On Numbers: On Complex Numbers (Unsourced)

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

“I have obtained these values by a singular analogy based on the passages from the real to the imaginary, passages that can be considered as a means of discovery.” (Pierre-Simon Laplace)

“I did not understand how such a quantity could be real, when imaginary or impossible numbers were used to express it.” (Gottfried W Leibniz) 

"Imagine a person with a gift of ridicule [He might say] First that a negative quantity has no logarithm [ln(-1)]; secondly that a negative quantity has no square root [√-1]; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter [π]." (Augustus De Morgan)

“In particular, in introducing new numbers, mathematics is only obliged to give definitions of them, by which such a definiteness and, circumstances permitting, such a relation to the older numbers are conferred upon them that in given cases they can definitely be distinguished from one another. As soon as a number satisfies all these conditions, it can and must be regarded as existent and real in mathematics. Here I perceive the reason why one has to regard the rational, irrational, and complex numbers as being just as thoroughly existent as the finite positive integers.” (Georg Cantor)

"Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence." (Keith Devlin)

"Mathematics is very much like poetry [...] what makes a good poem - a great poem - is that there is a large amount of thought expressed in very few words. In this insense formulas like e^iπ + 1 = 0 [...] are poems." (Lipman Bers)

"Much of the final resistance to complex numbers faded as it became clear that their behavior posed no threat to the rules and operations of algebra. On the contrary, quite often the complex realm opened paths that made already existing results easier to prove." (David Perkins, "φ, π, e & i", 2017) of ridicule [He might say] First that a negative quantity has no logarithm [ln(-1)]; secondly that a negative quantity has no square root [√-1]; thirdly that the first non-existent is to the second as the circumference of a circle is to the diameter [π]." (Augustus De Morgan) [attributed]

"That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question." (Carl Friedrich Gauss)

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

“The reason and the immediate purpose for the introduction of complex quantities into mathematics lie in the theory of uniform relations between variable quantities which are expressed by simple mathematical formulas. Using these relations in an extended sense, by giving complex values to the variable quantities involved, we discover in them a hidden harmony and regularity that would otherwise remain hidden.” (Bernhard Riemann, “Gesammelte Mathematische Werke”)

"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)

"Therefore one has taken everywhere the opposite road, and each time one encounters manifolds of several dimensions in geometry, as in the doctrine of definite integrals in the theory of imaginary quantities, one takes spatial intuition as an aid. It is well known how one gets thus a real overview over the subject and how only thus are precisely the essential points emphasized." (Bernhard Riemann)

"We have shown the symbol √-1 to be void of meaning, or rather self-contradictory and absurd. Nevertheless, by means of such symbols, a part of algebra is established which is of great utility. It depends upon the fact, which must be verified by experience, that the common rules of algebra may be applied to these expressions without leading to any false results." (Augustus De Morgan)