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. Into the late 18th and the 19th 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 $a, b$ real) with the point $(a, b)$ on a plane. In his 1799 doctoral dissertation he proved that every polynomial with real coefficients factors into real linear and quadratic factors (the real-coefficient case of the fundamental theorem of algebra); a gap in its geometric argument was filled by Ostrowski in 1920, and Argand (1806) is credited as the first to state the theorem for complex coefficients. 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 Danish–Norwegian surveyor Caspar Wessel in 1797 (published 1799), and later shown independently by Argand (1806). Wessel's paper, written in Danish, went largely unnoticed for about a century. That the view 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 one simple geometric answer: a complex number corresponds naturally to a point on a plane. Matching numbers with points does not by itself fix the multiplication, however; the rigorous construction of complex numbers as pairs of reals $(a, b)$ with defined addition and multiplication is due to Hamilton in the 1830s.
§3 The geometric meaning of operations
Seen geometrically, the operations on complex numbers can be understood in a natural way:
- Addition: the sum of position vectors from the origin (parallelogram)
- Multiplication: multiplying by $w \neq 0$ rotates about the origin by $\arg w$ and multiplies the distance from the origin by $|w|$ (rotation and 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." In the complex analysis and function theory that followed, the complex plane became the basic stage on which functions of a complex variable are studied (especially in Riemann's geometric view).
§4 Summary
- The geometric representation came first from Wessel (presented 1797, published 1799); 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 Argand, a mathematician from Geneva, published a similar diagram, so in the English-speaking world it is called the Argand diagram. Gauss also held this view from early on, but his explicit account was published only in 1831.
How can complex numbers be represented?
Besides the Cartesian form $z = a + bi$, the polar representation $z = r e^{i\theta}$ is widely used (for $z \neq 0$; $r = |z|$, and the argument $\theta$ is determined up to integer multiples of $2\pi$). A further development is the Riemann sphere: the complex plane with one point at infinity added, identified with a sphere.
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 1799 doctoral dissertation was a proof of the fundamental theorem of algebra (in the real-coefficient case).