# 1 (number)

This article discusses the number one. For the year AD 1, see 1. For other uses of 1, see 1 (disambiguation)
 0 10 20 30 40 50 60 70 80 90 >> Cardinal 1 one Ordinal 1st first Numeral system unary Factorization $\displaystyle 1$ Divisors 1 Roman numeral I Unicode representation of Roman numeral Ⅰ, ⅰ prefixes mono- (from Greek) uni- (from Latin) Binary 1 Octal 1 Duodecimal 1 Hexadecimal 1 Hebrew א (Alef)
Look up one in Wiktionary, the free dictionary.

1 (one) is a number, numeral, and the name of the glyph representing that number. It is the natural number following 0 and preceding 2. It represents a single entity. One is sometimes referred to as unity, and unit is sometimes used as an adjective in this sense. (For example, a line segment of "unit length" is a line segment of length 1.)

## History

Some Ancient Greeks did not consider one as a number: they considered it to be the unit, two being the first proper number as it represented a multiplicity.

## In mathematics

For any number x:

x·1 = 1·x = x (This expresses the fact that 1 is the multiplicative identity.) As a consequence of this, 1 is a 1-automorphic number in any place-based numbering system.
x/1 = x (see division)
x1 = x, 1x = 1, and for nonzero x, x0 = 1 (see exponentiation)
x↑↑1 = x and 1↑↑x = 1 (see tetration).

Using ordinary addition, we have 1 + 1 = 2; depending on the interpretation of the symbol "+" and the numeral system used, the expression can have many different meanings, listed at one plus one.

One cannot be used as the base of a positional numeral system in the ordinary way. Sometimes tallying is referred to as "base 1", since only one mark (the tally) is needed, but this doesn't work in the same way as other positional numeral systems. Related to this, one cannot take logarithms with base 1, since the "exponential function" with base 1 is the constant function 1.

In the Von Neumann representation of natural numbers, 1 is defined as the set {0}. This set has cardinality 1 and hereditary rank 1. Sets like this with a single element are called singletons.

In a multiplicative group or monoid, the identity element is sometimes denoted "1", but "e" (from the German Einheit, unity) is more traditional. However, "1" is especially common for the multiplicative identity of a ring. (Note that this multiplicative identity is also often called "unity".)

One is its own factorial, and its own square and cube (and so on, as 1 × 1 × ... × 1 = 1). As a consequence of its being its own square, one is also a Kaprekar number. One is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number to name just a few.

It is also the first and second numbers in the Fibonacci sequence, and is the first number in a lot of mathematical sequences. As a matter of convention, Sloane's early Handbook of Integer Sequences added an initial 1 to any sequence that didn't already have it, and considered these initial 1's in its lexicographic ordering. Sloane's later Encyclopedia of Integer Sequences and its Web counterpart, the On-Line Encyclopedia of Integer Sequences, ignore initial ones in their lexicographic ordering of sequences, because such initial ones often correspond to trivial cases.

One is the empty product.

One is a harmonic divisor number.

One is most often used for representing 'true' as a Boolean datatype in computer science.

One is currently considered neither a prime number, nor a composite number - although it used to be considered prime. Defining a prime as a number that is only divisible by one and itself, one is a prime. However, for purposes of factorization and especially the fundamental theorem of arithmetic, it is more convenient to not think of one as a prime factor, or to think of it as an implicit factor that's always there but need not be written down. To exclude the number one from the list of prime numbers, primality is defined as a number having exactly two distinct positive divisors, one and itself. The last professional mathematician to publicly label 1 a prime number was Henri Lebesgue in 1899, although Carl Sagan included one in a list of prime numbers in his book Contact in 1985.

One is one of three possible return values of the Möbius function. Passed an integer that is square-free with an even number of distinct prime factors, the Möbius function returns one.

One is the only odd number that is in the range of Euler's totient function φ(x), in the cases x = 1 and x = 2.

One is the only 1-perfect number (see multiply perfect number).

One is equal to the sum of its digits in any place-based numbering system, making it an all-Harshad number.

One is the number of n × n magic squares for n = 1, 3.

One is the number of n-queens problem solutions for n = 1.

One is a meandric number, a semi-meandric number, and an open meandric number.

By definition, 1 is the magnitude or absolute value of a unit vector and a unit matrix.

One is the value of the sine and cosine at π/2 and 0 radians, respectively.

One is the most common leading digit in many sets of data, a consequence of Benford's law.

## The Arabic glyph

The glyph used today in the Western world to represent the number 1, a vertical line, often with a little serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Brahmin Indians, who wrote 1 as a horizontal line (in Chinese today this is the way it is written). The Gupta wrote it as a curved line, and the Nagari 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 Punjabi scripts). The Nepali 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 Roman numeral I.

In fonts with text figures, 1 is typically the same height as a lowercase X, for example, File:TextFigs148.png.

## In science

One is:

• set equal to celerity (c), the speed of light, in Heaviside notation to simplify calculations.
• the factor in ratios for unit conversions.
• the total density ratio for a flat universe.
• The atomic number of hydrogen

In astronomy,

Messier object M1, a magnitude 7.0 supernova remnant in the constellation Taurus, also known as the Crab Nebula.
The New General Catalogue object NGC 1, a 13th magnitude 13 spiral galaxy in the constellation Pegasus
The Saros number of the solar eclipse series which began on -2872 June 4 and ended on -1592 July 11 . The duration of Saros series 1 was 1280.1 years, and it contained 73 solar eclipses.
The Saros number of the lunar eclipse series which began on -2588 March 2 and ended on -1272 April 30. The duration of Saros series 1 was 1316.2 years, and it contained 74 lunar eclipses.

## In human society

Many human cultures have given the concept of one-ness symbolic meanings. Many religions consider God to be a perfect example of one-ness. See monad for a detailed discussion of other types of one-ness.

One represents unity, togetherness, and absence of separation or discrimination, e.g. "We are all one" or "everyone".

Something is unique if it is the only one of its kind. More loosely and exaggeratingly (especially in advertising) the term is used for something very special.

One is also an (archaic) expression of the first person singular ("one is not amused") and of the second person singular ("does one take sugar?").

In Western culture, it is believed by many that the maximum number of girlfriends or boyfriends one may have at one time is 1. Also, it is strongly believed that you can be married to only 1 person at any time - this is called monogamy. Being married to more than one person at any time is called bigamy or polygamy. This is illegal in many Western societies.

Among children, or when otherwise calling for subtlety, the phrase "number 1" can refer to the act of urination. This can derive from a traditional U.S. elementary school practice of holding up one or two fingers to indicate the approximate time of a requested absence.

"Number 1" can also refer to oneself, or that something is first in its class, the latter being used often as a cheer in sports games.

On a clock, 1:00 signals that one full hour has passed since the last change of the "AM" or "PM" meridian.

## In film

File:Poster for one - large.jpg
One...The Movie, released in 2005.

One is the title of a few different films:

1. One (1956, TV movie)
2. Eins (English title One) (1971)
3. Jedno (1993, direct to video, English title One)
4. One (1994)
5. One (1998)
6. One (2000)
7. One (2001)
8. One... The Movie (2005) A documentary about religion, includes an interview with the Dalai Lama.

For a more complete listing of films with the number 1 in the title, see List of films: numbers#1.

## In music

In harmonic analysis of tonal music, the tonic chord is referred to as I.

"One is the loneliest number that you'll ever do", according to the first line of "One" by Three Dog Night. The number appears in the title of songs by Metallica, U2, Creed, Marvin Hamlisch (in the musical A Chorus Line), Alanis Morissette, Harry Nilsson, Three Dog Night, and the Bee Gees. Also the title of a best-selling compilation album of all Beatles songs that reached number 1 in the UK or US charts (see List of Number 1 Hits (USA)).

The song Green grow the rushes, O has the line "One and all alone and evermore shall be so."

Juan Crisóstomo Arriaga and César Franck each composed one symphony.

Ludwig van Beethoven, Johannes Brahms, Peter Ilyich Tchaikovsky, Antonín Dvořák, Jean Sibelius, and Edward Elgar each composed one concerto for violin and orchestra.

Robert Schumann, Edvard Grieg, and George Gershwin each wrote one concerto for piano and orchestra.

Giuseppe Verdi, Claude Debussy, and Maurice Ravel each composed one string quartet.

Anton Bruckner composed one quintet for strings.

## In religion

There is one surat al-Fatiha in the Qur'an.

All is One according to Monism and Theosophy.

## In sports

In some sports, one is the number of a specific position: in rugby union, the number of the loosehead prop; in baseball, the number representing the pitcher's position; in (soccer), the number of the goalkeeper.

It is the retired baseball jersey number of Ozzie Smith and Richie Ashburn.

In 2004, fans of the Philadelphia Eagles NFL team used the phrase "One" to show support for the team as they inched closer to the Super Bowl. The full text of the phrase was "One Team. One City. One Dream."

After a victory, especially the championship finals, the winning team will often pose for a team photo. They usually crowd around the trophy and raise their index finger while holding up the number one. This proves that they are the champions and they are number one. Typically, this type of celebration is considered a default.

## In technology

One is the DVD region of the United States and Canada.

In the DOS Shell and many Windows programs, the function key F1 calls up online help.

On most standard phones, the 1 key is not associated with any letters the way other number keys are, but on the BlackBerry, 1 is also the key for the letters E and R.

Some cellular phones associate the "1" key with various symbols (i.e. the pound sign, the ampersand, etc.) when users engage in text messaging.

In the Rich Text Format specification, 1 is the language code for the Arabic language. All codes for dialects of Arabic are congruent to 1 mod 256.

1 is a punctuation mark indicating exclamation, or the letter "l" in "leetspeak".

One is: