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Complex Magnitudes

In a 1673 letter to the mathematician and physicist Christian Huygens (1629-1695), Gottfreid Leibniz (1646-1916) noted with surprise:

"I do not remember," he commented, "to have noted a more singular and paradoxical fact in all of analysis: for I think I am the first one to have reduced irrational roots, imaginary in form, real values."

Descartes had rejected complex roots and coined the derogatory term "imaginary" to describe the square root of negative one, , but Leibniz thought that "The divine spirit found a sublime outlet in that wonder of analysis, that portent of the ideal world, that amphibian between being and non-being, which we call the imaginary root of negative unity.''

Gauss invented the "complex plane" (show below) to represent these quantities. He suggested that complex magnitudes be called "lateral" instead of "imaginary" magnitudes since they represent a dimensional extension of the continuum.

Gauss also proposed that complex magnitudes be awarded "full civil rights."

In the language of Plato's allegory of the cave, complex numbers represent "forms" from a higher dimension casting "shadows" on the real number line.

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