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= 1 =
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Introduction
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1 (one, also called unit, unity, and (multiplicative) identity) is a
number, and a numerical digit used to represent that number in
numerals. It represents a single entity, the unit of counting or
measurement. For example, a line segment of 'unit length' is a line
segment of length 1. It is also the first of the infinite sequence of
natural numbers, followed by 2.
Etymology
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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 Church Slavonic '-inu' and 'ino-', Lithuanian
'vienas', Old Irish 'oin' and Breton 'un' (one).
As a number
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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. As a result, 1 is its own factorial, its
own square and square root, its own cube and cube root, 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 instead considered a
unit (meaning of ring theory).
As a digit
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The glyph used today in the Western world to represent the number 1, a
vertical line, often with a serif at the top and sometimes a short
horizontal line at the bottom, traces its roots back to the Brahmic
script of ancient India, where it was a simple vertical line. It was
transmitted to Europe via Arabic during the Middle Ages.
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 for seven in other countries. Where the 1
is written with a long upstroke, the number 7 has a horizontal stroke
through the vertical line.
While the shape of the 1 character has an ascender in most modern
typefaces, in typefaces with text figures, the character usually is of
x-height, as, for example, in alt=Horizontal guidelines with a one
fitting within lines, a four extending below guideline, and an eight
poking above guideline.
Many older typewriters do not have a separate symbol for '1' and use
the lowercase letter 'l' instead. It is possible to find cases when
the uppercase 'J' is used, while it may be for decorative purposes.
Mathematics
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Mathematically, 1 is:
*in arithmetic (algebra) and calculus, the natural number that follows
0 and the multiplicative identity element of the integers, real
numbers and complex numbers;
*more generally, in algebra, the multiplicative identity (also called
'unity'), usually of a group or a ring.
Tallying is often referred to as "base 1", since only one mark - the
tally itself - is needed. This is more formally referred to as a unary
numeral system. Unlike base 2 or base 10, this is not a positional
notation.
Since the base 1 exponential function (1'x') always equals 1, its
inverse does not exist (which would be called the logarithm base 1 if
it did exist).
There are two ways to write the real number 1 as a recurring decimal:
as 1.000..., and as 0.999....
Formalizations of the natural numbers have their own representations
of 1:
*in the Peano axioms, 1 is the successor of 0;
*in 'Principia Mathematica', 1 is defined as the set of all singletons
(sets with one element);
*in the Von Neumann cardinal assignment of natural numbers, 1 is
defined as the set {0}.
In a multiplicative group or monoid, the identity element is sometimes
denoted 1, but 'e' (from the German 'Einheit', "unity") is also
traditional. However, 1 is especially common for the multiplicative
identity of a ring, i.e., when an addition and 0 are also present.
When such a ring has characteristic 'n' not equal to 0, the element
called 1 has the property that (where this 0 is the additive identity
of the ring). Important examples are finite fields.
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, vectors are often normalized to give unit
vectors, that is 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 equal 1.
It is also the first and second number in the Fibonacci sequence (0 is
the zeroth) and is the first number in many other mathematical
sequences.
1 is 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".)
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 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). 1 was
formerly considered prime by some mathematicians, using the definition
that a prime is divisible only by 1 and itself. However, this
complicates the fundamental theorem of arithmetic, so modern
definitions exclude units.
By definition, 1 is the magnitude, absolute value, or norm of a unit
complex number, unit vector, and a unit matrix (more usually called an
identity matrix). 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 almost certain
to occur.
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}{1-x} = 1+x+x^2+x^3+ \ldots
This power series converges and has finite value if and only if, |x|
< 1 .
In category theory, 1 is sometimes used to denote the terminal object
of a category.
In number theory, 1 is the value of Legendre's constant, which was
introduced in 1808 by Adrien-Marie Legendre in expressing the
asymptotic behavior of the prime-counting function. Legendre's
constant was originally conjectured to be approximately 1.08366, but
was proven to equal exactly 1 in 1899.
Table of basic calculations
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Multiplication !1 !2 !3 !4 !5 !6 !7 !8
!9 !10 !11 !12 !13 !14 !15 !16
!17 !18 !19 !20 !21 !22 !23 !24 !25 !50
!100 !1000
|**1 � 'x**'
|**1**
|2 |3 |4 |5 |6 |7 |8 |9 |10 !
|11 |12 |13 |14 |15 |16 |17 |18 |19 |20
! |21 |22 |23 |24 |25 ! |50 |100 |1000
Division !1 !2 !3 !4 !5 !6 !7 !8
!9 !10 !11 !12 !13 !14 !15
|**1 ÷ 'x**'
|**1**
|0.5 |0. |0.25 |0.2 |0.1 |0. |0.125 |0. |0.1 !
|0. |0.08 |0. |0.0
|0.0
|**'x' ÷ 1**
|**1**
|2 |3 |4 |5 |6 |7 |8 |9 |10 !
|11 |12 |13 |14 |15
Exponentiation !1 !2 !3 !4 !5 !6 !7 !8
!9 !10 !11 !12 !13 !14 !15 !16
!17 !18 !19 !20
|**1('x')**
|**1**
|**1**
|**1**
|**1**
|**1**
|**1**
|**1**
|**1**
|**1**
|**1**
|**1**
|**1**
|**1**
|**1**
|**1**
|**1**
|**1**
|**1**
|**1**
|**1**
!
|**'x'(1)**
|**1**
|2 |3 |4 |5 |6 |7 |8 |9 |10 !
|11 |12 |13 |14 |15 |16 |17 |18 |19 |20
In technology
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* The resin identification code used in recycling to identify
polyethylene terephthalate.
*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 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 true in many programming languages.
*1 is the ASCII code of "Start of Header".
In science
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*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 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
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In the philosophy of Plotinus and a number 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 [i.66]).
In literature
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*Number One is a character in the book series Lorien Legacies by
Pittacus Lore.
*Number 1 is also a character in the series "Artemis Fowl" by Eoin
Colfer.
*In a 1968 song by Harry Nilsson and recorded by Three Dog Night, the
number one is identified as "the loneliest number".
*'We Are Number One' is a 2014 song from the children's TV show
LazyTown, which gained popularity as a meme.
In comics
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*A character in the Italian comic book Alan Ford (authors Max Bunker
and Magnus), very old disabled man, the supreme leader of the group
TNT.
*A character in the Italian comic series PKNA and its sequels, an
artificial intelligence as an ally of the protagonist Paperinik
In sports
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*In baseball scoring, the number 1 is assigned to the pitcher.
*In association football (soccer) the number 1 is often given to the
goalkeeper.
*In most competitions of rugby league (though not the Super League,
which uses static squad numbering), the starting fullback wears jersey
number 1.
*In rugby union, the starting loosehead prop wears the jersey number
1.
*1 is the lowest number permitted for use by players of the National
Hockey League (NHL); the league prohibited the use of 00 and 0 in the
late 1990s. (The highest number permitted is 98.)
*1 is the lowest number permitted for use at most levels of American
football. Under National Football League policy, it can only be used
by a quarterback or kicking player (during preseason play,
restrictions are looser, and players of other positions can wear the
number and can also, if no other options exist, wear 0).
*In Formula One, the previous year's world champion is allowed to use
the number 1.
In film
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*'One A.M.' (1916), starring Charlie Chaplin.
*'One More Time' (1970), directed by Jerry Lewis and starring Sammy
Davis Jr. and Peter Lawford.
*' One Day' (2011), starring Anne Hathaway and Jim Sturgess.
In other fields
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*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 dialing code for Greater London
*PRS One, a German paraglider design
*+1 is the code for international telephone calls to countries in the
North American Numbering Plan
See also
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*�1
*+1 (disambiguation)
*One (word)
*Root of unity
External links
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*[
https://web.archive.org/web/20140201161542/http://numdic.com/1 The
Number 1]
*[
http://www.positiveintegers.org/1 The Positive Integer 1]
*[
http://primes.utm.edu/curios/page.php/1.html Prime curiosities: 1]
License
=========
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License URL:
http://creativecommons.org/licenses/by-sa/3.0/
Original Article:
http://en.wikipedia.org/wiki/1
