We are all familiar with splitting natural numbers into prime components. This decomposition is unique, except for the order of the factors. We can apply the idea of prime components to many more general sets of numbers.

The **Gaussian integers** are all the complex numbers with integer real and imaginary parts, that is, all numbers in the set

The set forms a two-dimensional lattice in the complex plane. For any element we consider the four numbers as *associates*. The associates of are known as units: .