Tag Archives: gaussian integers

Euclidean Algorithm for Complex Integers

We got into the Euclidean Algorithm at the beginning of this course and one of the questions I had at the end of the lab was how the algorithm applied to other numbers systems beyond the integers.  I was curious how it worked with polynomials and complex numbers.  Imagine my delight to discover this lab.

In this lab, we investigate the $Gaussian$ $Integers$, which take the form $a+bi$ and are named after Karl Friedrich Gauss who was the first person to discover the correct way to generalize the Euclidean Algorithm for these numbers.

Much like we did with the first Euclid lab, we will investigate using complex numbers in the division algorithm.  This lab also defines a new term.  In the complex plane, the distance a point is from the origin is called its $modulus$  and the $norm$ of a complex number $z$ is denoted by $N(z)$, which is the square of its modulus.

The Euclidean Algorithm is very fascinating and I’m excited to see how else it can be applied to other number systems.