Gaussian Integer

In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z. The Gaussian integers are a special case of the quadratic integers. This domain does not have a total ordering that respects arithmetic.

Formally, Gaussian integers are the set

Note that when they are considered within the complex plane the Gaussian integers may be seen to constitute the 2-dimensional integer lattice.

The norm of a Gaussian integer is the natural number defined as

(Where the overline over "a+bi" refers to the complex conjugate.)

The norm is multiplicative, i.e.

The units of Z are therefore precisely those elements with norm 1, i.e. the elements 1, −1, i and −i.