1 (one,unit,unity) is anumber,numeral, andglyph. It is the first and smallestpositive integer of the infinite sequence ofnatural numbers. 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 theunit ofcounting ormeasurement, a determiner for singular nouns, and a gender-neutral pronoun. Historically, the representation of 1 evolved from ancient Sumerian and Babylonian symbols to the modern Arabic numeral.
In mathematics, 1 is the multiplicative identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered aprime number. Indigital technology, 1 represents the "on" state inbinary code, the foundation ofcomputing. Philosophically, 1 symbolizes the ultimate reality or source of existence in various traditions.
In mathematics
The number 1 is the first natural number after0. Eachnatural number, including 1, is constructed bysuccession, that is, by adding 1 to the previous natural number. The number 1 is themultiplicative identity of theintegers,real numbers, andcomplex numbers, that is, any number multiplied by 1 remains unchanged (). As a result, thesquare (),square root (), and any other power of 1 is always equal to 1 itself.[1] 1 is its ownfactorial (), and 0! is also 1. These are a special case of theempty product.[2] Although 1 meets the naïve definition of aprime number, being evenly divisible only by 1 and itself (also 1), by modern convention it is regarded as neither aprime nor acomposite number.[3]
Different mathematical constructions of the natural numbers represent 1 in various ways. InGiuseppe Peano's original formulation of thePeano 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.[4][5] Peano later revised his axioms to begin the sequence with 0.[4][6] In theVon Neumann cardinal assignment of natural numbers, where each number is defined as aset that contains all numbers before it, 1 is represented as thesingleton, a set containing only the element 0.[7]Theunary numeral system, as used intallying, 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 forcounting due to its difficult readability.[8][9]
In many mathematical and engineering problems, numeric values are typicallynormalized to fall within theunit interval ([0,1]), where 1 represents the maximum possible value. For example, by definition 1 is theprobability of an event that is absolutely oralmost certain to occur.[10] Likewise,vectors are often normalized intounit vectors (i.e., vectors of magnitude one), because these often have more desirable properties. Functions are often normalized by the condition that they haveintegral one, maximum value one, orsquare integral one, depending on the application.[11]
1 is the most common leading digit in many sets of real-world numerical data. This is a consequence ofBenford’s law, which states that the probability for a specific leading digit is. 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.[15]
One originates from theOld English wordan, derived from theGermanic root*ainaz, from theProto-Indo-European root*oi-no- (meaning "one, unique").[16] Linguistically,one is acardinal number used for counting and expressing the number of items in a collection of things.[17]One is most commonly adeterminer used withsingular countablenouns, as inone day at a time.[18] The determiner has two senses: numerical one (I have one apple) and singulative one (one day I'll do it).[19]One is also a gender-neutralpronoun used to refer to an unspecifiedperson or to people in general as inone should take care of oneself.[20]
Words that derive their meaning fromone includealone, which signifiesall one in the sense of being by oneself,none meaningnot one,once denotingone time, andatone meaning to becomeat one with the someone. Combiningalone withonly (implyingone-like) leads tolonely, conveying a sense of solitude.[21] Other commonnumeral prefixes for the number 1 includeuni- (e.g.,unicycle, universe, unicorn),sol- (e.g., solo dance), derived from Latin, ormono- (e.g.,monorail, monogamy, monopoly) derived from Greek.[22][23]
Among the earliest known records of a numeral system, is theSumerian decimal-sexagesimal system onclay tablets dating from the first half of thethird millennium BCE.[24] Archaic Sumerian numerals for 1 and 60 both consisted of horizontal semi-circular symbols,[25] byc. 2350 BCE, the older Sumerian curviform numerals were replaced withcuneiform symbols, with 1 and 60 both represented by the same mostly vertical symbol.
The Sumerian cuneiform system is a direct ancestor to theEblaite andAssyro-BabylonianSemitic cuneiformdecimal systems.[26] Surviving Babylonian documents date mostly from Old Babylonian (c. 1500 BCE) and the Seleucid (c. 300 BCE) eras.[24] The Babylonian cuneiform script notation for numbers used the same symbol for 1 and 60 as in the Sumerian system.[27]
The most commonly used glyph in the modern Western world to represent the number 1 is theArabic numeral, a vertical line, often with aserif at the top and sometimes a short horizontal line at the bottom. It can be traced back to theBrahmic script of ancient India, as represented byAshoka as a simple vertical line in hisEdicts of Ashoka in c. 250 BCE.[28] This script's numeral shapes were transmitted to Europe via theMaghreb andAl-Andalus during the Middle Ages[29] The Arabic numeral, and other glyphs used to represent the number one (e.g., Roman numeral (I ), Chinese numeral (一)) arelogograms. These symbols directly represent the concept of 'one' without breaking it down into phonetic components.[30]
Modern typefaces
This Woodstock typewriter from the 1940s lacks a separate key for the numeral 1.
Hoefler Text, a typeface designed in 1991, usestext figures and represents the numeral 1 as similar to a small-caps I.
In moderntypefaces, the shape of the character for the digit 1 is typically typeset as alining figure with anascender, such that the digit is the same height and width as acapital letter. However, in typefaces withtext figures (also known asOld style numerals ornon-lining figures), the glyph usually is ofx-height and designed to follow the rhythm of the lowercase, as, for example, in.[31] Inold-style typefaces (e.g.,Hoefler Text), the typeface for numeral 1 resembles asmall caps version ofI, featuring parallel serifs at the top and bottom, while the capitalI retains a full-height form. This is a relic from theRoman numerals system whereI represents 1.[32] Many oldertypewriters do not have a dedicated key for the numeral 1, requiring the use of the lowercase letterL or uppercaseI as substitutes.[33][34][35][36]
The 24-hour tower clock inVenice, usingJ as a symbol for 1
The lower case "j" can be considered aswash variant of a lower-case Roman numeral "i", often employed for the finali of a "lower-case" Roman numeral. It is also possible to find historic examples of the use ofj orJ as a substitute for the Arabic numeral 1.[37][38][39][40] 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 forseven 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.[41]
In other fields
In digital technology, data is represented bybinary code, i.e., abase-2 numeral system with numbers represented by a sequence of 1s and0s. Digitised data is represented in physical devices, such ascomputers, as pulses of electricity through switching devices such astransistors orlogic gates where "1" represents the value for "on". As such, the numerical value oftrue is equal to 1 in manyprogramming languages.[42][43] Inlambda calculus andcomputability theory, natural numbers are represented byChurch encoding as functions, where the Church numeral for 1 is represented by the function applied to an argument once(1).[44]
In philosophy, the number 1 is commonly regarded as a symbol of unity, often representing God or the universe inmonotheistic traditions.[49] 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.[49] TheNeopythagorean philosopherNicomachus of Gerasa's number treatise, as recovered byBoethius in the Latin translationIntroduction to Arithmetic, affirmed that one is not a number, but the source of number.[50] In the philosophy ofPlotinus (and that of otherneoplatonists), 'The One' is the ultimate reality and source of all existence.[51]Philo of Alexandria (20 BC – AD 50) regarded the number one as God's number, and the basis for all numbers.[52]
Emsley, John (2001).Nature's Building Blocks: An A-Z Guide to the Elements (illustrated, reprint ed.). Oxford, UK: Oxford University Press.ISBN0198503415.
Glick, David; Darby, George; Marmodoro, Anna (2020).The Foundation of Reality: Fundamentality, Space, and Time. Oxford University Press.ISBN978-0198831501.
Guastello, Stephen J. (2023).Human Factors Engineering and Ergonomics: A Systems Approach (3rd ed.). CRC press.ISBN978-1000822045.
Huddleston, Rodney D.; Pullum, Geoffrey K. (2002).The Cambridge grammar of the English language. Cambridge, UK; New York: Cambridge University Press.ISBN978-0-521-43146-0.
Peano, Giuseppe (1889).Arithmetices principia, nova methodo exposita [The principles of arithmetic, presented by a new method]. An excerpt of the treatise where Peano first presented his axioms, and recursively defined arithmetical operations. Turin: Fratres Bocca. pp. xvi,1–20.JFM21.0051.02.
Polt, Richard (2015).The Typewriter Revolution: A Typist's Companion for the 21st Century. The Countryman Press.ISBN978-1581575873.
Schubring, Gert (2008). "Processes of Algebraization".Semiotics in Mathematics Education: Epistemology, History, Classroom, and Culture. By Radford, Luis; Schubring, Gert; Seeger, Falk. Kaiser, Gabriele (ed.). Semiotic Perspectives in the Teaching and Learning of Math Series. Vol. 1. Netherlands: Sense Publishers.ISBN978-9087905972.
Stewart, Ian (2024)."Number Symbolism".Brittanica.Archived from the original on 2008-07-26. Retrieved2024-08-21.
