1 (one, also called unit, and unity) is a number
and a numerical digit
used to represent that number in numeral
s. It represents a single entity, the unit
. 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
of natural number
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 returns that 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
; although universal today, this was a matter of some controversy until the mid-20th century.
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
''eins'' and Old Norse
Compare the Proto-Indo-European root ''*oi-no-'' (which means "one, single"
) to Greek
''oinos'' (which means "ace" on dice
), Old Persian
, Old Church Slavonic
''-inu'' and ''ino-'', Lithuanian
''vienas'', Old Irish
''oin'' and Breton
As a number
One, sometimes referred to as unity,
is the first non-zero natural number
. It is thus the integer
Any number multiplied by one remains that number, as one is the identity
. 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
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
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. Whereas the digit 1 is written with a long upstroke, the digit 7 has a horizontal stroke through the vertical line.
While the shape of the character for the digit 1 has an ascender
in most modern typeface
s, in typefaces with text figures
, the glyph usually is of x-height
, as, for example, in
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.
Mathematically, 1 is:
) and calculus
, the natural number
that follows 0
and the multiplicative identity element
of the integer
s, real number
s and complex number
*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
of natural numbers, it is defined as the set
In a multiplicative group
, 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 field
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 absolutely or almost certain
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.
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...
. 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 into unit vector
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
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
that has all coefficients 1 is given by
This power series converges and has finite value if and only if
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
'' 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
* The resin identification code
used in recycling to identify polyethylene terephthalate
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 computer
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
are also known as quantities of dimension one.
*1 is the atomic number of hydrogen
*+1 is the electric charge
s 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
*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 the philosophy of Plotinus
(and that of other neoplatonist
s), 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
*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
* ''1'' (Beatles album)
, a compilation album by the Beatles.
, a 1991 song by Irish rock band U2
*A character in the Italian comic book Alan Ford
(authors Max Bunker
), 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
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 being 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.
'' (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
*''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
, the first year of the Common Era
*01, the former dialing code for Greater London
, a German paraglider design
*+1 is the code for international telephone calls to countries in the North American Numbering Plan
*List of mathematical constants
*Root of unity
*List of highways numbered 1
The Number 1The Positive Integer 1