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1 ( one, also called unit, unity, and (multiplicative) identity) is a , numeral, and . It represents a single entity, the unit of or . For example, a of unit length is a line segment of  1. It is also the first of the infinite sequence of , followed by 2.


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 *ainaz. The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-.

Compare the Proto-Germanic root *ainaz to an, ains, en, een, eins and einn.

Compare the Proto-Indo-European root *oi-no- (which means "one, single") to oinos (which means "ace" on dice), unus (one), aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, oin and un (one).


As a number
One, sometimes referred to as unity,Skoog, Douglas. Principles of Instrumental Analysis. Brooks/Cole, 2007, p. 758. is the first non-zero . It is thus the before and after , and the first positive .

Any number multiplied by one remains that number, as one is the for . As a result, 1 is its own , its own square, its own cube, and so on. One is also the result of the , as any number multiplied by one is itself. It is also the only natural number that is neither nor with respect to division, but instead considered a unit (meaning of ).


As a digit
The glyph used today in the Western world to represent the number 1, a vertical line, often with a at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Indians, who wrote 1 as a horizontal line, much like the character 一. The wrote it as a curved line, and the sometimes added a small circle on the left (rotated a quarter turn to the right, this 9-look-alike became the present day numeral 1 in the Gujarati and scripts). The also rotated it to the right but kept the circle small. This eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the . In some countries, the little 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 , in typefaces with , the character usually is of , 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.


Mathematics
Mathematically, 1 is:

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 does not exist (which would be called the 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 , 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 , the 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 .

1 is the first 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 from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters. Likewise, are often normalized to give , that is vectors of magnitude one, because these often have more desirable properties. Functions, too, are often normalized by the condition that they have 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 sequence (0 is the zeroth) and is the first number in many other mathematical sequences.

1 is neither a nor a , but a unit (meaning of ), like −1 and, in the Gaussian integers, and − i. The fundamental theorem of arithmetic guarantees 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 . 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, , or norm of a unit complex number, , and a (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 of an event that is 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 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 , 1 is sometimes used to denote the of a category.

In , 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
1 × x12345678910 !11121314151617181920 !2122232425 !501001000

1 ÷ x10.50.0.250.20.10.0.1250.0.1 !0.0.080.0.00.0
x ÷ 112345678910 !1112131415

11111111111 !1111111111
x12345678910 !11121314151617181920


In technology
  • 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 is a sequence of 1 and 0 that is used in for representing any kind of .
  • 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 code of "".


In science
  • Dimensionless quantities are also known as quantities of dimension one.
  • 1 is the atomic number of .
  • +1 is the of and protons.
  • Group 1 of the consists of the .
  • Period 1 of the periodic table consists of just two elements, and .
  • 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 or (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 and a number of other , 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
  • Number One is a character in the book series by Pittacus Lore.
  • Number 1 is also a character in the series "Artemis Fowl" by .
  • In a 1968 song by and recorded by Three Dog Night, the number one is identified as "the loneliest number".


In comics
  • A character in the Italian comic book Alan Ford (authors and ), very old disabled man, the supreme leader of the group TNT.
  • A character in the Italian comic series and its sequels, an artificial intelligence as an ally of the protagonist Paperinik


In sports
  • In scoring, the number 1 is assigned to the .
  • In association football (soccer) the number 1 is often given to the goalkeeper.
  • In most competitions of (though not the European , which uses static squad numbering), the starting fullback wears jersey number 1.
  • In , 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 or kicking player (during , restrictions are looser, and players of other positions can wear the number and can also, if no other options exist, wear 0).
  • In , the previous year's world champion is allowed to use the number 1.


In other fields
  • 1 is the value of an in many playing card games, such as .
  • List of highways numbered 1
  • List of public transport routes numbered 1
  • 1 is often used to denote the Gregorian calendar month of .
  • 1 CE, the first year of the
  • 01, the former dialing code for
  • , a German paraglider design
  • +1 is the code for international telephone calls to countries in the North American Numbering Plan


See also
  • −1
  • +1 (disambiguation)
  • One (word)
  • Root of unity


External links

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