"The seeming absence of any ascertained organizing principle in the distribution or the succession of the primes had bedeviled mathematicians for centuries and given Number Theory much of its fascination. Here was a great mystery indeed, worthy of the most exalted intelligence: since the primes are the building blocks of the integers and the integers the basis of our logical understanding of the cosmos, how is it possible that their form is not determined by law? Why isn't 'divine geometry' apparent in their case?" (Apostolos Doxiadis, "Uncle Petros and Goldbach's Conjecture", 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."
"Imaginary numbers are not imaginary and the theory of complex numbers is no more complex than the theory of real numbers." (Mordechai Ben-Ari, "Just a Theory: Exploring the Nature of Science", 2005)
"We divide math up into separate areas (analysis, mechanics, algebra, geometry, electromagnetism, number theory, quantum mechanics, etc.) to clarify the study of each part; but the equally valuable activity of integrating the components into a working whole is all too often neglected. Without it, the stated aim of ‘taking something apart to see how it ticks’ degenerates imperceptibly into ‘taking it apart to ensure it never ticks again’." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)
"Transcendental numbers then are numerous but exceedingly slippery. As a rule of thumb, a number that arises in mathematics is almost always transcendental unless it is obvious that it is not. However, showing that a particular number is transcendental can be exceedingly difficult. Number theory throws up endless problems of this kind where everyone feels sure what the answer must be but at the same time no-one has any real idea how it could ever by proved." (Peter M. Higgins, "Number Story: From Counting to Cryptography", 2008)
"There are highly symmetric tiling patterns in hyperbolic geometry. For each of them, we can construct complex functions that repeat the same values on every tile. These are known as modular functions, and they are natural generalisations of elliptic functions. Hyperbolic geometry is a very rich subject, and the range of tiling patterns is much more extensive than it is for the Euclidean plane. So complex analysts started thinking seriously about non-Euclidean geometry. A profound link between analysis and number theory then appeared. Modular functions do for elliptic curves what trigonometric functions do for the circle." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)
"The significance of Fourier’s theorem to music cannot be overstated: since every periodic vibration produces a musical sound" (provided, of course, that it lies within the audible frequency range), it can be broken down into its harmonic components, and this decomposition is unique; that is, every tone has one, and only one, acoustic spectrum, its harmonic fingerprint. The overtones comprising a musical tone thus play a role somewhat similar to that of the prime numbers in number theory: they are the elementary building blocks from which all sound is made." (Eli Maor, "Music by the Numbers: From Pythagoras to Schoenberg", 2018)
"While number theory looks for patterns in sequences of numbers, dynamical systems actually produce sequences of numbers [...]. The two merge when mathematicians look for number-theoretic patterns hidden in those sequences." (Kelsey Houston-Edwards, "Mathematicians Set Numbers in Motion to Unlock Their Secrets", Quanta Magazine, 2021)
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