1 (number)

From Academic Kids

This article discusses the number one. For the year 1 AD, see 1. For other uses of 1, see 1 (disambiguation)
<tr><td>Hebrew<td>א (Alef)
Template:Numbers (digits)
Cardinal 1
one
Ordinal 1st
first
Numeral system unary
Factorization <math> 1 <math>
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

1 (one) is a number, numeral, and glyph. 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.)

Contents

Evolution of the glyph

Image:Evolution1glyph.png

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, Missing image
TextFigs148.png
Image:TextFigs148.png

.

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:

x1 = 1x = 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 Einigkeit, 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 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 Mbius function. Passed an integer that is square-free with an even number of distinct prime factors, the Mbius 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.

See also -1.

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 (http://www.ngcic.org/) NGC 1, a 13th magnitude 13 spiral galaxy in the constellation Pegasus
The Saros number (http://sunearth.gsfc.nasa.gov/eclipse/SEsaros/SEsaros1-175.html) 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 (http://sunearth.gsfc.nasa.gov/eclipse/LEsaros/LEsaros1-175.html) 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."

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.

In other fields

One is:

ca:Un da:1 (tal) de:Eins es:Uno eo:Unu fr:1 (nombre) ko:1 ia:1 it:Uno he:1 (מספר) nah:Ce nl:Een ja:1 no:1 (tall) nn:Talet in pl:Jeden pt:Um ru:Единица simple:One sl:1 (število) fi:1 (luku) sv:1 (tal) th:1 (จำนวน) zh:1

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