Painting - <I>Multiplication through Imaginary Numbers (Gauss)</I>
date: label: date made
content: 1967
name: label: painter
content: Johnson, Crockett
notes: label: Description
content: 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.
label: Description
content: The construction of this painting echoes a figure in an article on Gauss by Eric Temple Bell in J. R. Newman's <I>The World of Mathematics</I> that illustrates the representation of points on a plane. This book was in Crockett Johnson's library, and the figure is annotated.
label: Description
content: 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.
label: Description
content: 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.
label: Description
content: References:
label: Description
content: James R. Newman, <I>The World of Mathematics</I> (1956), p. 308. This volume was in Crockett Johnson's library. The figure on this page is annotated.
label: Location
content: Currently not on view
setName: label: See more items in
content: Medicine and Science: Mathematics
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content: Science & Mathematics
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content: Crockett Johnson
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content: Art
creditLine: label: Credit Line
content: Ruth Krauss in memory of Crockett Johnson
dataSource: label: Data Source
content: National Museum of American History
identifier: label: ID Number
content: 1979.1093.28
label: catalog number
content: 1979.1093.28
label: accession number
content: 1979.1093
objectType: label: Object Name
content: painting
physicalDescription: label: Physical Description
content: masonite (substrate material)
label: Physical Description
content: wood (frame material)
label: Measurements
content: overall: 126 cm x 126 cm x 3.8 cm; 49 5/8 in x 49 5/8 in x 1 1/2 in