By John C. Stillwell
This e-book explores the historical past of arithmetic from the point of view of the artistic rigidity among good judgment and the "impossible" because the writer follows the invention or invention of recent recommendations that experience marked mathematical development: - Irrational and Imaginary Numbers - The Fourth size - Curved area - Infinity and others the writer places those creations right into a broader context concerning similar "impossibilities" from paintings, literature, philosophy, and physics. through imbedding arithmetic right into a broader cultural context and during his shrewdpermanent and enthusiastic explication of mathematical principles the writer broadens the horizon of scholars past the slim confines of rote memorization and engages those who find themselves interested in where of arithmetic in our highbrow panorama
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Extra info for Yearning for the impossible : the surprising truths of mathematics
Thus the octave and the fifth have no “common measure” and the Pythagorean attempt to divide the octave into natural steps using rational numbers was doomed from the start—though the Pythagoreans may never have noticed. If one wants a scale with the sweetest harmonies, given by the octave and the fifth, then it is impossible to include both by dividing the octave into equal steps. 7. Equal Temperament 21 is close to seven octaves. 1) the difference between 12 fifths and seven octaves is a noticeable fraction of a step, so some compromise is necessary.
2. 4), though they would no doubt have balked at setting a = 0 to get (−b)2 = b 2 . (The initial figure is a square of side a plus a square of side b at its bottom left, hence its area is a 2 + b 2 . 2 Imaginary Numbers From our experience thus far, numbers with negative squares seem completely uncalled for. Why not just say, as we did in the previous section, that no real number has a negative square and leave it at that? If someone asks us to solve the equation x 2 = −1, we are perfectly entitled to say that there is no solution.
It suggests that parallels control the behavior of angles and, presumably, the behavior of lengths and areas as well. This is indeed the case. Here are just a few of the consequences of the parallel axiom that may be found in the Elements : • Rectangles exist. • The Pythagorean theorem. • The angles of any triangle sum to two right angles.
Yearning for the impossible : the surprising truths of mathematics by John C. Stillwell