On Complex Numbers XX

"If we refused to use complex numbers out of stubbornness disguised as some kind of bogus philosophical objection, a solution to a whole range of important problems would remain forever out of reach.[...] The plane of the complex numbers is the natural arena of discourse for much if not most of mathematics." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Nonetheless, some hesitation persisted. After all, the very word imaginary betrays ambivalence, and suggests that in our heart of hearts we do not believe these numbers exist. On the other hand, by calling every number representable by a decimal expansion real, we are making the psychological distinction more stark. Indeed the adjective imaginary is a somewhat unfortunate one - although an intriguing name, some students’ perceptions are so colored by the word that they consequently fail to come to grips with the idea." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Perhaps the greatest legacy of the solution of the cubic was the arrival, without invitation, of the imaginary number i into the world of mathematics." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Analyticity can often be exploited to advantage in the study of problems of approximation, even when the objects to be approximated are functions of a real variable." (Peter D Lax & Lawrence Zalcman, "Complex proofs of real theorems", 2012)

"It has been said that the three most effective problem-solving devices in mathematics are calculus, complex variables, and the Fourier transform." (Peter D Lax & Lawrence Zalcman, "Complex proofs of real theorems", 2012)

"Nothing illustrates the extraordinary power of complex function theory better than the ease and elegance with which it yields results which challenged and often baffled the very greatest mathematicians of an earlier age." (Peter D Lax & Lawrence Zalcman, "Complex proofs of real theorems", 2012)

"In fact the term ‘real number’ was invented after the discovery of its complex extension as a means of distinguishing between the two types of number. The terminology, in retrospect, is unfortunate. The concrete representation of √ −1 either as a π/2 -radian anticlockwise rotation of the plane about the origin or as a point in the plane neatly conceals its troubled history. The conceptual crisis faced by the sixteenth century mathematicians is clear: the other ‘new numbers’ of history: zero; negative numbers; irrational numbers (all of these will be formally introduced shortly) are at least interpretable as a magnitude of some sort, or as a directed length, whereas √ −1 seemed, at first, to come from another realm entirely." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"The words 'imaginary' and 'complex' again demonstrate how difficult it is to make a major change in conceptual systems - a difficulty that we already encountered with negative numbers, fractions, zero, and irrational numbers. The word 'imaginary' tells us that these numbers are unreal from the perspective of someone grounded in the real number system." (William Byers, "Deep Thinking: What Mathematics Can Teach Us About the Mind", 2015)