Painting - Multiplication through Imaginary Numbers (Gauss)

This painting was inspired by ideas of Carl Friedrich Gauss (1777–1855). In his 1797 doctoral thesis, Gauss proved what is now called the fundamental theorem of algebra. He showed that every polynomial with real coefficients must have at least one real or complex root. A complex number has the form a+bi, where a and b are real numbers and i represents the square root of negative one. The French mathematician René Descartes (1596–1650) called such numbers "imaginary", which explains the reference in the title. Gauss demonstrated that, just as real numbers can be represented by points on a coordinate line, complex numbers can be represented by points in the coordinate plane.
The construction of this painting echoes a figure in an article on Gauss by Eric Temple Bell in J. R. Newman's The World of Mathematics that illustrates the representation of points on a plane. This book was in Crockett Johnson's library, and the figure is annotated.
In Bell's figure, real numbers c and -c are plotted on the x axis, the imaginary numbers ci and -ci are plotted on the y axis, and the complex number a+bi is shown in the first quadrant. The figure is meant to show that if a complex number a+bi is multiplied by the imaginary number i then the product is a complex number on the same circle but rotated ninety degrees counterclockwise. That is, i(a+bi) = ai+bi² = -b+ai. Thus, this complex number lies in the second quadrant. If this process is repeated the next product is -a-bi, which lies in the third quadrant. It is unknown why Johnson did not illustrate the fourth product.
The colors of opposite quadrants of the painting are related. Those in quadrant three echo those of quadrant one and those of quadrant four echo those of quadrant two.This oil painting is #40 in the series. It is signed: CJ67.
James R. Newman, The World of Mathematics (1956), p. 308. This volume was in Crockett Johnson's library. The figure on this page is annotated.
Currently not on view
Object Name
date made
Johnson, Crockett
Physical Description
masonite (substrate material)
wood (frame material)
overall: 126 cm x 126 cm x 3.8 cm; 49 5/8 in x 49 5/8 in x 1 1/2 in
ID Number
catalog number
accession number
Science & Mathematics
Crockett Johnson
See more items in
Medicine and Science: Mathematics
Crockett Johnson
Data Source
National Museum of American History, Kenneth E. Behring Center
Credit Line
Ruth Krauss in memory of Crockett Johnson
Additional Media

Visitor Comments

Add a comment about this object

**Please read before submitting the form**

Have a comment or question about this object to share with the community? Please use the form below. Selected comments will appear on this page and may receive a museum response (but we can't promise). Please note that we generally cannot answer questions about the history, rarity, or value of your personal artifacts.

Have a question about anything else, or would you prefer a personal response? Please visit our FAQ or contact page.

Personal information will not be shared or result in unsolicited e-mail. See our privacy policy.

Enter the characters shown in the image.