Gauss and the Complex Plane
The birth of seeing complex numbers as points on a plane
Goal of this page
Understand how revolutionary the now-commonplace view "a complex number = a point on a plane" once was.
Prerequisites
- Basics of the coordinate plane
- The discovery of imaginary numbers
Contents
§1 The need for a geometric interpretation
Ever since Cardano and Bombelli, imaginary numbers had been used as a "computational convenience," but the essential question "what is a number?" remained unanswered. Until the end of the 18th century, the philosophical debate "do imaginary numbers really exist?" continued.
§2 Gauss's planar representation
Carl Friedrich Gauss held, from early on, the view that identifies a complex number $z = a + bi$ with the point $(a, b)$ on a plane. In his 1799 doctoral dissertation he gave an important proof of the fundamental theorem of algebra (every degree-$n$ polynomial with complex coefficients has a root among the complex numbers). However, the paper in which he explicitly treated complex numbers as points on a plane was not published until 1831.
The same geometric representation was presented by the Dane Caspar Wessel in 1797 (published 1799), and later shown independently by Argand (1806). That the view nonetheless spread under the name "Gaussian plane" owes much to Gauss's great influence.
In this way, the metaphysical problem "what is an imaginary number?" was given a simple geometric answer: a complex number corresponds naturally to a point on a plane.
§3 The geometric meaning of operations
Seen geometrically, the operations on complex numbers can be understood in a natural way:
- Addition: vector addition (parallelogram)
- Multiplication: rotation + scaling
- Conjugation: reflection across the real axis
- Modulus: distance from the origin
With this, the algebra of complex numbers turned from "abstract symbol manipulation" into "concrete geometric operations." The complex analysis and function theory that followed are built on this visual understanding.
§4 Summary
- The geometric representation came first from Wessel (1797); Gauss's explicit account was published in 1831
- The view that puts complex numbers in correspondence with points on a plane
- The basic operations correspond to geometric operations
- One of the important foundations underlying later complex analysis
Frequently Asked Questions
Why is it also called the "Argand diagram"?
In 1806 the Swiss mathematician Argand published a similar diagram, so in the English-speaking world it is called the Argand diagram. Gauss knew of it earlier, but the publication of his work was delayed.
Are there representations other than the complex plane?
The polar representation $z = r e^{i\theta}$ is widely used. There is also a further development called the Riemann sphere (a sphere with one extra point added).
What other contributions did Gauss make?
An extremely wide range: number theory, geometry, statistics, physics, and more. He is called the "Prince of Mathematicians." His proof of the fundamental theorem of algebra (about polynomials with complex coefficients, 1799) was also his doctoral dissertation.