One

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The concept of one has many synonyms

• Unity
• Unique
• Monad
• ...

The number one

One has sometimes not been considered a number in some cultures in the past (because it does not involve plurality). The number one is represented by the numeral 1. 1 is the first positive integer and the first positive odd number.

Although 1 has been considered prime until the beginning of the 20th century (former definition: no divisors apart from 1 and itself, itself not necessarily distinct from 1) the unit (i.e. multiplicatively invertible element) 1 is now widely known as the empty product (defined as the multiplicative identity, i.e. 1) of primes, where an integer

n

is now considered prime if and only if it has exactly two divisors, a unit and a nonunit, where associates of

k

(i.e. product of some unit

u

with

k

) are not considered distinct divisors. This is the definition (resulting from the development of abstract algebra at the turn of the 20th century) now accepted by most mathematicians.

Properties of 1

• Multiplicative identity (the empty product is defined as the multiplicative identity).
• Unit (has a multiplicative inverse) in any number set.

• 0 0 = 0! = 1

  (the empty product, defined as the multiplicative identity).

The identity matrix

The

n  × n

identity matrix

In

is a square matrix that has 1’s along the main diagonal and 0’s for all other entries. This matrix is often written simply as

I

, and is special in that it acts like 1 in matrix multiplication.

Examples:

     

I1  =  1 , I2  =  1   0 0   1 , I3  =  1   0   0 0   1   0 0   0   1 , I4  =  1   0   0   0 0   1   0   0 0   0   1   0 0   0   0   1 , I5  =  1   0   0   0   0 0   1   0   0   0 0   0   1   0   0 0   0   0   1   0 0   0   0   0   1 , ...

See also

• Roots of unity
• Units
• Zero (0)

External links

• Weisstein, Eric W., One, from MathWorld—A Wolfram Web Resource.
• 1—Wikipedia.org.