1 (number)

This article discusses the number one. For the year AD 1, see 1. For other uses of 1, see 1 (disambiguation)

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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)

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, unit or Integer is sometimes used as an adjective in this sense. (For example, a line segment of "unit length" is a line segment of length 1.)

Written out

one : English
一 : Chinese
un : French
ein : German
ένας : Greek
uno : Italian and Spanish
1 つ : Japanese
1개 : Korean
um : Portuguese
одно : Russian

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, 1^x = 1, and for nonzero x, x⁰ = 1 (see exponentiation)

x↑↑1 = x and 1↑↑x = 1 (see tetration).

1=0.999... (see Proof that 0.999... equals 1)

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.

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 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 the smallest positive odd number.

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.

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 some European countries (e.g., Germany) 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.

In science

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

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 religion

Many religions ascribe symbolic meanings to the concept of one-ness.

In Judaism's credo, the Shema Yisrael, God is described as being "one" (Deut. 6:4)
In Christianity, Jesus is God's only begotten Son
In many Gnostic systems and heresiologies, God is known as the Monad, the One
There is one surat al-Fatiha in the Qur'an.
All is One according to Monism and Theosophy.

In culture

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 modern culture, 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 exaggeratedly (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 twelve-hour clock, 1:00 signals that one full hour has passed since the last change of the "AM" or "PM" meridian. On a twenty-four-hour clock, it signals that one full hour has passed since midnight.

One is used as a pronoun in English grammar to refer to an unspecified person: see generic you.

In film

One is the title of a few different films:

One (1956, TV movie)
Eins (English title One) (1971)
Jedno (1993, direct to video, English title One)
One (1994)
One (1998)
One (2000)
One (2001)
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 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", reads the first line of "One" by Three Dog Night. The number also appears in the title of songs by U2 (voted greatest song in history by a Channel4 poll in 2001), Metallica, Creed, Marvin Hamlisch (in the musical A Chorus Line), Alanis Morissette, Harry Nilsson, and the Bee Gees. #1 Zero is a song by Audioslave. It was 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."

The last George Jones and Tammy Wynette duet album was called "One" and included a love song of the same name.

In politics

ONE Campaign an antipoverty coalition.

as Air Force One is the callsign of any United States Air Force aircraft carrying the President of the United States.
the house number of Number One Observatory Circle, the US Vice-President's residence.

In currency

the denomination of U.S. dollar bill with George Washington's portrait, and the denomination of coin with Sacagawea's portrait. It is also the denomination of the older Eisenhower and Susan B. Anthony tender coins and the American Silver Eagle bullion coin.
the denomination of Canadian dollar coin with a swimming loon on the reverse, hence its universally employed nickname, the "loonie"
in cents of a U.S. dollar, the denomination of coin with Abraham Lincoln's portrait, commonly known as a penny.
in cents of a Canadian dollar, the denomination of coin with two maple leaves on the reverse, also known as a penny.

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 football, 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 symbolic gesture is used to demonstate a team's sense of pride in itself. Typically, this type of celebration is considered a default.

In Formula 1, the number one is used to designate the car of the previous year's champion.

In technology

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

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

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

One is the code for international direct dial phone calls to countries participating in the North American Numbering Plan, such as the United States and Canada.

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".

In ASCII (American Standard Code for Information Interchange), 1 is a non printing character representing 'Start of Heading' or 'SOH'

In other fields

One is:

the designation of many roads, listed at Route 1
the address of Apsley House, known simply as Number 1, London
a subway service in New York City. See 1 (New York City Subway service)
the name of a train operating company in East Anglia, England. See: 'one'
an enneagram personality type
part of a nickname for the U.S. Army's First Infantry Division, "The Big Red One"
as a word in all capitals, ONE, a trademark of Nestlé Purina PetCare for a line of pet food products (acronym for the "optimum nutritional effectiveness" they are claimed to possess)
In Astrology, Aries is the first sign of the Zodiac
In Tarot, card No. 1 is "the Magician"

Historical years

AD 1 or the immediately previous year 1 BC, 1901, 2001, etc.

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