The simple complex numbers

TL;DR

This paper introduces a new geometric interpretation of complex numbers as vector operations, offering a simpler and more natural framework that enhances understanding in geometry and physics, especially for rotations.

Contribution

It presents a novel interpretation of complex numbers as vector operations, differing from the traditional point-based view, and clarifies its relation to the complex plane and applications.

Findings

Provides a more natural geometric interpretation of complex numbers

Demonstrates improved understanding of rotations in a plane

Shows applications in physics such as mechanics and optics

Abstract

A new simple geometrical interpretation of complex numbers is presented. It differs from their usual interpretation as points in the complex plane. From the new point of view the complex numbers are rather operations on vectors than points. Moreover, in this approach the real, imaginary and complex numbers have similar interpretation. They are simply some operations on vectors. The presented interpretation is simpler, more natural, and better adjusted to possible applications in geometry and physics than the usual one, especially for describing rotations in a plane. The relation of the new approach to the usual interpretation and especially to the notion of complex plane is also clarified in the paper. The new interpretation of complex numbers gives new insight into their applications in physics, which is demonstrated by some elementary examples in mechanics and optics