This article contains special characters. Without proper rendering support, you may see question marks, boxes, or other symbols. Natural number ← 0 1 2 → −1 0 1 2 3 4 5 6 7 8 9 → List of numbersIntegers← 0 10 20 30 40 50 60 70 80 90 →CardinaloneOrdinal1st(first)Numeral systemunaryFactorization∅Divisors1Greek numeralΑ´Roman numeralI, iGreek prefixmono-/haplo-Latin prefixuni-Binary12Ternary13Senary16Octal18Duodecimal112Hexadecimal116Greek numeralα'Arabic, Kurdish, Persian, Sindhi, Urdu١Assamese & Bengali১Chinese numeral一/弌/壹Devanāgarī१Santali᱑Ge'ez፩GeorgianႠ/ⴀ/ა(Ani)HebrewאJapanese numeral一/壱Kannada೧Khmer១ArmenianԱMalayalam൧Meitei꯱Thai๑Tamil௧Telugu೧Babylonian numeral𒐕Egyptian hieroglyph, Aegean numeral, Chinese counting rod𓏤Mayan numeral•Morse code. _ _ _ _
1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural 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 the unit of counting or measurement, 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 a prime number. In 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 mathematicsThe 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 {\displaystyle n} multiplied by 1 remains unchanged ( 1 × n = n × 1 = n {\displaystyle 1\times n=n\times 1=n} ). As a result, the square ( 1 2 = 1 {\displaystyle 1^{2}=1} ), square root ( 1 = 1 {\displaystyle {\sqrt {1}}=1} ), and any other power of 1 is always equal to 1 itself.[1] 1 is its own factorial ( 1 ! = 1 {\displaystyle 1!=1} ), and 0! is also 1. These are a special case of the empty product.[2] 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.[3]
multiplied by 1 remains unchanged (
1
×
n
=
n
×
1
=
n
{\displaystyle 1\times n=n\times 1=n}
). As a result, the square (
1
2
=
1
{\displaystyle 1^{2}=1}
), square root (
1
=
1
{\displaystyle {\sqrt {1}}=1}
), and any other power of 1 is always equal to 1 itself.[1] 1 is its own factorial (
1
!
=
1
{\displaystyle 1!=1}
), and 0! is also 1. These are a special case of the empty product.[2] 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.[3]
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.[4][5] Peano later revised his axioms to begin the sequence with 0.[4][6] 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 { 0 } {\displaystyle \{0\}} , a set containing only the element 0.[7] 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.[8][9]
, a set containing only the element 0.[7]
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.[8][9]
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.[10] Likewise, vectors are often normalized into unit vectors (i.e., vectors of magnitude one), because these often he more desirable properties. Functions are often normalized by the condition that they he integral one, maximum value one, or square integral one, depending on the application.[11]
1 is the value of Legendre's constant, introduced in 1808 by Adrien-Marie Legendre to express the asymptotic behior of the prime-counting function.[12] The Weil's conjecture on Tamagawa numbers states that the Tamagawa number τ ( G ) {\displaystyle \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').[13][14]
, 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').[13][14]
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 {\displaystyle d} is log 10 ( d + 1 d ) {\textstyle \log _{10}\left({\frac {d+1}{d}}\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.[15]
As a word See also: One (pronoun)
is
log
10
(
d
+
1
d
)
{\textstyle \log _{10}\left({\frac {d+1}{d}}\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.[15]
One originates from the Old English word an, derived from the Germanic root *ainaz, from the Proto-Indo-European root *oi-no- (meaning "one, unique").[16] Linguistically, one is a cardinal number used for counting and expressing the number of items in a collection of things.[17] One is most commonly a determiner used with singular countable nouns, as in one day at a time.[18] The determiner has two senses: numerical one (I he one apple) and singulative one (one day I'll do it).[19] 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.[20]
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.[21] 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.[22][23]
Symbols and representation History See also: History of the Hindu–Arabic numeral systemAmong 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.[24] Archaic Sumerian numerals for 1 and 60 both consisted of horizontal semi-circular symbols, [25] by c. 2350 BCE, the older Sumerian curviform numerals were replaced with cuneiform symbols, with 1 and 60 both represented by the same mostly vertical symbol.
The Sumerian cuneiform system is a direct ancestor to the Eblaite and Assyro-Babylonian Semitic cuneiform decimal 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 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.[28] This script's numeral shapes were transmitted to Europe via the Maghreb and Al-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)) are logograms. 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, 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 
.[31] 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 I.[32][33] Many older typewriters do not he a dedicated key for the numeral 1, requiring the use of the lowercase letter L or uppercase I as substitutes.[34][35][36][37]
The 24-hour tower clock in Venice, using J as a symbol for 1
The lower case "j" can be considered a swash variant of a lower-case Roman numeral "i", often employed for the final 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.[38][39][40][41] 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.[42]
In other fieldsIn 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.[43][44] 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 {\displaystyle f} applied to an argument x {\displaystyle x} once (1 f x = f x {\displaystyle fx=fx} ).[45]
applied to an argument
x
{\displaystyle x}
once (1
f
x
=
f
x
{\displaystyle fx=fx}
).[45]
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.[46] Dimensionless quantities are also known as 'quantities of dimension one'.[47] In quantum mechanics, the normalization condition for wefunctions requires the integral of a wefunction's squared modulus to be equal to 1.[48] 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.[49]
In philosophy, the number 1 is commonly regarded as a symbol of unity, often representing God or the universe in monotheistic traditions.[50] 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.[50] 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.[51] In the philosophy of Plotinus (and that of other neoplatonists), 'The One' is the ultimate reality and source of all existence.[52] Philo of Alexandria (20 BC – AD 50) regarded the number one as God's number, and the basis for all numbers.[53]
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Retrieved February 24, 2022. ^ Conway & Guy 1996, p. 4. ^ a b Conway & Guy 1996, p. 17. ^ Chrisomalis 2010, p. 241. ^ Chrisomalis 2010, p. 244. ^ Chrisomalis 2010, p. 249. ^ Acharya, Eka Ratna (2018). "Evidences of Hierarchy of Brahmi Numeral System". Journal of the Institute of Engineering. 14 (1): 136–142. doi:10.3126/jie.v14i1.20077. ^ Schubring 2008, pp. 147. ^ Crystal 2008, pp. 289. ^ Cullen 2007, p. 93. ^ Hendel, Richard (2013). Aspects of Contemporary Book Design. University of Iowa Press. p. 146. ISBN 9781609381752. ^ Katz, Joel (2012). Designing Information: Human Factors and Common Sense in Information Design. John Wiley & Sons. p. 82. ISBN 9781118420096. ^ "Why Old Typewriters Lack A "1" Key". Post Haste Telegraph Company. April 2, 2017. ^ Polt 2015, pp. 203. ^ Chicago 1993, pp. 52. ^ Guastello 2023, pp. 453. ^ Köhler, Christian (November 23, 1693). "Der allzeitfertige Rechenmeister". p. 70 – via Google Books. ^ "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 ..." by Jan ten Hoorn. November 23, 1679. p. 341 – via Google Books. ^ "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". Heußler. November 23, 1586. p. 3. Archived from the original on November 13, 2024. Retrieved December 2, 2023 – via Google Books. ^ August (Herzog), Braunschweig-Lüneburg (November 23, 1624). "Gusti 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". Johann & Heinrich Stern. p. 285 – via Google Books. ^ Huber & Headrick 1999, pp. 181. ^ Woodford 2006, p. 9. ^ Godbole 2002, p. 34. ^ Hindley & Seldin 2008, p. 48. ^ Glick, Darby & Marmodoro 2020, pp. 99. ^ Mills 1995, pp. 538–539. ^ McWeeny 1972, pp. 14. ^ Emsley 2001. ^ a b Stewart 2024. ^ British Society for the History of Science (July 1, 1977). "From Abacus to Algorism: Theory and Practice in Medieval Arithmetic". The British Journal for the History of Science. 10 (2). Cambridge University Press: Abstract. doi:10.1017/S0007087400015375. S2CID 145065082. Archived from the original on May 16, 2021. Retrieved May 16, 2021. ^ Halfwassen 2014, pp. 182–183. ^ "De Allegoriis Legum", ii.12 [i.66] Sources Blokhintsev, D. I. (2012). 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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. JFM 21.0051.02. Peano, Giuseppe (1908). Formulario Mathematico [Mathematical Formulary] (V ed.). Turin: Fratres Bocca. pp. xxxvi, 1–463. JFM 39.0084.01. Pintz, Janos (1980). "On Legendre's Prime Number Formula". The American Mathematical Monthly. 87 (9): 733–735. doi:10.2307/2321863. ISSN 0002-9890. JSTOR 2321863. Polt, Richard (2015). The Typewriter Revolution: A Typist's Companion for the 21st Century. The Countryman Press. ISBN 978-1581575873. 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. 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Retrieved 2016-03-24. vteIntegers−2, −10s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100s 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200s 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 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1023 1024 1089 1093 1105 1234 1289 1458 1510 1728 1729 1980 1987 2000 2016 2520 3000 3511 4000 4104 5000 5040 6000 6174 7000 7744 7825 8000 8128 8192 9000 9855 9999 10,000 16,807 20,000 30,000 40,000 50,000 60,000 64,079 65,535 65,536 65,537 70,000 80,000 90,000 100,000 142,857 144,000 1,000,000 10,000,000 43,112,609 100,000,000 1,000,000,000 2,147,483,647 4,294,967,295 10,000,000,000 100,000,000,000 1,000,000,000,000 10,000,000,000,000 vteNumber theoryFields Algebraic number theory (class field theory, non-abelian class field theory, Iwasawa theory, Iwasawa–Tate theory, Kummer theory) Analytic number theory (analytic theory of L-functions, probabilistic number theory, sieve theory) Geometric number theory Computational number theory Transcendental number theory Diophantine geometry (Arakelov theory, Hodge–Arakelov theory) Arithmetic combinatorics (additive number theory) Arithmetic geometry (anabelian geometry, p-adic Hodge theory) Arithmetic topology Arithmetic dynamics Key concepts Numbers 0 Natural numbers Unity Prime numbers Composite numbers Rational numbers Irrational numbers Algebraic numbers Transcendental numbers p-adic numbers (p-adic analysis) Arithmetic Modular arithmetic Chinese remainder theorem Arithmetic functions Advanced concepts Quadratic forms Modular forms L-functions Diophantine equations Diophantine approximation Irrationality measure Simple continued fractions
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