On Numbers: On Complex Numbers (1980-1989)

"If explaining minds seems harder than explaining songs, we should remember that sometimes enlarging problems makes them simpler! The theory of the roots of equations seemed hard for centuries within its little world of real numbers, but it suddenly seemed simple once Gauss exposed the larger world of so-called complex numbers. Similarly, music should make more sense once seen through listeners' minds." (Marvin Minsky, "Music, Mind, and Meaning", 1981)

“The origin and immediate purpose of the introduction of complex magnitudes into mathematics lie in the theory of simple laws of dependence between variable magnitudes expressed by means of operations on magnitudes. If we enlarge the scope of applications of these laws by assigning to the variables they involve complex values, then there appears an otherwise hidden harmony and regularity.” (Heinz-Dieter Ebbinghaus et al., “Numbers”, 1983)

“What could be more beautiful than a deep, satisfying relation between whole numbers. How high they rank, in the realms of pure thought and aesthetics, above their lesser brethren: the real and complex numbers.” (Manfred Schroeder, “Number Theory in Science and Communication”, 1984)

“The attitudes of mathematicians can be found not only in what they wrote, but in what they did not write. It is possible to divide mathematicians into those who gave complex numbers some kind of coverage, and those who sometimes or always ignored them.” (Diana Willment, “Complex Numbers from 1600 to 1840” [Masters thesis], 1985)

“The lack of a visual representation for √-1 had a profound influence on attitudes to it, and complex numbers were not widely accented until after the invention of the Argand diagram.” (Diana Willment, “Complex Numbers from 1600 to 1840” [Masters thesis], 1985)

“The square roots of negative numbers! If negative numbers were false, absurd or fictitious, it is hardly to be wondered at that their square roots were described as 'imaginary'.” (David Wells, “The Penguin Dictionary of Curious and Interesting Numbers”, 1986)