"If contradictory attributes be assigned to a concept, I say, that mathematically the concept does not exist. So, for example, a real number whose square is -1 does not exist mathematically." (David Hilbert, [address to the International Congress of Mathematicians], 1900)
"When we get used to playing with these complex numbers, we cease to think of a + ib as a pair of things, namely the two real numbers a and b, but we think of a+ib as an entire thing on its own, and we could use a single letter, say:, to denote the whole complex number z = a+ib. It may be checked that all the normal rules of algebra are satisfied by complex numbers. In fact, all this is a good deal more straightforward than checking everything for real numbers. […] From this point of view, it seems rather extraordinary that complex numbers were viewed with suspicion for so long, whereas the much more complicated extension from the nationals to the reals had, after ancient Greek times, been generally accepted without question." (Sir Roger Penrose, "The Road to Reality: A Complete Guide to the Laws of the Universe", 2004)
"Apparent Impossibilities that Are New Truths […] irrational numbers, imaginary numbers, points at infinity, curved space, ideals, and various types of infinity. These ideas seem impossible at first because our intuition cannot grasp them, but they can be captured with the help of mathematical symbolism, which is a kind of technological extension of our senses." (John Stillwell,"Yearning for the impossible: the surpnsing truths of mathematics", 2006)
"In fact, complex numbers are not much more complicated than reals, and many structures built on the complex numbers actually have simpler behavior than the corresponding structures built on the real numbers." (John Stillwell,"Yearning for the impossible: the surpnsing truths of mathematics", 2006)
"It is impossible for √-1 to be a real number, since its square is negative. This implies that √-1 is neither greater nor less than zero, so we cannot see √-1 on the real line. However, √-1 behaves like a number with respect to + and x. This prompts us to look elsewhere for it, and indeed we find it on another line (the imaginary axis) perpendicular to the real line." (John Stillwell,"Yearning for the impossible: the surpnsing truths of mathematics", 2006)
"Consider for example the complex numbers x + iy, where you of course ask what is i = √ −1 when you first encounter this mathematical construction. But that uncomfortable feeling of what this strange imaginary unit really is fades away as you get more experienced and learn that C is a field of numbers that is extremely useful, to say the least. You no longer care what kind of object i is but are satisfied only to know that i^2 = −1, which is how you calculate with i." (Andreas Rosén,"Geometric Multivector Analysis: From Grassmann to Dirac", 2019)
"Therefore one has taken everywhere the opposite road, and each time one encounters manifolds of several dimensions in geometry, as in the doctrine of definite integrals in the theory of imaginary quantities, one takes spatial intuition as an aid. It is well known how one gets thus a real overview over the subject and how only thus are precisely the essential points emphasized." (Bernhard Riemann)
"Imaginary numbers have an intuitive explanation: they 'rotate' numbers, just like negatives make a 'mirror image' of a number. […] Seeing imaginary numbers as rotations gives us a new mindset to approach problems; the 'plug and chug' formulas can make intuitive sense, even for a strange topic like complex numbers." (Kalid Azal, Math, Better Explained)
"Zero is such a weird idea, having 'something' represent 'nothing', and it eluded the Romans. Complex numbers are similar - it’s a new way of thinking. But both zero and complex numbers make math much easier. If we never adopted strange, new number systems, we’d still be counting on our fingers." (Kalid Azal, Math,"Better Explained")
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