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# One

(Redirected from 1)

The concept of one has many synonyms

• Unity
• Unique
• ...

## 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.

## 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
, ...