On Complex Numbers XIX

"At this point it may be useful to observe that a certain type of intellect is always worrying itself and others by discussion as to the applicability of technical terms. Are the incommensurable numbers properly called numbers? Are the positive and negative numbers really numbers? Are the imaginary numbers imaginary, and are they numbers?-are types of such futile questions. Now, it cannot be too clearly understood that, in science, technical terms are names arbitrarily assigned, like Christian names to children. There can be no question of the names being right or wrong. They may be judicious or injudicious; for they can sometimes be so arranged as to be easy to remember, or so as to suggest relevant and important ideas. But the essential principle involved was quite clearly enunciated in Wonderland to Alice by Humpty Dumpty, when he told her, apropos of his use of words, 'I pay them extra and make them mean what I like.' So we will not bother as to whether imaginary numbers are imaginary, or as to whether they are numbers, but will take the phrase as the arbitrary name of a certain mathematical idea, which we will now endeavour to make plain." (Alfred N Whitehead, "Introduction to Mathematics", 1911)

"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 unrecognisable form, and not in the theory of quadratic equations, where our present textbooks introduce them." (Dirk J Struik, A Concise History of Mathematics", 1948)

"There are many useful connections between these two disciplines [geometry and algebra]. Many applications of algebra to geometry and of geometry to algebra were known in antiquity; nearer to our time there appeared the important subject of analytical geometry, which led to algebraic geometry, a vast and rapidly developing science, concerned equally with algebra and geometry. Algebraic methods are now used in projective geometry, so that it is uncertain whether projective geometry should be called a branch of geometry or algebra. In the same way the study of complex numbers, which arises primarily within the bounds of algebra, proved to be very closely connected with geometry; this can be seen if only from the fact that geometers, perhaps, made a greater contribution to the development of the theory than algebraists." (Isaak M Yaglom, "Complex Numbers in Geometry", 1968)

"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)

"Likewise, complex functions are actually better behaved than real functions, and the subject of complex analysis is known for its regularity and order, while real analysis is known for wildness and pathology A smooth complex function is predictable, in the sense that the values of the function in an arbitrarily small region determine its values everywhere. A smooth real function can be completely unpredictable for example, it can be constantly zero for a long interval, then smoothly change to the value 1." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"The word 'complex' was introduced m a well-meaning attempt to dispel the mystery surrounding 'imaginary' or 'impossible' numbers, and (presumably) because two dimensions are more complex than one Today, 'complex' no longer seems such a good choice of word. It is usually interpreted as 'complicated', and hence is almost as prejudicial as its predecessors. Why frighten people unnecessarily? If you are not sure what 'analysis' is, you won't want to know about 'complex analysis' - but it is the best part of analysis." (John Stillwell, "Yearning for the Impossible: The Surprising Truths of Mathematics", 2006)

"Complex numbers do not fit readily into many people’s schema for ‘number’, and students often reject the concept when it is first presented. Modern mathematicians look at the situation with the aid of an enlarged schema in which the facts make sense." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"When we extend the system of natural numbers and counting to embrace infinite cardinals, the larger system need not have all of the properties of the smaller one. However, familiarity with the smaller system leads us to expect certain properties, and we can become confused when the pieces don’t seem to fit. Insecurity arose when the square of a complex number violated the real number principle that all squares are positive. This was resolved when we realised that the complex numbers cannot be ordered in the same way as their subset of reals." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"Much of the final resistance to complex numbers faded as it became clear that their behavior posed no threat to the rules and operations of algebra. On the contrary, quite often the complex realm opened paths that made already existing results easier to prove." (David Perkins, "φ, π, e & i", 2017)