On Complex Numbers I

“[…] neither the true roots nor the false are always real; sometimes they are, however, imaginary; namely, whereas we can always imagine as many roots for each equation as I have predicted, there is still not always a quantity which corresponds to each root so imagined. Thus, while we may think of the equation x^3 - 6xx + 13x - 10 = 0 as having three roots, yet there is just one real root, which is 2, and the other two [2+i and 2-i]], however, increased, diminished, or multiplied them as we just laid down, remain always imaginary.” (René Descartes, “Gemetry”, 1637)

“We have before had occasion (in the Solution of some Quadratick and Cubick Equations) to make mention of Negative Squares, and Imaginary Roots, (as contradistinguished to what they call Real Roots, whether affirmative or Negative) […].These ‘Imaginary’ Quantities (as they are commonly called) arising from ‘Supposed’ Root of a Negative Square, (when they happen) are reputed to imply that the Case proposed is Impossible.” (John Wallis, "A Treatise of Algebra, Both Historical and Practical", 1673)

“Imaginary numbers are a fine and wonderful refuge of the divine spirit almost an amphibian between being and non-being.” (Gottfried Leibniz, 1702)

"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)

“All such expressions as √-1, √-2, etc., are consequently impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible.” (Euler, Algebra, 1770)

“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)

“One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?” (Stephen W Hawking, “The Universe in a Nutshell”, 2001).

“The shortest path between two truths in the real domain passes through the complex domain." (Jacque Hadamard [misquoted])

“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)

“The more science I studied, the more I saw that physics becomes metaphysics and numbers become imaginary numbers. The farther you go into science, the mushier the ground gets. You start to say, 'Oh, there is an order and a spiritual aspect to science.’” (Dan Brown)

"Adam and Eve are like imaginary number, like the square root of minus one… If you include it in your equation, you can calculate all manners of things, which cannot be imagined without it." (Philip Pullman)

See also:
5 Books 10 Quotes: Complex Numbers
More Quotes on Complex Numbers III
More Quotes on Complex Numbers II
More Quotes on Complex Numbers I