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	. _ _ _ _

1 (one, unit, unity) is a number, numeral, and glyph. 1 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; this was not universally accepted until the mid-20th century. 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 mathematics

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 ${\displaystyle n}$ multiplied by 1 remains unchanged (${\displaystyle 1n=n1=n}$). As a result, 1 own square (${\displaystyle 1^{2}=1}$) and square root (${\displaystyle {\sqrt {1}}=1}$), and one raised to any power is always equal to 1. ^[1] 1 is its own factorial (${\displaystyle 1!=1}$), and 0! is also one 1. These are a special case of the empty product. ^[2]

Different constructions of the natural numbers have different representations of 1. For example, in the original formulation of the Peano axioms, 1 serves as the starting point in the sequence of natural numbers. ^[3] Peano later revised his axioms to state 1 is the successor of 0. ^[4] In the Von Neumann cardinal assignment of natural numbers, numbers are defined as the set containing all preceding numbers, with 1 represented as the singleton {0}. ^[5] 1 is both the first and second number in the Fibonacci sequence (0 being the zeroth) and is the first number in many other mathematical sequences.

The number 1 can be represented in decimal form by two recurring notations: 1.000..., where the digit 0 repeats infinitely after the decimal point, and 0.999..., which contains an infinite repetition of the digit 9 after the decimal point. The latter arises from the definition of decimal numbers as the limits of their summed components, such that "0.999..." and "1" represent exactly the same number. ^[6]

The simplest way to represent the natural numbers is by the unary numeral system, as used in tallying. ^[7] This is an example of a "base-1" number system, since only one mark – the tally itself – is needed, although base-1 is rarely used as a practical base for counting due to its difficult readability. ^[8]

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. ^[9] 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. ^{[10] [11]}

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. ^{[12] [13]}

1 is the most common leading digit in many sets of data (occurring about 30% of the time), a consequence of Benford's law. ^[14]

1 is the only known Tamagawa number for all simply connected algebraic groups over a number field. ^{[15] [16]}

Primality

Main article: Prime number § Primality of one

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 nor a composite number. ^[17] Some mathematicians of the Middle Ages and Renaissance considered 1 a prime number. ^[18] The 18th century mathematician Christian Goldbach listed 1 as prime in his correspondence with Leonhard Euler, ^[19] and many 19th century mathematicians still considered 1 to be prime. ^[18] By the early 20th century, mathematicians started to agree that 1 should not be classified as a prime number. ^[20] However, published lists of primes continued to include 1 as recently as 1956. ^{[21] [22]}

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		11	12	13	14	15	16	17	18	19	20
1x	1	1	1	1	1	1	1	1	1	1		1	1	1	1	1	1	1	1	1	1
x1	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15	16	17	18	19	20

As a word

See also: One (pronoun)

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"). ^[23]

Modern usage

Linguistically, one is a cardinal number used for counting and expressing the number of items in a collection of things. ^[24] One is most commonly a determiner used with singular countable nouns, as in one day at a time. ^[25] The determiner has two senses: numerical one (I have one apple) and singulative one (one day I'll do it). ^[26]

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. ^[27]

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. ^[28] 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. ^{[29] [30]}

The various glyphs used to represent the number one, including Arabic numerals (1), Roman numerals (I), and Chinese numerals (一), are logograms. These symbols directly represent the concept of 'one' without breaking it down into phonetic components. ^[31]

Symbols and representation

History

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. ^[32] The Archaic Sumerian numerals for 1 and 60 both consisted of horizontal semi-circular symbols. ^[33] 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. ^[34] Surviving Babylonian documents date mostly from Old Babylonian (c. 1500 BCE) and the Seleucid (c. 300 BCE) eras. ^[32] The Babylonian cuneiform script notation for numbers used the same symbol for 1 and 60 as in the Sumerian system. ^[35]

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. ^[36] This script's numeral shapes were transmitted to Europe via the Maghreb and Al-Andalus during the Middle Ages ^[37]

Modern typefaces

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 . ^[38] 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. ^[39] Many older typewriters do not have dedicated key for the numeral 1, requiring the use of the lowercase letter l or uppercase I as substitutes. ^{[40] [41] [42] [43]} 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. ^{[44] [45] [46] [47]} 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. ^[48]

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. ^{[49] [50]} 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 ${\displaystyle f}$ applied to an argument ${\displaystyle x}$ once (1${\displaystyle fx=fx}$). ^[51]

In science

Physics

Dimensionless quantities are also known as quantities of dimension one. ^[52] In physics, selected physical constants are set to 1 in natural unit systems (for example in Planck units the speed of light c=1) to simplify the form of equations. ^[53] In quantum mechanics, the normalization condition wavefunctions requires the integral of a wavefunction's squared modulus to be equal to 1. ^[54]

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. ^[55]

In philosophy and religion

Main article: Numerology

The number 1 is commonly regarded as a symbol of unity, often representing God or the universe in monotheistic traditions. ^[56] 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. ^[56] 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. ^[57] In the philosophy of Plotinus (and that of other neoplatonists), The One is the ultimate reality and source of all existence. ^[58] Philo of Alexandria (20 BC – AD 50) regarded the number one as God's number, and the basis for all numbers. ^[59]

See also

• Mathematics portal

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

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