Friday, September 11, 2009

An Argand diagram is a Wessel diagram is a Gauss diagram

An Argand diagram is a two-dimensional coordinate system for the presentation of complex numbers. The scaling on the horizontal axis gives the real part and the scaling on the vertical axis gives the imaginary part of a complex number. Although Jean-Robert Argand (1768-1822) is usually credited for geometrically picturing complex numbers this way, Marcus du Sautoy provides a detailed look into the origin of complex number presentation [1]:
Gauss [Carl Friedrich, (1777-1855)] had actually used a picture of the imaginary numbers as a mathematical tool in his proof [of the Fundamental Theorem of Algebra], but he kept it hidden for many years, fearing he would be laughed at by a mathematical establishment still wedded to the language of equations and formulae. But because the image was so powerful and gave imaginary numbers a physical reality, it was only a matter of time before others hit upon the idea. Two amateur mathematicians, the Dane [/Norwegian] Caspar Wessel [(1745-1818)] and the Swiss Jean Argand, independently proposed similar pictures in pamphlets they published. Argand, who was the last of the three to have the idea, is the person whose name became attached to the picture we now call the Argand diagram. Credit is rarely just.

Hence, the terms Wessel diagram or Gauss diagram would as well be justified.

Instead of using the word diagram for a figure, that illustrates complex numbers and their relationships graphically, the word plane is often used: Argand plane instead of Argand diagram. Other synonymous terms for Argand plane are complex plane or z-plane; giving credit to the complex numbers z = a + bi again!

Keywords: mathematics, geometry, algebra, history, graphical illustration, synonymous expressions

[1] Marcus du Sautoy: Symmetry. A Journey into the Patterns of Nature. First Harper Perennial Edition, Harper Collins Publishers, New York, 2009; page 153.

[2] Orlando Merino: A Short History of Complex Numbers. University of Rhode Island, 2006; PDF-file.

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