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1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or
measurement 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 of natural numbers, 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; this was not universally accepted until the mid-20th century. Additionally, 1 is the smallest possible difference between two distinct natural numbers. The unique mathematical properties of the number have led to its unique uses in other fields, ranging from science to sports. It commonly denotes the first, leading, or top thing in a group.


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 ''an'', Gothic ''ains'', Danish ''en'', Dutch ''een'', German ''eins'' and Old Norse ''einn''. Compare the Proto-Indo-European root ''*oi-no-'' (which means "one, single") to Greek ''oinos'' (which means "ace" on dice), Latin ''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 ''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. It is thus the integer after zero. Any number multiplied by one remains that number, as one is the identity 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 and square root, 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, as any number multiplied by one is itself. It is also the only natural number that is neither composite nor prime with respect to division, but is instead considered a unit (meaning of ring theory).


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 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. 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 typefaces, in typefaces with text figures, the glyph usually is of x-height, 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 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) and calculus, the natural number that follows 0 and the multiplicative identity element of the integers, real numbers and complex numbers; *more generally, in algebra, the multiplicative identity (also called ''unity''), usually of a group or a ring. Formalizations of the natural numbers have their own representations of 1. In the Peano axioms, 1 is the successor of 0. In '' Principia Mathematica'', it is defined as the set of all singletons (sets with one element), and in the
Von Neumann cardinal assignment 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 . In a multiplicative group or monoid, the identity element is sometimes denoted 1, but ''e'' (from the German ''Einheit'', "unity") is also traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. When such a ring has characteristic ''n'' not equal to 0, the element called 1 has the property that (where this 0 is the additive identity of the ring). Important examples are finite fields. 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 of a unit complex number, unit vector, and a unit matrix (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 of an event that is absolutely or almost certain 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, 1 is the value of Legendre's constant, which was introduced in 1808 by Adrien-Marie Legendre in expressing the asymptotic behavior of the
prime-counting function 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 â€“ the tally itself â€“ is needed. This is more formally referred to as a unary numeral system. Unlike base 2 or base 10, this is not a
positional notation 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 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, 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 from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters. Likewise, vectors are often normalized into unit vectors (i.e., vectors of magnitude one), because these often have more desirable properties. Functions, too, are often normalized by the condition that they have integral one, maximum value one, or square integral one, depending on the application. Because of the multiplicative identity, if ''f''(''x'') is a multiplicative function, then ''f''(1) must be equal to 1. 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 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 \frac = 1+x+x^2+x^3+ \ldots This power series converges and has finite value if and only if , x, < 1 .


Primality

1 is by convention neither a prime number nor a composite number, but a unit (meaning of ring theory) like −1 and, in the Gaussian integers, '' i'' and −''i''. The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units. For example, , 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 is divisible by all positive integers.


Table of basic calculations


In technology

* 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 country code for the North American Numbering Plan 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 many physical devices, 1 represents the value for "on", which means that electricity is flowing. *The numerical value of true in many programming languages. *1 is the ASCII code of " Start of Header".


In science

* Dimensionless quantities are also known as quantities of dimension one. *1 is the atomic number of hydrogen. *+1 is the electric charge 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 and helium. *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 or minor planet (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 (and that of other neoplatonists), The One is the ultimate reality and source of all existence. Philo of Alexandria (20 BC â€“ 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 was recovered by Boethius in his Latin translation of Nicomachus's treatise '' Introduction to Arithmetic''.


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 goalkeeper in association football (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, the starting loosehead prop in rugby union and the previous year's world champion in Formula One. 1 may be the lowest possible player number, like in the American–Canadian National Hockey League (NHL) since the 1990s or in American football.


In other fields

*''Number One'' is Royal Navy 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 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 *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), 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. * 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) * Root of unity * List of highways numbered 1


References


External links


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