On Complex Numbers II

“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) 

“But it is just that the Roots of Equations should be often impossible (complex), lest they should exhibit the cases of Problems that are impossible as if they were possible." (Isaac Newton, “Universal Mathematic” 2nd Ed., 1728)

"Complete knowledge of the nature of an analytic function must also include insight into its behavior for imaginary values of the arguments. Often the latter is indispensable even for a proper appreciation of the behavior of the function for real arguments. It is therefore essential that the original determination of the function concept be broadened to a domain of magnitudes which includes both the real and the imaginary quantities, on an equal footing, under the single designation complex numbers." (Carl F Gauss, cca. 1831)

“[…] such numbers, which by their natures are impossible, are ordinarily called imaginary or fanciful numbers, because they exist only in the imagination.”  (Leohnard Euler, 1732)

“We completely repudiate the symbol √-1, abandoning it without regret because we do not know what this alleged symbolism signifies nor what meaning to give to it.” (Augustin-Louis Cauchy, 1847)

“Analysis […] would lose immensely in beauty and balance and would be forced to add very hampering restrictions to truths which would hold generally otherwise, if […] imaginary quantities were to be neglected.” (Garrett Birkhoff, 1973)

"It is a curious fact that the first introduction of the imaginaries occurred in the theory of cubic equations, in the case where it was clear that real solutions existed though in an unrecognizable form, and not in the theory of quadratic equations, where our present textbooks introduce them." (Dirk J Struik, “A Concise History of Mathematics” Vol. I, 1948)

"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)

"The word ‘imaginary’ is the great algebraical calamity, but it is too well established for mathematicians to eradicate. It should never have been used. Books on elementary algebra give a simple interpretation of imaginary numbers in terms of rotations. […] Although the interpretation by means of rotations proves nothing, it may suggest that there is no occasion for anyone to muddle himself into a state of mystic wonderment over nothing about the grossly misnamed ‘imaginaries’." (Philip E B Jourdain, "The Nature of Mathematics" in [James R Newman, “The World of Mathematics” Vol. I, 1956])

See also:
5 Books 10 Quotes: Complex Numbers V
Complex Numbers IV
Complex Numbers III
Complex Numbers I