This tag is for questions relating to the Gaussian integer, which is a complex number $~z=~a~+i~b~$ whose real part $~a~$ and imaginary part $~b~$ are both integers.

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as $\mathbb {Z}[i]$. This integral domain is a particular case of a commutative ring of quadratic integers. It does not have a total ordering that respects arithmetic.

Formally, Gaussian integers are the set

$$\mathbb {Z}[~i~]=\{~a~+i~b~|~a,~b~\in \mathbb {Z}\} \qquad \text{where$~i=\sqrt{-1}$}$$

In particular, since either or both of $~x~$ and $~y~$ are allowed to be $~0~$, every ordinary integer is also a Gaussian integer.

When considered within the complex plane, the Gaussian integers constitute the $~2-$dimensional integer lattice.

The conjugate of a Gaussian integer $~a~ +~ i~b~$ is the Gaussian integer $~a~ -~ i~b~$.

Gaussian integers can be uniquely factored in terms of other Gaussian integers (known as Gaussian primes) up to powers of $~i~$ and rearrangements.