1

This article is about the number. For the year AD 1, and other uses, see One (disambiguation) and Number One (disambiguation).

← 0 1 2 →
−1 0 1 2 3 4 5 6 7 8 9 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	one
Ordinal	1st (first)
Numeral system	unary
Factorization	∅
Divisors	1
Greek numeral	Α´
Roman numeral	I, i
Greek prefix	mono-/haplo-
Latin prefix	uni-
Binary	12
Ternary	13
Senary	16
Octal	18
Duodecimal	112
Hexadecimal	116
Greek numeral	α'
Arabic, Kurdish, Persian, Sindhi, Urdu	١
Assamese & Bengali	১
Chinese numeral	一/弌/壹
Devanāgarī	१
Ge'ez	፩
Georgian	Ⴀ/ⴀ/ა(Ani)
Hebrew	א
Japanese numeral	一/壱
Kannada	೧
Khmer	១
Armenian	Ա
Malayalam	൧
Meitei	꯱
Thai	๑
Tamil	௧
Telugu	೧
Babylonian numeral	𒐕
Egyptian hieroglyph, Aegean numeral, Chinese counting rod	𓏤
Mayan numeral	•
Morse code	. _ _ _ _
British Sign Language

1 (one, unit, unity) is a number representing a single unit of counting or measurement. 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number. It commonly denotes the first, leading, or top thing in a group.

1 is usually a multiplicative identity. Most if not all properties of 1 stem from this property. 1 is by convention not considered a prime number.

One commonly denotes the first, leading, or top thing in a group.

As a word

Etymology

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").

Modern usage

Linguistically, one is a cardinal number used for counting and expressing the number of items in a collection of things. One is commonly used as a determiner for singular countable nouns, as in one day at a time. 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. 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. 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.

Symbols and representation

The 24-hour tower clock in Venice, using J as a symbol for 1

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.

See also: History of the Hindu–Arabic numeral system

Among the earliest known record of a numeral system, is the Sumerian decimal-sexagesimal system on clay tablets dating from the first half of the third millennium BCE. The Archaic Sumerian numerals for 1 and 60 both consisted of horizontal semi-circular symbols. By c. 2350 BCE, the older Sumerian curviform numerals were replaced with cuneiform symbols, with 1 and 60 both represented by the same symbol . The Sumerian cuneiform system is a direct ancestor to the Eblaite and Assyro-Babylonian Semitic cuneiform decimal systems. Surviving Babylonian documents date mostly from Old Babylonian (c. 1500 BCE) and the Seleucid (c. 300 BCE) eras. The Babylonian cuneiform script notation for numbers used the same symbol for 1 and 60 as in the Sumerian system.

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. This script's numeral shapes were transmitted to Europe via the Maghreb and Al-Andalus during the Middle Ages, through scholarly works written in Arabic. 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 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.

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 . In old-style typefaces (e.g., Hoefler Text), the typeface for numeral 1 resembles a small caps version of I, featuring parallel serifs at the top and bottom, while the capital I retains a full-height form. This is a relic from the Roman numerals system where I represents 1. 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. 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.

In mathematics

In normal arithmetic (algebra), the number 1 is the first natural number after 0.

In algebraic structures such as multiplicative groups and monoids the identity element is often denoted 1, but e (from the German Einheit, "unity") is also traditional. 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present.

Any number $n$ multiplied or divided by 1 remains unchanged. This makes it a mathematical unit, and for this reason, 1 is often called unity.

A matrix of ones or all-ones matrix is defined as a matrix composed entirely of 1s.

The simplest way to represent the natural numbers is by the unary numeral system, as used in tallying. This is often referred to as "base 1", since only one mark – the tally itself – is needed.

Primality

Main article: Prime number § Primality of one

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 nor a 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. However, the prevailing and enduring mathematical consensus has been to exclude due to its impact upon the fundamental theorem of arithmetic and other theorems related to prime numbers. For example, the fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units, i.e., 4 = 2² represents a unique factorization. However, if units are included, 4 can also be expressed as (−1)⁶ × 1²³ × 2², among infinitely many similar "factorizations". Furthermore, Euler's totient function and the sum of divisors function are different for prime numbers than they are for 1.

Other mathematical attributes and uses

In many mathematical and engineering problems, numeric values are typically normalized to fall within the 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 of an event that is absolutely or almost certain to occur. Likewise, vectors are often normalized into unit vectors (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 one, maximum value one, or square integral one, depending on the application.

In category theory, 1 is the terminal object of a category if there is a unique morphism. In number theory, 1 is the value of Legendre's constant, which was introduced in 1808 by Adrien-Marie Legendre in expressing the asymptotic behavior of the 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 Charles Jean de la Vallée Poussin.

1 is the most common leading digit in many sets of data (occurring about 30% of the time), a consequence of Benford's law.

1 is the only known Tamagawa number for a simply connected algebraic group over a number field.

Table of basic calculations

Multiplication	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	50	100	1000
1 × x	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	50	100	1000

Division	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
1 ÷ x	1	0.5	0.3	0.25	0.2	0.16	0.142857	0.125	0.1	0.1	0.09	0.083	0.076923	0.0714285	0.06
x ÷ 1	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

Exponentiation	1	2	3	4	5	6	7	8	9	10	20
1x	1	1	1	1	1	1	1	1	1	1	1
x1	1	2	3	4	5	6	7	8	9	10	20

In technology

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.

In science

• Dimensionless quantities are also known as quantities of dimension one.
• Hydrogen, the first element of the periodic table, has an atomic number of 1.
• Group 1 of the periodic table consists of the alkali metals.
• Period 1 of the periodic table consists of just two elements, hydrogen and helium.

In philosophy

In the philosophy of Plotinus (and that of other neoplatonists), The One is the ultimate reality and source of all existence. 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]).

The Neopythagorean philosopher 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 otherness. His number theory was recovered by Boethius in his Latin translation of Nicomachus's treatise Introduction to Arithmetic.

See also

• −1
• +1 (disambiguation)
• List of mathematical constants
• One (word)
• Root of unity

Sources

• Awodey, Steve (2010). Category Theory (2 ed.). Oxford, UK: Oxford University Press. pp. xv, 1–336. ISBN 978-0-19-958-736-0. Zbl 1291.00036.
• Blokhintsev, D. I. (2012). Quantum Mechanics.
• Caldwell, Chris K.; Xiong, Yeng (2012). "What is the smallest prime?". Journal of Integer Sequences. 15 (9, Article 12.9.7). Waterloo, CA: University of Waterloo David R. Cheriton School of Computer Science: 1–14. MR 3005530. Zbl 1285.11001.
• Chrisomalis, Stephen (2010). Numerical Notation: A Comparitive History. New York: Cambridge University Press. doi:10.1017/CBO9780511676062. ISBN 978-0-521-87818-0.
• Conway, John H.; Guy, Richard K. (1996). The Book of Numbers. New York: Copernicus Publications. doi:10.1007/978-1-4612-4072-3. ISBN 0614971667.
• Cullen, Kristin (2007). Layout Workbook: A Real-World Guide to Building Pages in Graphic Design. Gloucester, MA: Rockport Publishers. pp. 1–240. ISBN 978-1-592-533-527.
• Curtiss, D. R. (1922). "On Kellogg's diophantine problem". American Mathematical Monthly. 29 (10): 380–387. doi:10.2307/2299023. JSTOR 2299023.
• Gaitsgory, Dennis; Lurie, Jacob (2019). Weil's Conjecture for Function Fields (Volume I). Annals of Mathematics Studies. Vol. 199. Princeton: Princeton University Press. pp. viii, 1–311. doi:10.2307/j.ctv4v32qc. ISBN 978-0-691-18213-1. MR 3887650. Zbl 1439.14006.
• Godbole, Achyut S. (2002). Data Comms & Networks. Tata McGraw-Hill Education. ISBN 978-1-259-08223-8.
• Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1988). Concrete Mathematics. Reading, MA: Addison-Wesley. ISBN 0-201-14236-8.
• Halmos, Paul R. (1974). Naive Set Theory. Undergraduate Texts in Mathematics. Springer. pp. vii, 1–104. doi:10.1007/978-1-4757-1645-0. ISBN 0-387-90092-6. MR 0453532.
• Hindley, J. Roger; Seldin, Jonathan P. (2008). Lambda-Calculus and Combinators: An Introduction (2nd ed.). Cambridge, UK: Cambridge University Press. pp. xi, 1–358. ISBN 978-1-139-473-248. MR 2435558.
• Hodges, Andrew (2009). One to Nine: The Inner Life of Numbers. New York, NY: W. W. Norton & Company. pp. 1–330. ISBN 9780385672665. S2CID 118490841.
• Horn, Roger A.; Johnson, Charles R. (2012). "0.2.8 The all-ones matrix and vector". Matrix Analysis. Cambridge University Press. p. 8. ISBN 9780521839402..
• Huddleston, Rodney D.; Pullum, Geoffrey K.; Reynolds, Brett (2022). A student's Introduction to English Grammar (2nd ed.). Cambridge: Cambridge University Press. pp. 1–418. ISBN 978-1-316-51464-1. OCLC 1255524478.
• Hurford, James R. (1994). Grammar: A Student's Guide. Cambridge, UK: Cambridge University Press. pp. 1–288. ISBN 978-0-521-45627-2. OCLC 29702087.
• Kottwitz, Robert E. (1988). "Tamagawa numbers". Annals of Mathematics. 2. 127 (3). Princeton, NJ: Princeton University & the Institute for Advanced Study: 629–646. doi:10.2307/2007007. JSTOR 2007007. MR 0942522.
• Miller, G. A. (1919). "Groups possessing a small number of sets of conjugate operators". Transactions of the American Mathematical Society. 20 (3): 260–270. doi:10.2307/1988867. JSTOR 1988867.
• Miller, Steven J., ed. (2015). Benford's law: theory and applications. Princeton, NJ: Princeton University Press. pp. xxvi, 1–438. ISBN 978-0-691-14761-1. MR 3408774.
• Nathanson, Melvyn B. (January 2023). "Underapproximation by Egyptian fractions". Journal of Number Theory. 242: 208–234. arXiv:2202.00191. doi:10.1016/j.jnt.2022.07.005.
• Olson, Roger (2017). The Essentials of Christian Thought: Seeing Reality through the Biblical Story. Grand Rapids, MI: Zondervan Academic. pp. 1–252. ISBN 9780310521563.
• 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. JFM 21.0051.02.
• Peano, Giuseppe (1908). Formulario Mathematico [Mathematical Formulary] (V ed.). Turin: Fratres Bocca. pp. xxxvi, 1–463. JFM 39.0084.01.
• Rosenman, Martin; Underwood, F. (1933). "Problem 3536". American Mathematical Monthly. 40 (3): 180–181. doi:10.2307/2301036. JSTOR 2301036.
• Salzer, H. E. (1947). "The approximation of numbers as sums of reciprocals". American Mathematical Monthly. 54 (3): 135–142. doi:10.2307/2305906. JSTOR 2305906. MR 0020339.
• Sandifer, C. Edward (2007). How Euler Did It. The MAA Euler Celebration. Vol. III. Washington, DC: Mathematical Association of America. pp. 1–237. ISBN 978-0-88385-563-8. MR 2321397.
• Sierpiński, Wacław (1988). Elementary Theory of Numbers. North-Holland Mathematical Library. Vol. 31 (2nd ed.). Elsevier. pp. 1–513. ISBN 978-0-08-096019-7. MR 0930670.
• Soundararajan, K. (2005). "Approximating 1 from below using n Egyptian fractions". arXiv:math.CA/0502247.
• Stillwell, John (1994). Elements of Algebra: Geometry, Numbers, Equations. Springer-Verlag. pp. xi, 1–181. ISBN 9783540942900. MR 1311026. Zbl 0832.00001.
• Sung, Kelvin; Smith, Gregory (2019). Basic Math for Game Development with Unity 3D: A Beginner's Guide to Mathematical Foundations.