“The original purpose and immediate objective in introducing complex numbers into mathematics is to express laws of dependence between variables by simpler operations on the quantities involved. If one applies these laws of dependence in an extended context, by giving the variables to which they relate complex values, there emerges a regularity and harmony which would otherwise have remained concealed.” (Heinz-Dieter Ebbinghaus et al, “Numbers”, 1990)
“The letter ‘i’ originally was meant to suggest the imaginary nature of this number, but with the greater abstraction of mathematics, it came to be realized that it was no more imaginary than many other mathematical constructs. True, it is not suitable for measuring quantities, but it obeys the same laws of arithmetic as do the real numbers, and, surprisingly enough, it makes the statement of various physical laws very natural.” (John A Paulos, “Beyond Numeracy”, 1991)
"The number ‘i’ is evidence that much real progress can result from the positing of imaginary entities. Theologians who have built elaborate systems on much flimsier analogies should perhaps take heart." (John A Paulos, “Beyond Numeracy”, 1991)
"Besides being essential in modern physics, the complex-number field provides pure mathematics with a multitude of brain-boggling theorems. It is worth keeping in mind that complex numbers, although they include the reals.as a subset, differ from real numbers in startling ways. One cannot, for example, speak of a complex number as being either positive or negative: those properties apply only to the reals and the pure imaginaries. It is equally meaningless to say that one complex number is larger or smaller than another." (Martin Gardner, "Fractal Music, Hypercards and More... Mathematical Recreations from Scientific American Magazine", 1992)
"The seemingly preposterous assumption that there is a square root of -1 was justified on pragmatic grounds: it simplified certain calculations and so could be used as long as 'real' values were obtained at the end. The parallel with the rules for using negative numbers is striking. If you are trying to determine how many cows there are in a field (that is, if you are working in the domain of positive integers), you may find negative numbers useful in the calculation, but of course the final answer must be in terms of positive numbers because there is no such thing as a negative cow." (Martin Gardner, "Fractal Music, Hypercards and More... Mathematical Recreations from Scientific American Magazine", 1992)
"[…] the words real and imaginary are picturesque relics of an age when the nature of complex numbers was not properly understood." (Harold S M Coxeter, "The Real Projective Plane" 3rd Ed, 1993)
"Such a close connection between trigonometric functions, the mathematical constant 'e', and the square root of -1 is already quite startling. Surely, such an identity cannot be a mere accident; rather, we must be catching a glimpse of a rich, complicated, and highly abstract mathematical pattern that for the most part lies hidden from our view." (Keith Devlin, "Mathematics: the Science of Patterns", 1994)
"All these questions he [the master?] does not pose. So we have to ask them: is the ‘Ring I’ a trap to catch the master or is the ‘Ring I’ a vessel of understanding? In quantum theory it specifies a formula which includes the irrational in a symbol of totality, in a holistic ‘cosmogramm’. But the formula has a catch. If one squares i = √-1, although a negative, one obtains a rationally understandable negative number -1. So one can make the irrational disappear through a slight of hand. This formula does not correspond to reality because the irrational that we call the collective unconscious or the objective psyche can never be rational. It remains always creatively spontaneous, not predictable, not manipulatable. Each holistic formula is in that sense also a trap, because it brings about the illusion that one has understood the whole." (Marie Louise von Franz, "Reflexionen zum ‘Ring I’", 1995)
"The real numbers are one of the most audacious idealizations made by the human mind, but they were used happily for centuries before anybody worried about the logic behind them. Paradoxically, people worried a great deal about the next enlargement of the number system, even though it was entirely harmless. That was the introduction of square roots for negative numbers, and it led to the 'imaginary' and 'complex' numbers. A professional mathematican should never leave home without them […]"
"The dictum that everything that people do is 'cultural' licenses the idea that every cultural critic can meaningfully analyze even the most intricate accomplishments of art and science. [...] It is distinctly weird to listen to pronouncements on the nature of mathematics from the lips of someone who cannot tell you what a complex number is!" (Norman Levitt, "The Flight from Science and Reason", Science, 1996)
"At this stage you might be thinking that there is no justification for calling something of the form a+bi a number, even if you are prepared to countenance i = √-1 in the first place. But remember, it is not what numbers are that matters, but how they behave. Provided the complex numbers have a workable and useful (either in mathematics itself or possibly in a wider context) arithmetic, possibly forming a field, then they have as much right to be called 'numbers' as do any others." (Keith Devlin, "Mathematics: The New Golden Age", 1998)
"In fact the complex numbers form a field. [...] So however strange you may feel the very notion of a complex number to be, it does turn out to provide a 'normal' type of arithmetic. In fact it gives you a tremendous bonus not available with any of the other number systems. [...] The fundamental theorem of algebra is just one of several reasons why the complex-number system is such a 'nice' one. Another important reason is that the field of complex numbers supports the development of a powerful differential calculus, leading to the rich theory of functions of a complex variable." (Keith Devlin, "Mathematics: The New Golden Age", 1998)
"The whole apparatus of the calculus takes on an entirely different form when developed for the complex numbers." (Keith Devlin, "Mathematics: The New Golden Age", 1998)
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