Misquoted: Jacque Hadamard on Complex Numbers

"The shortest path between two truths in the real domain passes through the complex domain." 

Probably this is one of the most known quotes on complex numbers as it easy to remember and reflects the fact that important problems in algebra, analysis, geometry, number theory and physics can be simplified by considering them into the complex plane. Even if the quote reflects pretty good the idea, the actual quote comes from Jacque Hadamard’s "An Essay on the Psychology of Invention in the Mathematical Field" published in 1945:

"It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one."

[French: "On a pu écrire depuis que la voie la plus courte et la meilleure entre deux vérités du domaine réel passe souvent par le domaine imaginaire." (Jacques Hadamard, "Essai sur la psychologie de l'invention dans le domaine mathématique", 1945)]

Here Hadamard refers to Paul Painlevé, who in his "Analyse des travaux scientifiques" published in 1900 wrote as follows:

"The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain."

Actually, "la voie" can be translated as "the way" as well as "the path", the latter being closer to Painlevé’s quote, to whom the metaphor can be attributed to. Painlevé is not the first who stressed this important advantage of the complex numbers over the real ones, however his metaphor captures this aspect the best. 

"At the beginning I would ask anyone who wants to introduce a new function in analysis to clarify whether he intends to confine it to real magnitudes (real values of the argument) and regard the imaginary values as just vestigial - or whether he subscribes to my fundamental proposition that in the realm of magnitudes the imaginary ones a+b√−1 = a+bi have to be regarded as enjoying equal rights with the real ones. We are not talking about practical utility here; rather analysis is, to my mind, a self-sufficient science. It would lose immeasurably in beauty and symmetry from the rejection of any fictive magnitudes. At each stage truths, which otherwise are quite generally valid, would have to be encumbered with all sorts of qualifications." (Carl F Gauss, [letter to Bessel] 1811)

"The origin and the immediate purpose for the introduction of complex number into mathematics is the theory of creating simpler dependency laws (slope laws) between complex magnitudes by expressing these laws through numerical operations. And, if we give these dependency laws an expanded range by assigning complex values to the variable magnitudes, on which the dependency laws are based, then what makes its appearance is a harmony and regularity which is especially indirect and lasting." (Bernhard Riemann, "Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen complexen Grösse", 1851)

"The conception of the inconceivable [imaginary], this measurement of what not only does not, but cannot exist, is one of the finest achievements of the human intellect. No one can deny that such imaginings are indeed imaginary. But they lead to results grander than any which flow from the imagination of the poet. The imaginary calculus is one of the master keys to physical science. These realms of the inconceivable afford in many places our only mode of passage to the domains of positive knowledge. Light itself lay in darkness until this imaginary calculus threw light upon light. And in all modern researches into electricity, magnetism, and heat, and other subtile physical inquiries, these are the most powerful instruments." (Thomas Hill, “The Imagination in Mathematics”, North American Review Vol. 85, 1857)

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