1

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

1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 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, following 0.

← 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
Malayalam	൧
Meitei	꯱
Thai	๑
Tamil	௧
Telugu	೧
Counting rod	𝍠

The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is the smallest possible difference between two distinct natural numbers.

The unique mathematical properties of the number have led to its unique uses in other fields, ranging from science to sports. It commonly denotes the first, leading, or top thing in a group.

As a word

Linguistically, one is a cardinal number used for counting and expressing the number of items in a collection of things.^[1] One is commonly used as a determiner for singular countable nouns, as in one day at a time.^[2] One is also a pronoun used to refer to an unspecified person or to people in general as in one should take care of oneself.^[3]

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").^[4]

As a number

The number 1 (often referred to as unity) is the first natural number, followed by 2, and the first integer after 0 (zero). 1 is a unique number that cannot be divided by any number greater than itself so that the result is a natural number, making it a mathematical unit. 1 is the multiplicative identity, and the result of the empty product, such that any number ${\displaystyle n}$  multiplied or divided by 1 remains unchanged (${\displaystyle n\times 1=n/1=n}$ ). As a result, 1 is its own factorial (${\displaystyle 1!=1}$ ), its own square (${\displaystyle 1^{2}=1}$ ) and square root (${\displaystyle {\sqrt {1}}=1}$ ), its own cube (${\displaystyle 1^{3}=1}$ ) and cube root (${\displaystyle {\sqrt[{3}]{1}}=1}$ ), and so on. 1 is the only natural number that is neither composite (a number with more than two distinct positive divisors) nor prime (a number with exactly two distinct positive divisors) with respect to division.^[5]

As a digit

 

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, represents the numeral 1 as similar to a small-caps I.

Main article: History of the Hindu–Arabic numeral system

The glyph used today in the Western world to represent the number 1, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Brahmic script of ancient India, where it was a simple vertical line.^{[citation needed]} It was transmitted to Europe via the Maghreb and Andalusia during the Middle Ages, through scholarly works written in Arabic.^{[citation needed]} In some countries, the serif at the top is sometimes extended into a long upstroke, sometimes as long as the vertical line, which can lead to confusion with the glyph used for seven in other countries. In styles in which the digit 1 is written with a long upstroke, the digit 7 is often written with a horizontal stroke through the vertical line, to disambiguate them. Styles that do not use the long upstroke on digit 1 usually do not use the horizontal stroke through the vertical of the digit 7 either.^{[citation needed]}

While the shape of the character for the digit 1 has an ascender in most modern typefaces, in typefaces with text figures, the glyph usually is of x-height, as, for example, in  .^{[citation needed]} Many older typewriters lack a separate key for 1, using the lowercase letter l or uppercase I instead. It is possible to find cases when the uppercase J is used, though it may be for decorative purposes. In some typefaces, different glyphs are used for I and 1, but the numeral 1 resembles a small caps version of I, with parallel serifs at top and bottom, with the capital I being full-height.^{[citation needed]}

Mathematics

Definitions

Mathematically, 1 is:

• in arithmetic (algebra) and calculus, the natural number that follows 0 and the multiplicative identity element of the integers, real numbers and complex numbers;
• more generally, in algebra, the multiplicative identity (also called unity), usually of a group or a ring.

Formalizations of the natural numbers have their own representations of 1. In the Peano axioms, 1 is the successor of 0. In Principia Mathematica, it is defined as the set of all singletons (sets with one element), and in the Von Neumann cardinal assignment of natural numbers, it is defined as the set {0}.

In a multiplicative group or monoid, the identity element is sometimes denoted 1, but e (from the German Einheit, "unity") is also traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. When such a ring has characteristic n not equal to 0, the element called 1 has the property that n1 = 1n = 0 (where this 0 is the additive identity of the ring). Important examples are finite fields.

By definition, 1 is the magnitude, absolute value, or norm of a unit complex number, unit vector, and a unit matrix (more usually called an identity matrix). The term unit matrix is sometimes used to mean a matrix composed entirely of 1s.

By definition, 1 is the probability of an event that is absolutely or almost certain to occur.

In category theory, 1 is sometimes used to denote the terminal object of a category.

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. Legendre's constant was originally conjectured to be approximately 1.08366, but was proven to equal exactly 1 in 1899.

Properties

Tallying is often referred to as "base 1", since only one mark – the tally itself – is needed. This is more formally referred to as a unary numeral system. Unlike base 2 or base 10, this is not a positional notation.

Since the base 1 exponential function (1^x) always equals 1, its inverse does not exist (which would be called the logarithm base 1 if it did exist).

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. 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.

Because of the multiplicative identity, if f(x) is a multiplicative function, then f(1) must be equal to 1.

There are two ways to write the real number 1 as a recurring decimal: as 1.000..., and as 0.999.... 1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few.

1 is also the first and second number in the Fibonacci sequence (0 being the zeroth) and is the first number in many other mathematical sequences.

The definition of a field requires that 1 must not be equal to 0. Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the field with one element, which is not a singleton and is not a set at all.

1 is the most common leading digit in many sets of data, a consequence of Benford's law.

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

The generating function that has all coefficients equal to 1 is a geometric series, given by

${\displaystyle {\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\ldots }$ 

The zeroth metallic mean is 1, with the golden section equal to the continued fraction [1;1,1,...], and the infinitely nested square root ${\displaystyle \scriptstyle {\sqrt {1+{\sqrt {{\text{ }}1+\cdots {\text{ }}}}}}.}$ 

The series of unit fractions that most rapidly converge to 1 are the reciprocals of Sylvester's sequence, which generate the infinite Egyptian fraction

${\displaystyle 1={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{43}}+\cdots }$ 

Primality

Main article: Prime number § Primality of one

1 is by convention neither a prime number nor a composite number, but a unit like −1 and, in the Gaussian integers, i and −i.

The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units. For example, 4 = 2², but if units are included, is also equal to, say, (−1)⁶ × 1²³ × 2², among infinitely many similar "factorizations".

1 appears to meet the naïve definition of a prime number, being evenly divisible only by 1 and itself (also 1). As such, some mathematicians considered it a prime number as late as the middle of the 20th century, but mathematical consensus has generally and since then universally been to exclude it for a variety of reasons (such as complicating the fundamental theorem of arithmetic and other theorems related to prime numbers).

1 is the only positive integer divisible by exactly one positive integer, whereas prime numbers are divisible by exactly two positive integers, composite numbers are divisible by more than two positive integers, and zero is divisible by all positive integers.

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

In technology

• A binary code is a sequence of 1 and 0 that is used in computers for representing any kind of data.
• In many physical devices, 1 represents the value for "on", which means that electricity is flowing.^[6][7]
• The numerical value of true 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.
• +1 is the electric charge of positrons and protons.
• 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.^[8] 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.^[9]

In other fields

• For Pythagorean numerology (a pseudoscience), the number 1 is the number that means beginning, new beginnings, new cycles, it is a unique and absolute number.

See also

•  Mathematics portal

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

References

1. Hurford 1994, p.23-24.
2. Huddleston, Pullum & Reynolds 2022, p. 117.
3. Huddleston, Pullum & Reynolds 2022, p. 140.
4. "Online Etymology Dictionary". etymonline.com. Douglas Harper. from the original on 2013-12-30. Retrieved 2013-12-30.
5. Weisstein, Eric W. "1". mathworld.wolfram.com. from the original on 2020-07-26. Retrieved 2020-09-22.
6. Woodford, Chris (2006), Digital Technology, Evans Brothers, p. 9, ISBN 978-0-237-52725-9, retrieved 2016-03-24
7. Godbole, Achyut S. (1 September 2002), Data Comms & Networks, Tata McGraw-Hill Education, p. 34, ISBN 978-1-259-08223-8
8. Olson, Roger (2017). The Essentials of Christian Thought: Seeing Reality through the Biblical Story. Zondervan Academic. ISBN 9780310521563.
9. 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. Cambridge University PRess. 10 (2): Abstract. doi:10.1017/S0007087400015375. S2CID 145065082. from the original on May 16, 2021. Retrieved May 16, 2021.

Sources

• Hurford, James R. (1994). Grammar: A Student's Guide. Cambridge University Press. ISBN 978-0-521-45627-2.
• Huddleston, Rodney D.; Pullum, Geoffrey K.; Reynolds, Brett (2022). A student's introduction to English grammar (2nd ed.). Cambridge, United Kingdom: Cambridge University Press. ISBN 978-1-316-51464-1. OCLC 1255524478.

 

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