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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 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'' (a ...
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. For example, a line segment of ''unit length'' is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the
infinite sequence 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 ...
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, 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 ...
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 Fri ...
''an'', Gothic ''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 a ...
''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 ''eins'' and
Old Norse Old Norse, Old Nordic, or Old Scandinavian, is a stage of development of North Germanic dialects before their final divergence into separate Nordic languages. Old Norse was spoken by inhabitants of Scandinavia and their overseas settlemen ...
''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 Church Slavonic ''-inu'' and ''ino-'', Lithuanian ''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 ''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 ...
. 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 languag ...
after
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
. Any number multiplied by one remains that number, as one is the identity for multiplication. As a result, 1 is its own factorial, 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 a ...
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 . ...
, its own cube and cube root, 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'' (a ...
(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 r ...
).


As a digit

The glyph used today in the Western world to represent the number 1, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the
Brahmic 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 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. E.Watson; Walter ...
. 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 thousands o ...
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 le ...
, 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 version of I, with parallel serifs at top and bottom, with the capital I being full-height.


Mathematics


Definitions

Mathematically, 1 is: *in arithmetic (
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 ...
) 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 ...
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 languag ...
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 ...
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 fo ...
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 ...
, the multiplicative identity (also called ''unity''), usually of a group (mathematics), group or a ring (mathematics), ring. Formalizations of the natural numbers have their own representations of 1. In the Peano axioms, 1 is the Successor function, successor of 0. In ''Principia Mathematica'', it is defined as the set of all singleton (mathematics), singletons (sets with one element), and in the Von Neumann cardinal assignment of natural numbers, it is defined as the Set (mathematics), set . In a multiplicative group (mathematics), group or monoid, 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 (algebra), 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 (mathematics), magnitude, absolute value, or Norm (mathematics), norm of a unit complex number, unit vector, and a identity matrix, unit matrix (more usually called an identity matrix). Note that the term ''unit matrix'' is sometimes used to mean something Matrix of ones, quite different. By definition, 1 is the probability of an event that is absolutely or almost certain to occur. In category theory, 1 is sometimes used to denote the terminal object of a category (mathematics), 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 analysis, asymptotic behavior of the prime-counting function. Legendre's constant was originally conjectured to be approximately 1.08366, but was proven to equal exactly 1 in 1899.


Properties

Tally mark, Tallying is often referred to as "base 1", since only one mark – the tally itself – is needed. This is more formally referred to as a unary numeral system. Unlike base 2 or base 10, this is not a positional notation. Since the base 1 exponential function (1''x'') always equals 1, its inverse function, inverse 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, 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, vector space, 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 integrable, 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 number, Fibonacci sequence (0 being the zeroth) and is the first number in many other Sequence, mathematical sequences. The definition of a field (mathematics), field requires that 1 must not be equal to zero, 0. Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the field with one element, which is not a singleton and is not a set at all. 1 is the most common leading digit in many sets of data, a consequence of Benford's law. 1 is the only known Tamagawa number for a simply connected algebraic group over a number field. The generating function that has all coefficients 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 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, 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'' (a ...
(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 r ...
) like −1 and, in the Gaussian integers, ''imaginary unit, i'' and −''i''. The fundamental theorem of arithmetic guarantees factorization, 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 0, zero is divisible by all positive integers.


Table of basic calculations


In technology

* The resin identification code used in recycling to identify polyethylene terephthalate. *The International Telecommunication Union, ITU country code for the North American Numbering Plan area, which includes the United States, Canada, and parts of the Caribbean. *A binary code is a sequence of 1 and 0 that is used in computers for representing any kind of data. *In many physical devices, 1 represents the value for "on", which means that electricity is flowing. *The numerical value of Boolean data type, true in many programming languages. *1 is the ASCII code of "Start-of-Header, 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 positrons and protons. *Group 1 of the periodic table consists of the alkali metals. *Period 1 of the periodic table consists of just two elements, hydrogen and helium. *The dwarf planet Ceres (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 (moon), 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), Plotinus#The One, The One is the ultimate reality and source of all existence. Philo#Numbers, Philo of Alexandria (20 BC – AD 50) regarded the number one as God's number, and the basis for all numbers ("De Allegoriis Legum," ii.12 [i.66]). The Neopythagorean philosopher Nicomachus, Nicomachus of Gerasa affirmed that one is not a number, but the source of number. He also believed the 2 (number), number two is the embodiment of the origin of Other (philosophy), 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 (association football), goalkeeper in association football (soccer), the starting Fullback (rugby league), fullback in most of rugby league, the starting Rugby union positions#Prop, 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. *1 CE, the first year of the Common Era *01, the former dialling code for Greater London (now 020) *For Pythagorean numerology (a pseudoscience), the number 1 is the number that means beginning, new beginnings, new cycles, it is a unique and absolute number. *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 house numbering, 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, 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) 1 (number), Integers