"All of this could have been said using notation that kept √-1 instead of the new representative i, which has the same virtual meaning. But i isolates the concept of rotation from the perception of root extraction, offering the mind a distinction between an algebraic result and an extension of the idea of number." (Joseph Mazur, "Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers", 2014)
"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)
"But the presence of √−1 (the rotation operator between the two perpendicular numbe rlines in the complex plane) in the exponent causes the expression e^ix to move from the real to the imaginary number line. As it does so, its real and imaginary parts oscillate in a sinusoidal fashion […] So the real and imaginary parts of the expression e^ix oscillate in exactly the same way as the real and imaginary components of the rotating phasor […]" (Daniel Fleisch & Laura Kinnaman, "A Student’s Guide to Waves", 2015)
"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)
"But the presence of √−1 (the rotation operator between the two perpendicular numbe rlines in the complex plane) in the exponent causes the expression e^ix to move from the real to the imaginary number line. As it does so, its real and imaginary parts oscillate in a sinusoidal fashion […] So the real and imaginary parts of the expression e^ix oscillate in exactly the same way as the real and imaginary components of the rotating phasor […]" (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)
"The association of multiplication with vector rotation was one of the geometric interpretation's most important elements because it decisively connected the imaginaries with rotary motion. As we'll see, that was a big deal." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
"The circle is a powerful symbol of generative force, associated over the ages with ideas of movement, rotation, transformation, cyclicality, and periodicity. A circle can be described as the curve drawn by a moving point revolving at a constant distance around a stationary point. This definition is central to the idea of rotation implicit in the circle and reinforced by one of the circle's inescapable manifestations, the wheel." (Manuel Lima, "The Book of Circle: Visualizing Spheres of Knowledge", 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)
"Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. The data sets themselves are explicitly linked as a form of representation to an observational or otherwise empirical domain of interest. 'Structure' has long been understood as symmetry which can take many forms with respect to any transformation, including point, translational, rotational, and many others. Symmetries directly point to invariants, which pinpoint intrinsic properties of the data and of the background empirical domain of interest. As our data models change, so too do our perspectives on analysing data." (Fionn Murtagh, "Data Science Foundations: Geometry and Topology of Complex Hierarchic Systems and Big Data Analytics", 2018)
"Mathematically, circles embody change without change. A point moving around the circumference of a circle changes direction without ever changing its distance from a center. It’s a minimal form of change, a way to change and curve in the slightest way possible. And, of course, circles are symmetrical. If you rotate a circle about its center, it looks unchanged. That rotational symmetry may be why circles are so ubiquitous. Whenever some aspect of nature doesn’t care about direction, circles are bound to appear. Consider what happens when a raindrop hits a puddle: tiny ripples expand outward from the point of impact. Because they spread equally fast in all directions and because they started at a single point, the ripples have to be circles. Symmetry demands it." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
No comments:
Post a Comment