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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