1#As a digit

The number 1 is the first natural number after 0. Each natural number, including 1, is constructed by succession, that is, by adding 1 to the previous natural number. The number 1 is the multiplicative identity of the integers, real numbers, and complex numbers, that is, any number $n$ multiplied by 1 remains unchanged ($1\backslash times\; n\; =\; n\backslash times\; 1\; =\; n$). As a result, the square ($1^2=1$), square root ($\backslash sqrt\{1\}\; =\; 1$), and any other power of 1 is always equal to 1 itself.{{sfn|Colman|1912|loc=chapt.2|pp=9–10}} 1 is its own factorial ($1!=1$), and 0! is also 1. These are a special case of the empty product.{{sfn|Graham|Knuth|Patashnik|1994|p=111}} 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 nor a composite number.{{sfn|Caldwell|Xiong|2012|pp=8–9}}

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 that contains all numbers before it, 1 is represented as the singleton $\backslash \{0\backslash \}$, a set containing only the element 0.{{sfn|Halmos|1974|p=32}}

The unary numeral system, as used in 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 to fall within the unit interval ([0,1]), 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, vectors are often normalized into unit vectors (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 integral one, depending on the application.{{sfn|Blokhintsev|2012|p=35}}

1 is the value of Legendre's constant, introduced in 1808 by Adrien-Marie Legendre to express the 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 $\backslash tau(G)$, a geometrical measure of a connected linear algebraic group over a global number field, is 1 for all simply connected groups (those that are path-connected with no 'holes').{{sfn|Gaitsgory|Lurie|2019|pp=204–307}}{{sfn|Kottwitz|1988}}

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 $d$ is $\backslash log\_\{10\}\; \backslash left(\backslash frac\{d+1\}\{d\}\; \backslash right)$. 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}}

{{See also|One (pronoun)}}

One originates from the Old English word an, derived from the Germanic root {{not a typo|{{wikt-lang|gem-x-proto|*ainaz}}}}, from the Proto-Indo-European root *oi-no- (meaning "one, unique").{{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 }} 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 determiner used with singular countable 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 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 prefixes 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.{{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 }}{{sfn|Conway|Guy|1996|p=4}}

{{see also|History of the Hindu–Arabic numeral system}}

Among the earliest known records of a numeral system, is the Sumerian decimal-sexagesimal system on clay tablets dating from the first half of the 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

The Sumerian cuneiform system is a direct ancestor to the Eblaite and Assyro-Babylonian Semitic cuneiform decimal systems.{{sfn|Chrisomalis|2010|p=244}} Surviving 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 commonly used glyph in the modern Western world to represent the number 1 is the 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 Brahmic script of ancient India, as represented by Ashoka as a simple vertical line in his Edicts of Ashoka in c. 250 BCE.{{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 }} 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 numeral, and other glyphs used to represent the number one (e.g., Roman numeral ({{rn|I}} ), Chinese numeral ({{zhi|c=一}})) are logograms. These symbols directly represent the concept of 'one' without breaking it down into phonetic components.{{sfn|Crystal|2008|pp=289}}

{{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 figures and represents the numeral 1 as similar to a small-caps I.

}}

In modern typefaces, 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 (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.{{sfn|Cullen|2007|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.{{Cite web|url=https://www.typography.com/|title=Fonts by Hoefler&Co.|website=www.typography.com|access-date=November 21, 2023|archive-date=November 23, 2024|archive-url=https://web.archive.org/web/20241123092348/https://www.typography.com/|url-status=live}} Many older typewriters do not have a dedicated key for the numeral 1, requiring the use of the lowercase letter L or uppercase I as substitutes.{{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}}{{sfn|Polt|2015|pp=203}}{{sfn|Chicago|1993|pp=52}}{{sfn|Guastello|2023|pp=453}}

File:Clock 24 J.jpg, using J as a symbol for 1]]

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.{{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}}{{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}}{{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}}{{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}} In 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 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 digital technology, data is represented by binary code, i.e., a base-2 numeral system with numbers represented by a sequence of 1s and 0s. Digitised data is represented in physical devices, such as computers, as pulses of electricity through switching devices such as transistors or logic gates where "1" represents the value for "on". As such, the numerical value of true is equal to 1 in many programming languages.{{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 $f$ applied to an argument $x$ once {{nobr|(1$fx=fx$)}}.{{sfn|Hindley|Seldin|2008|p=48}}

In physics, selected physical constants 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 wavefunctions 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 abundant element in the known universe, has an atomic number of 1. Group 1 of the periodic table consists of hydrogen and the alkali metals.{{sfn|Emsley|2001}}

In philosophy, the number 1 is commonly regarded as a symbol of unity, often representing God or the universe in monotheistic traditions.{{sfn|Stewart|2024}} The 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 Neopythagorean philosopher 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.{{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}} In the philosophy of Plotinus (and that of other neoplatonists), 'The One' is the ultimate reality and source of all existence.{{sfn|Halfwassen|2014|pp=182–183}} Philo of Alexandria (20 BC – AD 50) regarded the number one as God's number, and the basis for all numbers."De Allegoriis Legum", ii.12 [i.66]

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• {{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 |issn=0002-9890 |jstor=2321863 |url-access=subscription }}
• {{cite book| last=Polt |first=Richard |year=2015| title=The Typewriter Revolution: A Typist's Companion for the 21st Century |publisher=The Countryman Press|isbn=978-1581575873}}
• {{Cite book |last1=Radford |first1=Luis |last2=Schubring |first2=Gert |last3=Seeger |first3=Falk |year=2008 |title=Semiotics in Mathematics Education: Epistemology, History, Classroom, and Culture |series=Semiotic Perspectives in the Teaching and Learning of Math Series |volume=1 |publisher=Sense Publishers |editor-last=Kaiser|editor-first=Gabriele |location=Netherlands |isbn=978-9087905972 | contributor-last = Schubring | contributor-first = Gert|contribution=Processes of Algebraization}}
• {{cite encyclopedia |title=Number Symbolism |encyclopedia=Brittanica |year=2024 |last=Stewart |first=Ian |url=https://www.britannica.com/topic/number-symbolism |access-date=2024-08-21 |archive-date=2008-07-26 |archive-url=https://web.archive.org/web/20080726140908/http://www.britannica.com/eb/article-248155/number-symbolism |url-status=live }}
• {{Cite book |first1=Chris |last1=Woodford |author1-link=Chris Woodford (author) |url={{GBurl|id=My7Zr0aP2L8C|p=9}} |title=Digital Technology |date=2006 |publisher=Evans Brothers |isbn=978-0-237-52725-9 |access-date=2016-03-24 }}

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Category:Integers