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1 (one, unit, unity) is a
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
representing a single or the only
entity An entity is something that exists as itself, as a subject or as an object, actually or potentially, concretely or abstractly, physically or not. It need not be of material existence. In particular, abstractions and legal fictions are usually r ...
. 1 is also a
numerical digit A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent number A number is a mathematical object used to count, measure, and label. The original example ...
and represents a single
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
of
counting Counting is the process of determining the number of Element (mathematics), elements of a finite set of objects, i.e., determining the size (mathematics), size of a set. The traditional way of counting consists of continually increasing a (mental ...
or
measurement Measurement is the quantification (science), quantification of variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determi ...
. For example, a
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct end Point (geometry), points, and contains every point on the line that is between its endpoints. The length of a line segment is give ...
of ''unit length'' is a line segment of
length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most Measurement system, systems of measurement a Base unit (measurement), base unit f ...
 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest
positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
. It is also sometimes considered the first of the
infinite sequence In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s, followed by  2, although by other definitions 1 is the second natural number, following  0. The fundamental mathematical property of 1 is to be a
multiplicative identity In mathematics, an identity element, or neutral element, of a binary operation operating on a Set (mathematics), set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in alge ...
, 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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...
; this was not universally accepted until the mid-20th century. Additionally, 1 is the smallest possible difference between two distinct
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s. 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.


Etymology

The word ''one'' can be used as a noun, an adjective, and a pronoun. It comes from the English word ''an'', which comes from the Proto-Germanic root . The Proto-Germanic root comes from the Proto-Indo-European root ''*oi-no-''. Compare the Proto-Germanic root to
Old Frisian Old Frisian was a West Germanic languages, West Germanic language spoken between the 8th and 16th centuries along the North Sea coast, roughly between the mouths of the Rhine and Weser rivers. The Frisian settlers on the coast of South Jutland ...
''an'', Gothic ''ains'', Danish ''en'', Dutch ''een'', German ''eins'' and
Old Norse Old Norse, Old Nordic, or Old Scandinavian, is a stage of development of North Germanic languages, North Germanic dialects before their final divergence into separate Nordic languages. Old Norse was spoken by inhabitants of Scandinavia and t ...
''einn''. Compare the Proto-Indo-European root ''*oi-no-'' (which means "one, single") to
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
''oinos'' (which means "ace" on dice),
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
''unus'' (one),
Old Persian Old Persian is one of the two directly attested Old Iranian languages (the other being Avestan language, Avestan) and is the ancestor of Middle Persian (the language of Sasanian Empire). Like other Old Iranian languages, it was known to its native ...
,
Old Church Slavonic Old Church Slavonic or Old Slavonic () was the first Slavic languages, Slavic literary language. Historians credit the 9th-century Byzantine Empire, Byzantine missionaries Saints Cyril and Methodius with Standard language, standardizing the lan ...
''-inu'' and ''ino-'', Lithuanian ''vienas'',
Old Irish Old Irish, also called Old Gaelic ( sga, Goídelc, Ogham, Ogham script: ᚌᚑᚔᚇᚓᚂᚉ; ga, Sean-Ghaeilge; gd, Seann-Ghàidhlig; gv, Shenn Yernish or ), is the oldest form of the Goidelic languages, Goidelic/Gaelic language for which ...
''oin'' and Breton ''un'' (one).


As a number

One, sometimes referred to as unity, is the first non-zero
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
. It is thus the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
after
zero 0 (zero) is a number, and the numerical digit used to represent that number in numeral system, numerals. It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. A ...
. Any number multiplied by one remains that number, as one is the identity for
multiplication Multiplication (often denoted by the Multiplication sign, cross symbol , by the mid-line #Notation and terminology, dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Op ...
. As a result, 1 is its own
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
, its own
square In Euclidean geometry, a square is a regular polygon, regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree (angle), degree angles, π/2 radian angles, or right angles). It can also be defined as a rec ...
and
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square (algebra), square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots o ...
, its own
cube In geometry, a cube is a three-dimensional space, three-dimensional solid object bounded by six square (geometry), square faces, Facet (geometry), facets or sides, with three meeting at each vertex (geometry), vertex. Viewed from a corner it i ...
and
cube root In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. Fo ...
, and so on. One is also the result of the
empty product In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplication, multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operat ...
, as any number multiplied by one is itself. It is also the only natural number that is neither composite nor
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...
with respect to
division Division or divider may refer to: Mathematics *Division (mathematics) Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and ...
, but is instead considered a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
(meaning of
ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure ...
).


As a digit

The glyph used today in the Western world to represent the number 1, a vertical line, often with a
serif In typography, a serif () is a small line or stroke regularly attached to the end of a larger stroke in a letter or symbol within a particular font or family of fonts. A typeface or "font family" making use of serifs is called a serif typeface ( ...
at the top and sometimes a short horizontal line at the bottom, traces its roots back to the
Brahmic The Brahmic scripts, also known as Indic scripts, are a family of abugida writing systems. They are used throughout the Indian subcontinent, Southeast Asia and parts of East Asia. They are descended from the Brahmi script of ancient India ...
script of ancient India, where it was a simple vertical line. It was transmitted to Europe via the Maghreb and Andalusia during the Middle Ages, through scholarly works written in
Arabic Arabic (, ' ; , ' or ) is a Semitic languages, Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C ...
. 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. While the shape of the character for the digit 1 has an ascender in most modern
typeface A typeface (or font family) is the design of lettering that can include variations in size, weight (e.g. bold), slope (e.g. italic), width (e.g. condensed), and so on. Each of these variations of the typeface is a font. There are list of type ...
s, in typefaces with
text figures Text figures (also known as non-lining, lowercase, old style, ranging, hanging, medieval, billing, or antique figures or numerals) are numerals designed with varying heights in a fashion that resembles a typical line of running text, hence the ...
, the glyph usually is of
x-height file:Typography Line Terms.svg, upright 2.0, alt=A diagram showing the line terms used in typography In typography, the x-height, or corpus size, is the distance between the baseline (typography), baseline and the mean line of lowercase letters in ...
, as, for example, in . 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 In typography, small caps (short for "small capitals") are grapheme, characters typeset with glyphs that resemble letter case, uppercase letters (capitals) but reduced in height and weight close to the surrounding letter case, lowercase letters ...
version of I, with parallel serifs at top and bottom, with the capital I being full-height.


Mathematics


Definitions

Mathematically, 1 is: *in
arithmetic Arithmetic () is an elementary part of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their chang ...
(
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
) and
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
that follows 0 and the multiplicative
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a Set (mathematics), set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in alge ...
of the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
s,
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s and
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
s; *more generally, in
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
, the multiplicative identity (also called ''unity''), usually of a
group A group is a number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, ...
or a ring. Formalizations of the natural numbers have their own representations of 1. In the
Peano axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have be ...
, 1 is the successor of 0. In ''
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'', 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 . In a multiplicative
group A group is a number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, ...
or
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ar ...
, the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a Set (mathematics), set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in alge ...
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 (where this 0 is the additive identity of the ring). Important examples are
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
s. By definition, 1 is the magnitude,
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign (mathematics), sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative number, negative (in which cas ...
, or norm of a unit complex number,
unit vector In mathematics, a unit vector in a normed vector space is a Vector_(mathematics_and_physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
, and a
unit matrix In linear algebra, the identity matrix of size n is the n\times n square matrix In mathematics, a square matrix is a Matrix (mathematics), matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of o ...
(more usually called an identity matrix). Note that the term ''unit matrix'' is sometimes used to mean something quite different. By definition, 1 is the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
of an event that is absolutely or
almost certain In probability theory, an event (probability theory), event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty ...
to occur. In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
, 1 is sometimes used to denote the
terminal object In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic top ...
of a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories'' (Aristotle) *Category (Kant) ...
. In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...
, 1 is the value of
Legendre's constant Legendre's constant is a mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an Letter (alphabet), alphabet letter), or by mathematicians' names ...
, which was introduced in 1808 by
Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials In physical science and mathemati ...
in expressing the
asymptotic behavior In mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, i ...
of the
prime-counting function In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
. 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 The unary numeral system is the simplest numeral system to represent natural numbers: to represent a number ''N'', a symbol representing 1 is repeated ''N'' times. In the unary system, the number 0 (zero) is represented by the empty string, that ...
. Unlike base 2 or base 10, this is not a
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a positional system is a numeral syste ...
. Since the base 1 exponential function (1''x'') always equals 1, its inverse does not exist (which would be called the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
base 1 if it did exist). There are two ways to write the real number 1 as a
recurring decimal A repeating decimal or recurring decimal is decimal representation of a number whose Numerical digit, digits are periodic function, periodic (repeating its values at regular intervals) and the infinity, infinitely repeated portion is not zero. It ...
: as 1.000..., and as 0.999.... 1 is the first
figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polyg ...
of every kind, such as
triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and Cube (algebra)#In integers, cube numbers. The th triangular numbe ...
,
pentagonal number A pentagonal number is a figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (poly ...
and
centered hexagonal number In mathematics and combinatorics, a centered hexagonal number, or hex number, is a centered polygonal number, centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagona ...
, to name just a few. In many mathematical and engineering problems, numeric values are typically ''normalized'' to fall within the
unit interval In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters. Likewise, vectors are often normalized into
unit vector In mathematics, a unit vector in a normed vector space is a Vector_(mathematics_and_physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
s (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 In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
one, maximum value one, or square integral one, depending on the application. Because of the multiplicative identity, if ''f''(''x'') is a
multiplicative function In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) ...
, then ''f''(1) must be equal to 1. It is also the first and second number in the
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talente ...
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 In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French–English pun, Fun. The name ...
, 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 Benford's law, also known as the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation that in many real-life sets of numerical data, the leading digit is likely to be small.Arno Berger and Theodore P ...
. 1 is the only known Tamagawa number for a simply connected algebraic group over a number field. The
generating function In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
that has all coefficients 1 is given by \frac = 1+x+x^2+x^3+ \ldots This power series converges and has finite value
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
, x, < 1 .


Primality

1 is by convention neither a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only wa ...
nor a
composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime nu ...
, but a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
(meaning of
ring theory In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure ...
) like −1 and, in the
Gaussian integers In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) o ...
, '' i'' and −''i''. The
fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
guarantees unique factorization over the integers only up to units. For example, , but if units are included, is also equal to, say, 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 0 (zero) is a number, and the numerical digit used to represent that number in numeral system, numerals. It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. A ...
is divisible by all positive integers.


Table of basic calculations


In technology

* The
resin identification code The ASTM International Resin Identification Coding System, often abbreviated RIC, is a set of symbols appearing on plastic products that identify the Synthetic resin, plastic resin out of which the product is made. It was developed in 1988 by t ...
used in recycling to identify
polyethylene terephthalate Polyethylene terephthalate (or poly(ethylene terephthalate), PET, PETE, or the obsolete PETP or PET-P), is the most common thermoplastic polymer resin of the polyester family and is used in synthetic fibre, fibres for clothing, packaging, conta ...
. *The
ITU The International Telecommunication Union is a specialized agency of the United Nations responsible for many matters related to information and communication technologies Information and communications technology (ICT) is an extensional ter ...
country code for the
North American Numbering Plan The North American Numbering Plan (NANP) is a telephone numbering plan for twenty-five regions in twenty countries, primarily in North America and the Caribbean. This group is historically known as World Zone 1 and has the international calling ...
area, which includes the United States, Canada, and parts of the Caribbean. *A
binary code A binary code represents plain text, text, instruction set, computer processor instructions, or any other data using a two-symbol system. The two-symbol system used is often "0" and "1" from the binary number, binary number system. The binary cod ...
is a sequence of 1 and 0 that is used in
computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as Computer program, pr ...
s for representing any kind of
data In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interp ...
. *In many physical devices, 1 represents the value for "on", which means that electricity is flowing. *The numerical value of
true True most commonly refers to truth, the state of being in congruence with fact or reality. True may also refer to: Places * True, West Virginia, an unincorporated community in the United States * True, Wisconsin, a town in the United States * Tr ...
in many programming languages. *1 is the
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because of ...
code of " Start of Header".


In science

*
Dimensionless quantities A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate di ...
are also known as quantities of dimension one. *1 is the atomic number of
hydrogen Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is a gas of diatomic molecules having the chemical ...
. *+1 is the
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electron ...
of
positron The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 ''elementary charge, e'', a spin (physics), spin of 1/2 (the same as the electron), and the same Electron rest ...
s and protons. *Group 1 of the
periodic table The periodic table, also known as the periodic table of the (chemical) elements, is a rows and columns arrangement of the chemical elements. It is widely used in chemistry, physics, and other sciences, and is generally seen as an Cultural i ...
consists of the
alkali metals The alkali metals consist of the chemical elements lithium (Li), sodium (Na), potassium (K),The symbols Na and K for sodium and potassium are derived from their Latin names, ''natrium'' and ''kalium''; these are still the origins of the names ...
. *Period 1 of the periodic table consists of just two elements,
hydrogen Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is a gas of diatomic molecules having the chemical ...
and
helium Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol (chemistry), symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert gas, inert, monatomic gas and the first in the noble gas gr ...
. *The dwarf planet Ceres has the minor-planet designation 1 Ceres because it was the first asteroid to be discovered. *The Roman numeral I often stands for the first-discovered satellite of a
planet A planet is a large, rounded Astronomical object, astronomical body that is neither a star nor its Stellar remnant, remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud colla ...
or
minor planet According to the International Astronomical Union (IAU), a minor planet is an astronomical object in direct orbit around the Sun that is exclusively classified as neither a planet nor a comet. Before 2006, the IAU officially used the term ''minor ...
(such as Neptune I, a.k.a. Triton). For some earlier discoveries, the Roman numerals originally reflected the increasing distance from the primary instead.


In philosophy

In the philosophy of
Plotinus Plotinus (; grc-gre, Πλωτῖνος, ''Plōtînos'';  – 270 CE) was a philosopher in the Hellenistic philosophy, Hellenistic tradition, born and raised in Roman Egypt. Plotinus is regarded by modern scholarship as the founder of Neop ...
(and that of other
neoplatonist Neoplatonism is a strand of Platonism, Platonic philosophy that emerged in the 3rd century AD against the background of Hellenistic philosophy and Hellenistic religion, religion. The term does not encapsulate a set of ideas as much as a chain of ...
s), The One is the ultimate reality and source of all existence.
Philo of Alexandria Philo of Alexandria (; grc, Φίλων, Phílōn; he, יְדִידְיָה, Yəḏīḏyāh (Jedediah); ), also called Philo Judaeus, was a Hellenistic Judaism, Hellenistic Jewish Jewish philosophy, philosopher who lived in Alexandria, in the ...
(20 BC – AD 50) regarded the number one as God's number, and the basis for all numbers ("De Allegoriis Legum," ii.12 .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 Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative intege ...
was recovered by
Boethius Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman Roman Senate, senator, Roman consul, consul, ''magister officiorum'', historian, and philosopher of the Early Middle Ages. He was ...
in his Latin translation of Nicomachus's treatise ''
Introduction to Arithmetic The book ''Introduction to Arithmetic'' ( grc-gre, Ἀριθμητικὴ εἰσαγωγή, ''Arithmetike eisagoge'') is the only extant work on mathematics by Nicomachus (60–120 AD). Summary The work contains both philosophical prose and ...
''.


In sports

In many professional sports, the number 1 is assigned to the player who is first or leading in some respect, or otherwise important; the number is printed on his sports uniform or equipment. This is the
pitcher In baseball, the pitcher is the player who throws ("pitches") the Baseball (ball), baseball from the pitcher's mound toward the catcher to begin each play, with the goal of out (baseball), retiring a batter (baseball), batter, who attempts to e ...
in
baseball Baseball is a bat-and-ball games, bat-and-ball sport played between two team sport, teams of nine players each, taking turns batting (baseball), batting and Fielding (baseball), fielding. The game occurs over the course of several Pitch ...
, the goalkeeper in
association football Association football, more commonly known as football or soccer, is a team sport played between two teams of 11 Football player, players who primarily use their feet to propel the Ball (association football), ball around a rectangular field ca ...
(soccer), the starting fullback in most of
rugby league Rugby league football, commonly known as just rugby league and sometimes football, footy, rugby or league, is a contact sport, full-contact sport played by two teams of thirteen players on a rectangular Rugby league playing field, field measur ...
, the starting loosehead prop in
rugby union Rugby union, commonly known simply as rugby, is a Contact sport#Terminology, close-contact team sport that originated at Rugby School in the first half of the 19th century. One of the Comparison of rugby league and rugby union, two codes of ru ...
and the previous year's world champion in
Formula One Formula One (also known as Formula 1 or F1) is the highest class of international racing for open-wheel single-seater formula racing cars sanctioned by the Fédération Internationale de l'Automobile (FIA). The World Drivers' Championship ...
. 1 may be the lowest possible player number, like in the American–Canadian
National Hockey League The National Hockey League (NHL; french: Ligue nationale de hockey—LNH, ) is a professional ice hockey sports league, league in North America comprising 32 teams—25 in the United States and 7 in Canada. It is considered to be the top ranke ...
(NHL) since the 1990s or in
American football American football (referred to simply as football in the United States and Canada), also known as gridiron, is a team sport played by two teams of eleven players on a rectangular American football field, field with goalposts at each end. Th ...
.


In other fields

*''Number One'' is
Royal Navy The Royal Navy (RN) is the United Kingdom's naval warfare force. Although warships were used by Kingdom of England, English and Kingdom of Scotland, Scottish kings from the early medieval period, the first major maritime engagements were foug ...
informal usage for the chief executive officer of a ship, the captain's deputy responsible for discipline and all normal operation of a ship and its crew. *1 is the value of an ace in many playing card games, such as cribbage. * List of highways numbered 1 * List of public transport routes numbered 1 *1 is often used to denote the
Gregorian calendar The Gregorian calendar is the calendar used in most parts of the world. It was introduced in October 1582 by Pope Gregory XIII as a modification of, and replacement for, the Julian calendar. The principal change was to space leap years diffe ...
month of
January January is the first month of the year in the Julian calendar, Julian and Gregorian calendars and is also the first of seven months to have a length of 31 days. The first day of the month is known as New Year's Day. It is, on average, the colde ...
. * 1 CE, the first year of the
Common Era Common Era (CE) and Before the Common Era (BCE) are year notations for the Gregorian calendar The Gregorian calendar is the calendar used in most parts of the world. It was introduced in October 1582 by Pope Gregory XIII as a modificatio ...
*01, the former dialling code for
Greater London Greater London is an Metropolitan and non-metropolitan counties of England#Greater London, administrative area in England governed by the Greater London Authority. It is organised into 33 Districts of England, local government districts: the ...
(now 020) *For Pythagorean
numerology Numerology (also known as arithmancy) is the belief in an occult, divine or mystical relationship between a number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1 ...
(a
pseudoscience Pseudoscience consists of statements, beliefs, or practices that claim to be both scientific and factual but are incompatible with the scientific method. Pseudoscience is often characterized by contradictory, exaggerated or falsifiability, unfa ...
), the number 1 is the number that means beginning, new beginnings, new cycles, it is a unique and absolute number. * PRS One, a German paraglider design *+1 is the code for international telephone calls to countries in the
North American Numbering Plan The North American Numbering Plan (NANP) is a telephone numbering plan for twenty-five regions in twenty countries, primarily in North America and the Caribbean. This group is historically known as World Zone 1 and has the international calling ...
. * In some countries, a street address of "1" is considered prestigious and developers will attempt to obtain such an address for a building, to the point of lobbying for a street or portion of a street to be renamed, even if this makes the address less useful for wayfinding. The construction of a new street to serve the development may also provide the possibility of a "1" address. An example of such an address is the
Apple Campus The Apple Campus is the former corporate headquarters Corporate headquarters is the part of a corporate structure that deals with important tasks such as strategic planning, corporate communications, taxes, law, books of record, marketing, fi ...
, located at 1 Infinite Loop,
Cupertino, California Cupertino ( ) is a city in Santa Clara County, California Santa Clara County, officially the County of Santa Clara, is the sixth-most populous county in the U.S. state of California, with a population of 1,936,259, as of the 2020 United S ...
.


See also

* −1 * +1 (disambiguation) * List of mathematical constants * One (word) *
Root of unity In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...
* List of highways numbered 1


References


External links


The Number 1The Positive Integer 1
{{DEFAULTSORT:1 (Number) Integers