Definition:Gaussian Integer

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A Gaussian integer is a complex number whose real and imaginary parts are both integers.

That is, a Gaussian integer is a number in the form:

$a + b i: a, b \in \Z$

The set of all Gaussian integers can be denoted $\Z \sqbrk i$, and hence can be defined as:

$\Z \sqbrk i = \set {a + b i: a, b \in \Z}$

Some sources use the symbol $J$ for the set $\Z \sqbrk i$.


The following are examples of Gaussian integers:

$2 - 3 i$
$1 + 2 i$

Also known as

A Gaussian integer can also be referred to as a complex integer.

Some sources render the eponym in lowercase: gaussian integer.

Also see

  • Results about Gaussian integers can be found here.

Source of Name

This entry was named for Carl Friedrich Gauss.

Historical Note

The concept of a Gaussian integer was introduced by Carl Friedrich Gauss in his papers of $1828$ and $1832$.