Information about One

0 1 2 3 4 5 6 7 8 9 → List of numbers — Integers 0 10 20 30 40 50 60 70 80 90 →
Cardinal	1 one
Ordinal	1st first
Numeral system	unary
Factorization
Divisors	1
Roman numeral	I
Roman numeral (Unicode)	Ⅰ, ?
Arabic	١
Bengali	১
Chinese numeral	一 , ?
Devanāgarī	१
Hebrew	א (Alef)
Khmer	?
prefixes	mono- (from Greek) uni- (from Latin)
Binary	1
Octal	1
Duodecimal	1
Hexadecimal	1

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 or unit as an adjective. For example, a line segment of "unit length" is a line segment of length 1.

In mathematics

For any number x:^[1]

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 positional numeral 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).

Using ordinary addition, we have 1 + 1 = 2. 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 real number system, 1 can be represented in two ways as a recurring decimal: as 1.000... and as 0.999... This identity is not immediately obvious to many people, and a full understanding of why it is true requires an understanding of the properties of the real numbers. See the article 0.999... for more details.

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 Principia Mathematica, 1 is defined as the set of all 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). 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 integer.

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. (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).

By definition, 1 is the magnitude or absolute value of a unit vector and a unit matrix (more usually called an identity matrix). Note that the term unit matrix is usually used to mean something quite different.

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.

Sequence of natural numbers always ends with the number 1 (Collatz conjecture).

See also −1.

List of basic calculations

Multiplication	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15	16	17	18	19	20		21	22	23	24	25		50	100	1000
	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15	16	17	18	19	20		21	22	23	24	25		50	100	1000

Division	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15
	1	0.5		0.25	0.2			0.125		0.1						| |1 |2 |3 |4 |5 |6 |7 |8 |9 |10 ! |11 |12 |13 |14 |15

Exponentiation	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15	16	17	18	19	20
	1	1	1	1	1	1	1	1	1	1		1	1	1	1	1	1	1	1	1	1 | |1 |2 |3 |4 |5 |6 |7 |8 |9 |10 ! |11 |12 |13 |14 |15 |16 |17 |18 |19 |20

Evolution of the glyph

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 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.^[2] 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.

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 .

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
• Group 1 in the Periodic Table consists of hydrogen and the alkali metals whose usual valence is +1.
• Period 1, the shortest row in the Periodic Table, consists only of hydrogen and helium.
• A haploid has one set of chromosomes in the nucleus.

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 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.
• The first space shuttle mission STS-1

In religion

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

• Islam's core belief is in the One God. Allah is the Arabic translation for the word God.
• In Judaism's credo, the Shema Yisrael, God is described as being "one" (Deut. 6:4)
• Christian "God is one" as expressed in scripture Mk 12:29, Jas 2:19 NASB. In many sects, Jesus is God's only begotten Son; part of the Holy Trinity "three persons in one God,". Others, including many Mormon sects, believe God, Jesus, and the Holy Spirit to be individual entities that make up a 'Godhead'.
• In Sikhism's philosophy, there is only one God.
• In many Gnostic systems and heresiologies, God is known as the Monad, the One
• 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, monogamy is a typical and socially encouraged relationship practice.

Among children, or when otherwise calling for subtlety, the phrase "number 1" can refer to the act of urination.

"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 music

• In harmonic analysis of tonal music, the tonic chord is referred to as I.
• In funk music, being "On The One" denotes a four-to-the-bar bass guitar-driven beat with emphasis on first note. The term can also describe being "plugged in" to the music, achieving a state of nirvana.
• "One is the loneliest number that you'll ever do", reads the first line of "One" by Harry Nilsson, which was covered by Three Dog Night. The number also appears in the title of songs by U2 (voted greatest song in history by a Channel 4 poll in 2001), Metallica, Korn, Apocalyptica, Creed, Marvin Hamlisch (in the musical A Chorus Line), Alanis Morissette, 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 last George Jones and Tammy Wynette duet album was called "One" and included a love song of the same name.

In politics

• ONE Campaign is 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 the Danish Krone coin, featuring a hole in the center.
• 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 league the number of the fullback; in rugby union, the number of the loosehead prop; in baseball, the number representing the pitcher's position; in basketball, the number representing the point guard's position; in football, the number of the goalkeeper.
• In Formula 1, the number one is used to designate the car of the previous year's champion.
• In Motocross, the number on the champion of last year's number plate (unless he chooses to have a red background or to not any symbol of winning).

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.

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

A common railway gauge is 1 m (3 ft 3³⁄₈ in).

In other fields

One is:

• the designation of many roads; see List of highways numbered 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)
• 'one' train operating company in the United Kingdom
• 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.

Etymology

The Old English án is in Old Frisian ân, ên, Old Saxon ên (Middle Dutch, Dutch een), Old High German (Middle High German, German) ein, Old Norse einn:–ein-r (Danish een, Sw. en), Gothic ain-s:–Old Teutonic *ain-oz:–pre-Teutonic *oinos = Latin ūnus (Old Latin oinos); Old Irish óen, Old Slavic inu, Lithuanian venas one; cf. Greek oἶνoς, oἴνη, ace. Old English án became in regular course in south and midland dialect on, exemplified before 1200. By the 15th century, on, oon, in southwest and west, had developed (through on, uon, uön, won, wun) an initial w (cf. the southwest wuk, wuts = oak, oats), which only occasionally appears in the spelling, but is now the standard pronunciation. The first orthoepist to refer to it was apparently Jones 1701: earlier grammarians, down to Christopher Cooper, 1685, give to one the sound that it has in alone, atone, and only; Thomas Dyche in 1710 has [ɒn] beside [wɒn]. In the north, ān was retained in Middle English; but through the narrowing of the originally long ā to [æ:, ɛ:, e:, ɛə, ɪə] ân has sunk in dialectal utterance through ane, to eane, eän, yan, yen, the development of [jɛn] in the north being the counterpart of that of [wʌn] in the south. In Old English, án had the full adjective inflexions, definite and indefinite, remains of which persisted in the south to ca. 1300, and in Kent still later; but, in north and midland England, the uninflected ān, ōn, with the definite form āne, ōne (Old English ána, áne), is found in the accusative and dative, as well as the nominative by 1200. Already also, ān, ōn were reduced before a consonant to ā, ō (oo), which did not die out till the 16th century.

In the north the separation of ân and â was more permanent; at the present day in Scots the full form ane, eane, etc., is only used absolutely or in the predicate, ae, eae, is the attributive form before consonants and vowels alike: ae day, ae yeir, we hae ane; so in north English dialects with yà and yàn. From the early an, a, pronounced proclitically without stress, arose the “indefinite article” an, a. In the northern dialect the numeral and article were long written alike, the stress or emphasis alone distinguishing them; in 16th century Scots both were written ane. By more or less permanent coalescence of a preceding thet, the collocations thet ane, thet one, thet a, thet o, became the tane, the tone, the ta, the to.

See also

• 0.999...

References