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 c ...
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 ...
. 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 numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits (Latin ...
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'' (alb ...
of
counting
Counting is the process of determining the number of elements of a finite set of objects, i.e., determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every ele ...
or
measurement
Measurement is the quantification of 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 determining how large or small a physical quantity is as compared ...
. For example, a
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
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 distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
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, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
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 is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
, 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 ways ...
; 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 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 (today's Northern Friesl ...
''an'',
Gothic
Gothic or Gothics may refer to:
People and languages
*Goths or Gothic people, the ethnonym of a group of East Germanic tribes
**Gothic language, an extinct East Germanic language spoken by the Goths
**Crimean Gothic, the Gothic language spoken b ...
''ains'',
Danish
Danish may refer to:
* Something of, from, or related to the country of Denmark
People
* A national or citizen of Denmark, also called a "Dane," see Demographics of Denmark
* Culture of Denmark
* Danish people or Danes, people with a Danish ance ...
''en'',
Dutch
Dutch commonly refers to:
* Something of, from, or related to the Netherlands
* Dutch people ()
* Dutch language ()
Dutch may also refer to:
Places
* Dutch, West Virginia, a community in the United States
* Pennsylvania Dutch Country
People E ...
''een'',
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
''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 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 the power of the ...
''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
Lithuanian may refer to:
* Lithuanians
* Lithuanian language
* The country of Lithuania
* Grand Duchy of Lithuania
* Culture of Lithuania
* Lithuanian cuisine
* Lithuanian Jews as often called "Lithuanians" (''Lita'im'' or ''Litvaks'') by other Jew ...
''vienas'',
Old Irish
Old Irish, also called Old Gaelic ( sga, Goídelc, Ogham script: ᚌᚑᚔᚇᚓᚂᚉ; ga, Sean-Ghaeilge; gd, Seann-Ghàidhlig; gv, Shenn Yernish or ), is the oldest form of the Goidelic/Gaelic language for which there are extensive writt ...
''oin'' and
Breton
Breton most often refers to:
*anything associated with Brittany, and generally
** Breton people
** Breton language, a Southwestern Brittonic Celtic language of the Indo-European language family, spoken in Brittany
** Breton (horse), a breed
**Ga ...
''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 ...
after
zero
0 (zero) is a number representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
.
Any number multiplied by one remains that number, as one is the
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
for
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
. As a result, 1 is its own
factorial
In mathematics, the factorial of a non-negative denoted is the 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 (n-1) \times (n-2) \t ...
, its own
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
and
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
E ...
, its own
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
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 multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...
, as any number multiplied by one is itself. It is also the only natural number that is neither
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials
...
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 ways ...
with respect to
division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting ...
, 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'' (alb ...
(meaning of
ring theory
In algebra, ring theory is the study of 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 of rings, their re ...
).
As a digit
![Clock 24 J](https://upload.wikimedia.org/wikipedia/commons/0/0e/Clock_24_J.jpg)
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
7 is a number, numeral, and glyph.
7 or seven may also refer to:
* AD 7, the seventh year of the AD era
* 7 BC, the seventh year before the AD era
* The month of
July
Music Artists
* Seven (Swiss singer) (born 1978), a Swiss recording artist ...
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
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 and the mean line of lowercase letters in a typeface. Typically, this is the height of the let ...
, as, for example, in
![TextFigs148](https://upload.wikimedia.org/wikipedia/commons/8/81/TextFigs148.svg)
.
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 characters typeset with glyphs that resemble uppercase letters (capitals) but reduced in height and weight close to the surrounding lowercase letters or text figures. This is technicall ...
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 that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
(
algebra
Algebra () is one of the 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 mathematics.
Elementary a ...
) and
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, 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 is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
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 ...
s,
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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 ...
s;
*more generally, in
algebra
Algebra () is one of the 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 mathematics.
Elementary a ...
, the multiplicative identity (also called ''unity''), usually of a
group or a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
.
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
The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set ''U'', we define its cardinal number to be the smallest ordinal number equinumerous to ''U'', using the von Neumann definition of an or ...
of natural numbers, it is defined as the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
.
In a multiplicative
group 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 ...
, the
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a 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 algebraic structures su ...
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 that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
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. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
, or
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
of a
unit complex number
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
,
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
, and a
unit matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere.
Terminology and notation
The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
(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, cate ...
, 1 is sometimes used to denote the
terminal object of a
category.
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 integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, 1 is the value of
Legendre's constant
Legendre's constant is a mathematical constant occurring in a formula conjectured by Adrien-Marie Legendre to capture the asymptotic behavior of the prime-counting function \pi(x). Its value is now known to be 1.
Examination of available n ...
, which was introduced in 1808 by
Adrien-Marie Legendre in expressing the
asymptotic behavior of the
prime-counting function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ).
History
Of great interest in number theory is t ...
. 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 base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
.
Since the base 1 exponential function (1
''x'') always equals 1, its
inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when ad ...
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 o ...
base 1 if it did exist).
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
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 numbers. The th triangular number is the number of dots in ...
,
pentagonal number and
centered hexagonal number
In mathematics and combinatorics, a centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice. The following ...
, to name just a few.
In many mathematical and engineering problems, numeric values are typically ''normalized'' to fall within the
unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...
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 (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
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
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 i ...
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.
It 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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
that has all coefficients 1 is given by
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 ...
.
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 ways ...
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, ...
, 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'' (alb ...
(meaning of
ring theory
In algebra, ring theory is the study of 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 of rings, their re ...
) like −1 and, in the
Gaussian integers, ''
i'' and −''i''.
The
fundamental theorem of arithmetic guarantees
unique factorization
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an ...
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 representing an empty quantity. In place-value notation
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
is divisible by all positive integers.
Table of basic calculations
In technology
![U+2673 DejaVu Sans](https://upload.wikimedia.org/wikipedia/commons/e/e1/U%2B2673_DejaVu_Sans.svg)
* The
resin identification code 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 fibres for clothing, containers for liquids and foods ...
.
*The
ITU
The International Telecommunication Union is a specialized agency of the United Nations responsible for many matters related to information and communication technologies. It was established on 17 May 1865 as the International Telegraph Unio ...
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 callin ...
area, which includes the United States, Canada, and parts of the Caribbean.
*A
binary code
A binary code represents text, 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 system. The binary code assigns a pattern of binary digits, also ...
is a sequence of 1 and
0 that is used in
computer
A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...
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 interpreted ...
.
*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 are also known as quantities of dimension one.
*1 is the atomic number of
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, an ...
.
*+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 electrons respe ...
of
positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collides ...
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 icon of ch ...
consists of the
alkali metals.
*Period 1 of the periodic table consists of just two elements,
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, an ...
and
helium
Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic table. ...
.
*The dwarf planet
Ceres
Ceres most commonly refers to:
* Ceres (dwarf planet), the largest asteroid
* Ceres (mythology), the Roman goddess of agriculture
Ceres may also refer to:
Places
Brazil
* Ceres, Goiás, Brazil
* Ceres Microregion, in north-central Goiás st ...
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 body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
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
Triton commonly refers to:
* Triton (mythology), a Greek god
* Triton (moon), a satellite of Neptune
Triton may also refer to:
Biology
* Triton cockatoo, a parrot
* Triton (gastropod), a group of sea snails
* ''Triton'', a synonym of ''Triturus' ...
). For some earlier discoveries, the Roman numerals originally reflected the increasing distance from the primary instead.
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
.66.
The Neopythagorean philosopher
Nicomachus of Gerasa
Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Greek. He was born in ...
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 integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
was recovered by
Boethius
Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, ''magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the tr ...
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 from the pitcher's mound toward the catcher to begin each play, with the goal of retiring a batter, who attempts to either make contact with the pitched ball or draw ...
in
baseball
Baseball is a bat-and-ball sport played between two teams of nine players each, taking turns batting and fielding. The game occurs over the course of several plays, with each play generally beginning when a player on the fielding tea ...
, the
goalkeeper
In many team sports which involve scoring goals, the goalkeeper (sometimes termed goaltender, netminder, GK, goalie or keeper) is a designated player charged with directly preventing the opposing team from scoring by blocking or intercepting o ...
in
association football
Association football, more commonly known as football or soccer, is a team sport played between two teams of 11 players who primarily use their feet to propel the ball around a rectangular field called a pitch. The objective of the game is ...
(soccer), the starting
fullback Fullback or Full back may refer to:
Sports
* A position in various kinds of football, including:
** Full-back (association football), in association football (soccer), a defender playing in a wide position
** Fullback (gridiron football), in Americ ...
in most of
rugby league
Rugby league football, commonly known as just rugby league and sometimes football, footy, rugby or league, is a full-contact sport played by two teams of thirteen players on a rectangular field measuring 68 metres (75 yards) wide and 112 ...
, the starting
loosehead prop in
rugby union
Rugby union, commonly known simply as rugby, is a close-contact team sport that originated at Rugby School in the first half of the 19th century. One of the two codes of rugby football, it is based on running with the ball in hand. In its m ...
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 league in North America comprising 32 teams—25 in the United States and 7 in Canada. It is considered to be the top ranked professional ...
(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 field with goalposts at each end. The offense, the team with ...
.
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 English and Scottish kings from the early medieval period, the first major maritime engagements were fought in the Hundred Years' War against F ...
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
An ace is a playing card, Dice, die or domino with a single Pip (counting), pip. In the standard French deck, an ace has a single suit (cards), suit symbol (a heart, diamond, spade, or club) located in the middle of the card, sometimes large a ...
in many playing card games, such as
cribbage.
*
List of highways numbered 1
*
List of public transport routes numbered 1
In public transport, Route 1 may refer to:
*Route 1 (MTA Maryland), a bus route in Baltimore, Maryland
*Barcelona Metro line 1
*Line 1 (Beijing Subway)
*Line 1 (Hangzhou Metro)
*Citybus Route 1 in Hong Kong
*KMB Route 1 in Hong Kong
*London Buses ...
*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 dif ...
month of
January
January is the first month of the year in the 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 coldest month of the ...
.
*
1 CE
__NOTOC__
AD 1 or 1 CE is the epoch year for the Anno Domini (AD) Christian calendar era and also the 1st year of the Common Era (CE) and the 1st millennium and of the 1st century of the Christian and the common era. It was a common year starti ...
, the first year of the
Common Era
Common Era (CE) and Before the Common Era (BCE) are year notations for the Gregorian calendar (and its predecessor, the Julian calendar), the world's most widely used calendar era. Common Era and Before the Common Era are alternatives to the or ...
*01, the former dialling code for
Greater London
Greater may refer to:
*Greatness, the state of being great
*Greater than, in inequality (mathematics), inequality
*Greater (film), ''Greater'' (film), a 2016 American film
*Greater (flamingo), the oldest flamingo on record
*Greater (song), "Greate ...
(now 020)
*For Pythagorean
numerology
Numerology (also known as arithmancy) is the belief in an occult, divine or mystical relationship between a number and one or more coinciding events. It is also the study of the numerical value, via an alphanumeric system, of the letters in ...
(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
The PRS One is a German single-place paraglider that was produced by Pilots Right Stuff (PRS) of Brannenburg, starting in 2000. It was designed with outside contract assistance. The glider is now out of production.
Design and development
The c ...
, 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 callin ...
.
* In some countries, a
street address
An address is a collection of information, presented in a mostly fixed format, used to give the location of a building, apartment, or other structure or a plot of land, generally using political boundaries and street names as references, along ...
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, located at 1 Infinite Loop,
Cupertino, California.
See also
*
−1
*
+1 (disambiguation)
*
List of mathematical constants
*
One (word)
''One'' is an English language, gender-neutral, indefinite pronoun that means, roughly, "a person". For purposes of verb agreement it is a third-person singular pronoun, though it sometimes appears with first- or second-person reference. It is ...
*
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