Definition:Gaussian Integer

Definition

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 $\Z \sqbrk i$.

Examples

The following are examples of Gaussian integers:

$2 - 3 i$

$5$

$-i$

$1 + 2 i$

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

• Definition:Lipschitz Quaternion
• Definition:Hurwitz Quaternion

• Units of Gaussian Integers
• Gaussian Integers form Integral Domain
• Definition:Algebraic Integer

• 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 $1832$.

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Direct Products
• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $5$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $3$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Gaussian integer
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Gaussian integer