"[...] gradually and unwittingly mathematicians began to introduce concepts that had little or no direct physical meaning. Of these, negative and complex numbers were most troublesome. It was because these two types of numbers had no 'reality' in nature that they were still suspect at the beginning of the nineteenth century, even though freely utilized by then. The geometrical representation of negative numbers as points or vectors in the complex plane, which, as Gauss remarked of the latter, gave them intuitive meaning and so made them admissible, may have delayed the realization that mathematics deals with man-made concepts. But then the introduction of quaternions, non-Euclidean geometry, complex elements in geometry, n-dimensional geometry, bizarre functions, and transfinite numbers forced the recognition of the artificiality of mathematics." (Morris Kline, "Mathematical Thought from Ancient to Modern Times", 1972)
"From a formal perspective, much about complex numbers and arithmetic seems arbitrary. From a purely algebraic point of view, i arises as a solution to the equation x^2+1=0. There is nothing geometric about this - no complex plane at all. Yet in the complex plane, the i-axis is 90° from the x-axis. Why? Complex numbers in the complex plane add like vectors. Why? Complex numbers have a weird rule of multiplication […]"
"The complex plane is just the 90° rotation plane-the rotation plane with the structure imposed by the 90° Rotation metaphor added to it. Multiplication by i is 'just' rotation by 90°. This is not arbitrary; it makes sense. Multiplication by-1 is rotation by 180°. A rotation of 180° is the result of two 90° rotations. Since i times i is -1, it makes sense that multiplication by i should be a rotation by 90°, since two of them yield a rotation by 180°, which is multiplication by -1." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)
"The equation e^πi =-1 says that the function w= e^z, when applied to the complex number πi as input, yields the real number -1 as the output, the value of w. In the complex plane, πi is the point [0,π) - π on the i-axis. The function w=e^z maps that point, which is in the z-plane, onto the point (-1, 0) - that is, -1 on the x-axis-in the w-plane. […] But its meaning is not given by the values computed for the function w=e^z. Its meaning is conceptual, not numerical. The importance of e^πi =-1 lies in what it tells us about how various branches of mathematics are related to one another - how algebra is related to geometry, geometry to trigonometry, calculus to trigonometry, and how the arithmetic of complex numbers relates to all of them." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being", 2000)
"Beyond the theory of complex numbers, there is the much greater and grander theory of the functions of a complex variable, as when the complex plane is mapped to the complex plane, complex numbers linking themselves to other complex numbers. It is here that complex differentiation and integration are defined. Every mathematician in his education studies this theory and surrenders to it completely. The experience is like first love." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)
"Every smoothly bounded domain in the complex plane has a Green’s function. The Green’s function is fundamental to the Poisson integral, the theory of harmonic functions, and to the broad panorama of complex function theory. [...] The Green’s function contains information about the geometry of its domain, and it also contains information about the harmonic analysis of the domain. We can rarely calculate the Green’s function explicitly, but we can obtain enough qualitative information so that it is a potent tool." (Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)
"[...] one of the fundamental intellectual breakthroughs in the historical understanding of just what i = √-1 means, physically, came with the insight that multiplication by a complex number is associated with a rotation in the complex plane. That is, multiplying the vector of a complex number by the complex exponential e^iθ rotates that vector counterclockwise through angle θ." (Paul J Nahin, "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills", 2006)
"The beauty of the complex plane is that we may finally carry out all our mathematical work in a single number arena. However, although there may be no pressing mathematical difficulty that is driving us further, we can ask the question whether or not it is possible to go beyond the complex plane into some larger realm of number." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)
"So a very useful way to think about i (√−1) is as an operator that produces a 90◦ rotation of any vector to which it is applied. Thus the two perpendicular number lines form the basis of what we know today as the complex plane. Unfortunately, since multiplication by √−1 is needed to get from the horizontal to the vertical number line, the numbers along the vertical number line are called 'imaginary'. We say 'unfortunately' because these numbers are every bit as real as the numbers along the horizontal number line. But the terminology is pervasive, so when you first learned about complex numbers, you probably learned that they consist of a 'real' and an 'imaginary' part." (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)
"Raising e to an imaginary-number power can be pictured as a rotation operation in the complex plane. Applying this interpretation to e raised to the 'i times π' power means that Euler’s formula can be pictured in geometric terms as modeling a half-circle rotation." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
"Wessel and his fellow explorers had discovered the natural habitat of Leibniz’s ghostly amphibians: the complex plane. Once the imaginaries were pictured there, it became clear that their meaning could be anchored to a familiar thing - sideways or rotary motion - giving them an ontological heft they’d never had before. Their association with rotation also meant that they could be conceptually tied to another familiar idea: oscillation." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
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