{{Short description|Natural number}} {{Hatnote|This article is about the number. For the year [[AD 1]], and other uses, see [[One (disambiguation)]] and [[Number One (disambiguation)]]}} {{pp-semi-indef|small=yes}} {{good article}} {{Contains special characters}} {{Infobox number |number=1 |numeral=[[Unary numeral system|unary]] |factorization=[[Empty product|∅]] |divisor=1 |roman =I, i |greek prefix=[[Wiktionary:mono-|mono-]]/[[Wiktionary:haplo-|haplo-]] |latin prefix=[[Wiktionary:uni-|uni-]] |lang1=[[Greek numeral]] |lang1 symbol=α' |lang2=[[Eastern Arabic numerals|Arabic]], [[Central Kurdish|Kurdish]], [[Persian language|Persian]], [[Sindhi language|Sindhi]], [[Urdu numerals|Urdu]] |lang2 symbol={{resize|150%|١}} |lang3=[[Assamese language|Assamese]] & [[Bengali language|Bengali]] |lang3 symbol={{resize|150%|১}} |lang4=[[Chinese numeral]] |lang4 symbol=一/弌/壹 |lang5=[[Devanāgarī]] |lang5 symbol={{resize|150%|१}} |lang6=[[Santali language|Santali]] |lang6 symbol={{resize|150%|᱑}} |lang7=[[Ge'ez alphabet|Ge'ez]] |lang7 symbol={{resize|150%|፩}} |lang8=[[Georgian numerals|Georgian]] |lang8 symbol={{resize|130%|Ⴀ/ⴀ/ა}}([[Ani (letter)|Ani]]) |lang9=[[Hebrew numerals|Hebrew]] |lang9 symbol=[[aleph|{{resize|150%|א}}]] |lang10=[[Japanese numeral]] |lang10 symbol=一/壱 |lang11=[[Kannada language|Kannada]] |lang11 symbol={{resize|150%|[[೧]]}} |lang12=[[Khmer numerals|Khmer]] |lang12 symbol={{resize|150%|១}} |lang13=[[Armenian numerals|Armenian]] |lang13 symbol=Ա |lang14=[[Malayalam]] |lang14 symbol=൧ |lang15=[[Meitei language|Meitei]] |lang15 symbol={{resize|150%|꯱}} |lang16=[[Thai numerals|Thai]] |lang16 symbol={{resize|150%|๑}} |lang17=[[Tamil language|Tamil]] |lang17 symbol={{resize|150%|௧}} |lang18=[[Telugu language|Telugu]] |lang18 symbol={{resize|150%|೧}} |lang19=[[Babylonian cuneiform numerals|Babylonian numeral]] |lang19 symbol=𒐕 |lang20=[[Egyptian numerals|Egyptian hieroglyph]], [[Aegean numerals|Aegean numeral]], [[counting rods|Chinese counting rod]] |lang20 symbol={{resize|350%|𓏤}} |lang21=[[Maya numerals|Mayan numeral]] |lang21 symbol=• |lang22=[[Morse code]] |lang22 symbol={{nowrap|{{resize|150%|. _ _ _ _}}}} |lang23=[[Cyrillic numerals]] |lang23 symbol=А }}
'''1''' ('''one''', '''unit''', '''unity''') is a [[number]], [[Numeral (linguistics)|numeral]], and [[grapheme]]. It is the first and smallest [[Positive number|positive integer]] of the infinite sequence of [[natural number]]s. This fundamental property has led to its unique uses in other fields, ranging from science to sports, where it commonly denotes the first, leading, or top thing in a group. 1 is the [[unit (measurement)|unit]] of [[counting]] or [[measurement]], and represents a single thing. The representation of 1 evolved from ancient Sumerian and Babylonian symbols to the modern Arabic numeral. Linguistically, in English, "one" is a determiner for singular nouns and a gender-neutral pronoun.
In mathematics, 1 is the multiplicative identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered a [[prime number]]. In [[Digital electronics|digital technology]], 1 represents the "on" state in [[binary code]], the foundation of [[computing]]. Philosophically, 1 symbolizes the ultimate reality or source of existence in various traditions.
== In mathematics == The number 1 is the first natural number after [[0]]. Each [[natural number]], including 1, is constructed by [[Successor function|succession]], that is, by adding 1 to the previous natural number. Although 1 meets the naïve definition of a [[prime number]], being evenly divisible only by 1 and itself (also 1), by modern convention it is regarded as neither a [[Prime number#Primality of one|prime]] nor a [[composite number]].{{sfn|Caldwell|Xiong|2012|pp=8–9}}
The number 1 is the [[multiplicative identity]] of the [[integer]]s, [[real number]]s, and [[complex number]]s, that is, any number <math>n</math> multiplied by 1 remains unchanged <math>1\times n = n\times 1 = n</math>. As a result, the [[Square (algebra)|square]] <math>1^2=1</math>, [[square root]] <math>\sqrt{1} = 1</math>, and any other power of 1 is always equal to 1 itself.{{sfn|Colman|1912|loc=chapt.2|pp=9–10}} More generally, in [[algebra]], it denotes the multiplicative identity in any [[unital ring]] or [[field (mathematics)|field]]. An element with a multiplicative inverse is called a [[unit (ring theory)|unit]], generalizing the role of 1. For any number <math>a</math>, the first power satisfies <math>a^1=a</math>, so that 1 is also the identity for any power semigroup.
1 is its own [[factorial]] <math>1!=1</math>. Moreover, the [[empty product]], that is the product of a set of zero numbers, is also 1; for example 0! is 1.{{sfn|Graham|Knuth|Patashnik|1994|p=111}}
Different mathematical constructions of the natural numbers represent 1 in various ways. In [[Giuseppe Peano]]'s original formulation of the [[Peano axioms]], a set of postulates to define the natural numbers in a precise and logical way, 1 was treated as the starting point of the sequence of natural numbers.{{sfn|Kennedy|1974|pp=389}}{{sfn|Peano|1889|p=1}} Peano later revised his axioms to begin the sequence with 0.{{sfn|Kennedy|1974|pp=389}}{{sfn|Peano|1908|p=27}} In the [[Von Neumann cardinal assignment]] of natural numbers, where each number is defined as a [[Set (mathematics)|set]] that contains all numbers before it, 1 is represented as the [[Singleton (mathematics)|singleton]] <math>\{0\}</math>, a set containing only the element 0.{{sfn|Halmos|1974|p=32}} The [[unary numeral system]], as used in [[Tally mark|tallying]], is an example of a "base-1" number system, since only one mark – the tally itself – is needed. While this is the simplest way to represent the natural numbers, base-1 is rarely used as a practical base for [[counting]] due to its difficult readability.{{sfn|Hodges|2009|p=14}}{{sfn|Hext|1990}}
In many mathematical and engineering problems, numeric values are typically [[Normalized solution (mathematics)|normalized]] to fall within the [[unit interval]] <math>[0,1]</math>, where 1 represents the maximum possible value. For example, by definition 1 is the [[probability]] of an event that is absolutely or [[almost certain]] to occur.{{sfn|Graham|Knuth|Patashnik|1994|p=381}} Likewise, [[vector space|vectors]] are often normalized into [[unit vector]]s (i.e., vectors of magnitude one), because these often have more desirable properties. Functions are often normalized by the condition that they have [[integral]] one, maximum value one, or [[square integrable|square integral]] one, depending on the application.{{sfn|Blokhintsev|2012|p=35}}
1 is the most common leading digit in many sets of real-world numerical data. This is a consequence of [[Benford’s law]], which states that the probability for a specific leading digit <math>d</math> is <math display="inline"> \log_{10} \left(\frac{d+1}{d} \right) </math>. The tendency for real-world numbers to grow exponentially or logarithmically biases the distribution towards smaller leading digits, with 1 occurring approximately 30% of the time.{{sfn|Miller|2015|pp=3-4}}
1 is the value of [[Legendre's constant]], introduced in 1808 by [[Adrien-Marie Legendre]] to express the [[Asymptotic analysis|asymptotic behavior]] of the [[prime-counting function]].{{sfn|Pintz|1980|pp=733-735}} The [[Weil's conjecture on Tamagawa numbers]] states that the [[Tamagawa number]] <math>\tau(G)</math>, a geometrical measure of a connected linear [[algebraic group]] over a global [[number field]], is 1 for all simply connected groups (those that are [[Homotopical connectivity|path-connected]] with no '[[Homotopical connectivity#Definition using holes|holes]]').{{sfn|Gaitsgory|Lurie|2019|pp=204–307}}{{sfn|Kottwitz|1988}}
== As a word == {{See also|One (pronoun)}} ''One'' originates from the [[Old English]] word ''an'', derived from the [[Germanic languages|Germanic]] root {{not a typo|{{wikt-lang|gem-x-proto|*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=December 30, 2013 |archive-date=December 30, 2013 |archive-url=https://web.archive.org/web/20131230234708/http://www.etymonline.com/index.php?term=one |url-status=live }}</ref> Linguistically, ''one'' is a [[cardinal number]] used for counting and expressing the number of items in a collection of things.{{sfn|Hurford|1994|pp=23–24}} ''One'' is most commonly a [[English determiners|determiner]] used with [[Grammatical number|singular]] countable [[English nouns|nouns]], as in ''one day at a time''.{{sfn|Huddleston|Pullum|Reynolds|2022|p=117}} The determiner has two senses: numerical one (''I have one apple'') and singulative one (''one day I'll do it'').{{sfn|Huddleston|Pullum|2002|pp=386}} ''One'' is also a gender-neutral [[One (pronoun)|pronoun]] used to refer to an unspecified [[person]] or to people in general as in ''one should take care of oneself''.{{sfn|Huddleston|Pullum|2002|p=426–427}}
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|pp=3–4}} Other common [[numeral prefix]]es for the number 1 include ''uni-'' (e.g., [[unicycle]], universe, unicorn), ''sol-'' (e.g., solo dance), derived from Latin, or ''mono-'' (e.g., [[monorail]], monogamy, 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=February 24, 2022 |archive-date=January 29, 2022 |archive-url=https://web.archive.org/web/20220129005307/https://phrontistery.info/numbers.html |url-status=live }}</ref>{{sfn|Conway|Guy|1996|p=4}}
==Symbols and representation== === History === {{see also|History of the Hindu–Arabic numeral system}} Among the earliest known records of a numeral system, is the [[Sumer]]ian decimal-[[sexagesimal]] system on [[clay tablet]]s dating from the first half of the [[3rd millennium BC|third millennium BCE]].{{sfn|Conway|Guy|1996|p=17}} Archaic Sumerian numerals for 1 and 60 both consisted of horizontal semi-circular symbols, {{sfn|Chrisomalis|2010|p=241}} by {{circa|2350 BCE|lk=no}}, the older Sumerian curviform numerals were replaced with [[cuneiform]] symbols, with 1 and 60 both represented by the same mostly vertical symbol. [[File:Babylonian 1.svg|50px|centre|class=skin-invert-image]] The Sumerian cuneiform system is a direct ancestor to the [[Eblaite]] and [[Assyro-Babylonian]] [[Semitic languages|Semitic]] cuneiform [[decimal]] systems.{{sfn|Chrisomalis|2010|p=244}} Surviving [[Babylonia|Babylonian]] documents date mostly from Old Babylonian ({{circa|1500 BCE|lk=no}}) and the Seleucid ({{circa|300 BCE|lk=no}}) eras.{{sfn|Conway|Guy|1996|p=17}} The Babylonian cuneiform script notation for numbers used the same symbol for 1 and 60 as in the Sumerian system.{{sfn|Chrisomalis|2010|p=249}}
The most common [[representative glyph]] used in the modern [[Western world]] for 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 [[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 |issue=1 |pages=136–142 |doi-access=free }}</ref> This script's numeral shapes were transmitted to Europe via the [[Maghreb]] and [[Al-Andalus]] during the Middle Ages {{sfn|Schubring|2008|pp=147}} The Arabic numerals, and other glyphs used to represent the number one (e.g., Roman numeral ({{rn|I}} ), Chinese numeral ({{zhi|c=一}})) are [[logogram]]s. These symbols directly represent the concept of 'one' without breaking it down into phonetic components.{{sfn|Crystal|2008|pp=289}}
=== Modern typefaces === {{multiple image | total_width = 400 | image1 = Woodstock typewriter, 1940s, daylight - keyboard.jpg | caption1 = This Woodstock typewriter from the 1940s lacks a separate key for the numeral 1. | image2 = Mediaevalziffern.svg | caption2 = [[Hoefler Text]], a typeface designed in 1991, uses [[text figure]]s and represents the numeral 1 as similar to a small-caps I. | class2 = skin-invert-image }} In modern [[typeface]]s, the shape of the character for the digit 1 is typically typeset as a ''lining figure'' with an [[Ascender (typography)|ascender]], such that the digit is the same height and width as a [[capital letter]]. However, in typefaces with [[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|class=skin-invert-image]].{{sfn|Cullen|2007|p=93}} In many typefaces with text figures, the numeral 1 features parallel serifs at the top and bottom, resembling a [[small caps]] version of the [[Roman numeral]] {{rn|[[I]]}}.<ref>{{cite book|title=Aspects of Contemporary Book Design|first=Richard|last=Hendel|publisher=University of Iowa Press|year=2013|isbn=9781609381752|page=[https://books.google.com/books?id=6GS8gKyxDzoC&pg=PA146 146]}}</ref><ref>{{cite book|title=Designing Information: Human Factors and Common Sense in Information Design|first=Joel|last=Katz|publisher=John Wiley & Sons|year=2012|isbn=9781118420096|page=[https://books.google.com/books?id=V3WcT7Ap3t4C&pg=PA82 82]}}</ref> Many older [[typewriter]]s do not have a dedicated key for the numeral 1, requiring the use of the lowercase letter ''[[L]]'' or uppercase ''[[I]]'' as substitutes.<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=|last=|date=April 2, 2017|work=Post Haste Telegraph Company}}</ref>{{sfn|Polt|2015|pp=203}}{{sfn|Chicago|1993|pp=52}}{{sfn|Guastello|2023|pp=453}}
[[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]] The lower case "{{rn|[[j]]}}" can be considered a [[Swash (typography)|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|page=70}}</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|page=341}}</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|page=3|access-date=December 2, 2023|archive-date=November 13, 2024|archive-url=https://web.archive.org/web/20241113172327/https://books.google.com/books?id=UJ-VoRZUhaYC&dq=JO+JJ&pg=PA3|url-status=live}}</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|page=285}}</ref> In [[German language|German]], 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.{{sfn|Huber|Headrick|1999|pp=181}}
== In other fields == In digital technology, data is represented by [[binary code]], i.e., a [[radix|base]]-2 numeral system with numbers represented by a sequence of 1s and [[0 (number)|0]]s. Digitised data is represented in physical devices, such as [[computer]]s, as pulses of electricity through switching devices such as [[transistor]]s or [[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]]s.{{sfn|Woodford|2006|p=9}}{{sfn|Godbole|2002|p=34}} In [[lambda calculus]] and [[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 {{nobr|(1<math>fx=fx</math>)}}.{{sfn|Hindley|Seldin|2008|p=48}}
In [[physics]], selected [[physical constant]]s are set to 1 in [[natural unit]] systems in order to simplify the form of equations; for example, in [[Planck units]] the [[speed of light]] equals 1.{{sfn|Glick|Darby|Marmodoro|2020|pp=99}} [[Dimensionless quantities]] are also known as 'quantities of dimension one'.{{sfn|Mills|1995|pp=538-539}} In [[quantum mechanics]], the normalization condition for [[wavefunction]]s requires the integral of a wavefunction's squared modulus to be equal to 1.{{sfn|McWeeny|1972|pp=14}} In chemistry, [[hydrogen]], the first element of the [[periodic table]] and the most [[Abundance of the elements|abundant element]] in the known [[universe]], has an [[atomic number]] of 1. Group 1 of the periodic table consists of hydrogen and the [[alkali metal]]s.{{sfn|Emsley|2001}}
In philosophy, the number 1 is commonly regarded as a symbol of unity, often representing God or the universe in [[Monotheism|monotheistic]] traditions.{{sfn|Stewart|2024}} The [[Pythagoreanism|Pythagoreans]] considered the numbers to be plural and therefore did not classify 1 itself as a number, but as the origin of all numbers. In their number philosophy, where odd numbers were considered male and even numbers female, 1 was considered neutral capable of transforming even numbers to odd and vice versa by addition.{{sfn|Stewart|2024}} The [[Neopythagoreanism|Neopythagorean]] philosopher [[Nicomachus|Nicomachus of Gerasa]]'s number treatise, as recovered by [[Boethius]] in the Latin translation ''[[Introduction to Arithmetic]]'', affirmed that one is not a number, but the source of number.<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|url-access=subscription}}</ref> In the philosophy of [[Plotinus]] (and that of other [[neoplatonist]]s), 'The One' is the ultimate reality and source of all existence.{{sfn|Halfwassen|2014|pp=182–183}} [[Philo#Numbers|Philo of Alexandria]] (20 BC – AD 50) regarded the number one as God's number, and the basis for all numbers.<ref>"De Allegoriis Legum", ii.12 [i.66]</ref>
== See also == *[[−1]] *{{annotated link|0.999...}}
== References == {{reflist|2}}
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{{DEFAULTSORT:1 (Number)}} [[Category:1 (number)| ]] [[Category:Integers]]