical Monthly]] |volume=87 |issue=9 |pages=733–735 |doi=10.2307/2321863 |jstor=2321863 |issn=0002-9890}}</ref>

The definition of a [[Field (mathematics)|field]] requires that 1 must not be equal to 0. Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the [[Field with one element|field with one element]], which is not a singleton and is not a set at all.{{citation needed|date=December 2023}}

In numerical [[Data|data]], 1 is the most common leading digit in many sets of data (occurring about 30% of the time), a consequence of [[Benford's law]].{{sfn|Miller|2015|loc=p. 4}}

1 is the only known [[Tamagawa number]] for a simply connected algebraic group over a number field.{{sfn|Kottwitz|1988}}{{sfn|Gaitsgory|Lurie|2019|loc=pp. 204–307}}

The [[Generating function|generating function]] that has all coefficients equal to 1 is a [[Geometric series|geometric series]], given by
<math> \frac{1}{1-x} = 1+x+x^2+x^3+ \ldots </math>{{citation needed|date=December 2023}}

The zeroth metallic mean is 1, with the [[Golden ratio#Continued fraction and square root|golden section]] equal to the [[Continued fraction#Motivation and notation|continued fraction]] [1;1,1,...], and the infinitely [[Nested radical#Infinitely nested radicals|nested square root]] <math>\scriptstyle\sqrt{1+\sqrt{\text{ }1+\cdots\text{ }}}.</math>{{citation needed|date=December 2023}}

The series of [[Unit fraction|unit fraction]]s that most rapidly converge to 1 are the [[Multiplicative inverse|reciprocals]] of [[Sylvester's sequence]], which generate the infinite [[Egyptian fraction]]
<math>1 = \frac12 + \frac13 + \frac17 + \frac1{43} + \cdots</math>.{{citation needed|date=December 2023}}

=== Table of basic calculations ===
{|class="wikitable" style="text-align: center; background: white"
|-
! style="width:105px;"|[[Multiplication]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
! style="width:5px;"|
!11

!12
!13
!14
!15
!16
!17
!18
!19
!20
! style="width:5px;"|
!21
!22
!23
!24
!25
! style="width:5px;"|
!50
!100
!1000
|-
|'''1 × ''x'''''
|'''1'''
|2
|3
|4
|5
|6
|7
|8
|9
|10
!
|[[11 (number)|11]]
|[[12 (number)|12]]
|[[13 (number)|13]]
|[[14 (number)|14]]
|[[15 (number)|15]]
|[[16 (number)|16]]
|[[17 (number)|17]]
|[[18 (number)|18]]
|[[19 (number)|19]]
|[[20 (number)|20]]
!
|[[21 (number)|21]]
|[[22 (number)|22]]
|[[23 (number)|23]]
|[[24 (number)|24]]
|[[25 (number)|25]]
!
|[[50 (number)|50]]
|[[100 (number)|100]]
|[[1000 (number)|1000]]
|}

{|class="wikitable" style="text-align: center; background: white"
|-
! style="width:105px;"|[[Division (mathematics)|Division]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
! style="width:5px;"|
!11
!12
!13
!14
!15
|-
|'''1 ÷ ''x'''''
|'''1'''
|0.5
|0.{{overline|3}}
|0.25
|0.2
|0.1{{overline|6}}
|0.{{overline|142857}}
|0.125
|0.{{overline|1}}
|0.1
!
|0.{{overline|09}}
|0.08{{overline|3}}
|0.{{overline|076923}}
|0.0{{overline|714285}}
|0.0{{overline|6}}
|-
|'''''x'' ÷ 1'''
|'''1'''
|2
|3
|4
|5
|6
|7
|8
|9
|10
!
|11
|12
|13
|14
|15
|}

{|class="wikitable" style="text-align: center; background: white"
|-
! style="width:105px;"|[[Exponentiation]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
! style="width:5px;"|
!11
!12
!13
!14
!15
!16
!17
!18
!19
!20
|-
|'''1{{sup|''x''}}'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
!
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|'''1'''
|-
|'''''x''{{sup|1}}'''
|'''1'''
|2
|3
|4
|5
|6
|7
|8
|9
|10
!
|11
|12
|13
|14
|15
|16
|17
|18
|19
|20
|}

== In technology ==
In digital technology, data is represented by [[Binary code|binary code]], i.e., a [[Radix|base]]-2 numeral system with numbers represented by a sequence of 1s and 0s. Digitised data is represented in physical devices, such as [[Computer|computer]]s, as pulses of electricity through switching devices such as [[Engineering:Transistor|transistor]]s or [[Engineering:Logic gate|logic gate]]s where "1" represents the value for "on". As such, the numerical value of [[Boolean data type|true]] is equal to 1 in many [[Programming language|programming language]]s.<ref name=Woodford>{{citation |first=Chris |last=Woodford |url={{GBurl|id=My7Zr0aP2L8C|p=9}} |title=Digital Technology |date=2006 |publisher=Evans Brothers |isbn=978-0-237-52725-9 |page=9 |access-date=2016-03-24 }}</ref>{{sfn|Godbole|2002|loc=pp. 34}}

== In science ==
*[[Physics:Dimensionless quantities|Dimensionless quantities]] are also known as quantities of dimension one.
*[[Chemistry:Hydrogen|Hydrogen]], the first element of the periodic table, has an [[Physics:Atomic number|atomic number]] of 1.
*Group 1 of the [[Chemistry:Periodic table|periodic table]] consists of the alkali metals.
*Period 1 of the periodic table consists of just two elements, [[Chemistry:Hydrogen|hydrogen]] and [[Chemistry:Helium|helium]].

== In philosophy ==
In the philosophy of [[Biography:Plotinus|Plotinus]] (and that of other neoplatonists), [[Biography:Plotinus#The One|The One]] is the ultimate reality and source of all existence.{{sfn|Olson|2017}} [[Biography:Philo#Numbers|Philo of Alexandria]] (20&nbsp;BC&nbsp;– AD&nbsp;50) regarded the number one as God's number, and the basis for all numbers ("De Allegoriis Legum", ii.12 [i.66]).

The Neopythagorean philosopher [[Biography:Nicomachus|Nicomachus of Gerasa]] affirmed that one is not a number, but the source of number. He also believed the number two is the embodiment of the origin of [[Philosophy:Other|otherness]]. His [[Number theory|number theory]] was recovered by [[Biography:Boethius|Boethius]] in his Latin translation of Nicomachus's treatise ''[[Introduction to Arithmetic]]''.<ref>{{cite journal|url=https://www.cambridge.org/core/journals/british-journal-for-the-history-of-science/article/abs/from-abacus-to-algorism-theory-and-practice-in-medieval-arithmetic/7DFF144C90C127E715CA40083254E601#access-block|title=From Abacus to Algorism: Theory and Practice in Medieval Arithmetic|journal=The British Journal for the History of Science|volume=10|issue=2|date=July 1, 1977|page=Abstract|doi=10.1017/S0007087400015375|publisher=Cambridge University Press|author=British Society for the History of Science|s2cid=145065082|access-date=May 16, 2021|archive-date=May 16, 2021|archive-url=https://web.archive.org/web/20210516110812/https://www.cambridge.org/core/journals/british-journal-for-the-history-of-science/article/abs/from-abacus-to-algorism-theory-and-practice-in-medieval-arithmetic/7DFF144C90C127E715CA40083254E601#access-block|url-status=live}}</ref>

== See also ==
{{Commons and category|1 (number)|1 (number)}}

*[[−1|−1]]
*+1 (disambiguation)
*[[List of mathematical constants]]
*One (word)
*[[Root of unity]]

== References ==
{{reflist|2}}

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{{refend}}

{{Number theory|hide}}


{{DEFAULTSORT:1 (Number)}}
[[Category:1 (number)| ]]
[[Category:Integers]]

{{Sourceattribution|1}}