2D, 3D, and 4D Geometric Algebras

Abstract

It is the belief that imaginary numbers appeared for the first time around 1540 when the mathematicians Tartaglia and Cardano represented real roots of a cubic equation in terms of conjugated complex numbers. A Norwegian surveyor, Caspar Wessel, was in 1798 the first one to represent complex numbers by points on a plane with its vertical axis imaginary and horizontal axis real. This diagram was later known as the Argand diagram, although the true Aragand’s achievement was an interpretation of $i=\sqrt{({-}1)}$ as a rotation by a right angle in the plane. Complex numbers received their name by Gauss, and their formal definition as pair of real numbers was introduced by Hamilton in 1835.