On Complex Numbers III

"[geometrical representation of complex numbers] completely established the intuitive meaning of complex numbers, and more is not needed to admit these quantities into the domain of arithmetic." (Carl F Gauss, 1831) 

“Nearly fifty years had passed without any progress on the question of analytic representation of an arbitrary function, when an assertion of Fourier threw new light on the subject. Thus a new era began for the development of this part of Mathematics and this was heralded in a stunning way by major developments in mathematical Physics.” (Bernhard Riemann, 1854)

"I recall my own emotions: I had just been initiated into the mysteries of the complex number. I remember my bewilderment: here were magnitudes patently impossible and yet susceptible of manipulations which lead to concrete results. It was a feeling of dissatisfaction, of restlessness, a desire to fill these illusory creatures, these empty symbols, with substance. Then I was taught to interpret these beings in a concrete geometrical way. There came then an immediate feeling of relief, as though I had solved an enigma, as though a ghost which had been causing me apprehension turned out to be no ghost at all, but a familiar part of my environment." (Tobias Dantzig, “The Two Realities”, 1930)

“[…] imaginary numbers made their own way into arithmetical calculation without the approval, and even against the desires of individual mathematicians, and obtained wider circulation only gradually and the extent to which they showed themselves useful.” (Felix Klein, “Elementary Mathematics from an Advanced Standpoint”, 1945)

“The sweeping development of mathematics during the last two centuries is due in large part to the introduction of complex numbers; paradoxically, this is based on the seemingly absurd notion that there are numbers whose squares are negative.” (Emile Borel, 1952)

"For centuries [the concept of complex numbers] figured as a sort of mystic bond between reason and imagination.” (Tobias Dantzig)

"The whole apparatus of the calculus takes on an entirely different form when developed for the complex numbers." (Keith Devlin) "There can be very little of present-day science and technology that is not dependent on complex numbers in one way or another." (Keith Devlin)

 “[…] to the unpreoccupied mind, complex numbers are far from natural or simple and they cannot be suggested by physical observations. Furthermore, the use of complex numbers is in this case not a calculational trick of applied mathematics but comes close to being a necessity in the formulation of quantum mechanics.” (Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, 1960)

“Nothing in our experience suggests the introduction of [complex numbers]. Indeed, if a mathematician is asked to justify his interest in complex numbers, he will point, with some indignation, to the many beautiful theorems in the theory of equations, of power series, and of analytic functions in general, which owe their origin to the introduction of complex numbers. The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius.” (Eugene P Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”, Communications in Pure and Applied Mathematics 13 (1), 1960) 

“The letter ‘i’ originally was meant to suggest the imaginary nature of this number, but with the greater abstraction of mathematics, it came to be realized that it was no more imaginary than many other mathematical constructs. True, it is not suitable for measuring quantities, but it obeys the same laws of arithmetic as do the real numbers, and, surprisingly enough, it makes the statement of various physical laws very natural.” (John A Paulos, “Beyond Numeracy”, 1991)

"The only reason that we like complex numbers is that we don't like real numbers." (Bernd Sturmfels)

See also:
5 Books 10 Quotes: Complex Numbers
Complex Numbers IV
Complex Numbers II
Complex Numbers I