Why Prove it Again?: Alternative Proofs in Mathematical by John W. Dawson Jr.
By John W. Dawson Jr.
This monograph considers numerous recognized mathematical theorems and asks the query, “Why turn out it again?” whereas interpreting substitute proofs. It explores different rationales mathematicians can have for pursuing and providing new proofs of formerly proven effects, in addition to how they pass judgement on no matter if proofs of a given consequence are diversified. whereas a few books have tested substitute proofs of person theorems, this can be the 1st that provides comparative case reviews of different tools for various assorted theorems.
The writer starts by way of laying out the standards for distinguishing between proofs and enumerates explanation why new proofs have, for therefore lengthy, performed a favorite function in mathematical perform. He then outlines quite a few reasons that replacement proofs may well serve. every one bankruptcy that follows presents an in depth case research of different proofs for specific theorems, together with the Pythagorean Theorem, the basic Theorem of mathematics, Desargues’ Theorem, the leading quantity Theorem, and the evidence of the irreducibility of cyclotomic polynomials.
Why end up It back? will attract a vast variety of readers, together with historians and philosophers of arithmetic, scholars, and working towards mathematicians. also, lecturers will locate it to be an invaluable resource of different tools of featuring fabric to their students.
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This monograph considers a number of recognized mathematical theorems and asks the query, “Why end up it back? ” whereas interpreting substitute proofs. It explores the various rationales mathematicians can have for pursuing and proposing new proofs of formerly demonstrated effects, in addition to how they pass judgement on even if proofs of a given consequence are diverse.
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Additional resources for Why Prove it Again?: Alternative Proofs in Mathematical Practice
Third, not all of the proofs given in Yanney’s and Calderhead’s articles or in Loomis’s book are correct! The most egregious example is Loomis’s ‘algebraic’ proof 16, apparently taken over uncritically from Yanney’s and Calderhead’s proof X. 1 Several other fallacious proofs from those two earlier sources are also cited on Bogomolny’s web site. Loomis’s book is problematical on other grounds as well. The very idea, for example, of distinguishing proofs according to the diagrams used to represent them seems fatally flawed, both because the same diagram, interpreted differently, may be used to represent conceptually distinct arguments, and because, conversely, some arguments can be represented by more than one distinct diagram.
The trigonometric form of the Pythagorean Theorem: Let Â be an acute angle in a right triangle. Then cos2 Â C sin2 Â D 1. Proof: Since ratios of sides are unaffected by scaling, it suffices to consider a right triangle ABC with hypotenuse of length 1 and right angle at A. 6). In triangle ACD, cos Â D jCDj=jACj D jCDj= cos Â, and in triangle BAD, sin Â D jBDj=jABj D jBDj= sin Â, since angle BAD D Â. So 1 D jCDj C jBDj D cos2 Â C sin2 Â. a y/ D b 2 , as in the earlier proof. Should the proof of 4.
The assumption that L is the least common multiple of p1 ; p2 ; : : : ; pk is unnecessary; any common multiple of p1 ; p2 ; : : : ; pk will do. That assumption does, however, reduce the size of the upper bound for q. If p1 ; p2 ; : : : ; pk are all distinct, then their least common multiple L is p1 p2 pk , and if p1 < p2 < < pk are all the primes up to pk , then pk < q. If the sequence p1 < p2 < < pk does not include all the primes up to pk , then in general one can only say that q Ä p1 p2 pk C 1.