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Read e-book online Charming Proofs: A Journey into Elegant Mathematics PDF

By Claudi Alsina, Roger B. Nelsen

ISBN-10: 1614442010

ISBN-13: 9781614442011

Theorems and their proofs lie on the middle of arithmetic. In conversing of the only aesthetic traits of theorems and proofs, G. H. Hardy wrote that during attractive proofs 'there is a truly excessive measure of unexpectedness, mixed with inevitability and economy'. captivating Proofs provides a set of outstanding proofs in basic arithmetic which are awfully stylish, filled with ingenuity, and succinct. through a shocking argument or a strong visible illustration, the proofs during this assortment will invite readers to benefit from the great thing about arithmetic, and to boost the power to create proofs themselves. The authors give some thought to proofs from issues comparable to geometry, quantity concept, inequalities, aircraft tilings, origami and polyhedra. Secondary institution and collage lecturers can use this ebook to introduce their scholars to mathematical beauty. greater than one hundred thirty routines for the reader (with recommendations) also are integrated.

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Extra resources for Charming Proofs: A Journey into Elegant Mathematics (Dolciani Mathematical Expositions)

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10. i C j 1/ D n3 . Proof. We represent the double sum as a collection of unit cubes and compute the volume of a rectangular box composed of two copies of the collection. 14. 14. i C j 1/ fit into a rectangular box with base n2 and height 2n, hence computing the volume of the box in two ways yields 2S D 2n3 , or S D n3 . 3 There are infinitely many primes Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece; but a mathematician offers the game: G.

3 6 9 ... 2 4 6 ... 3. There are infinitely many primes 1 2 3 ✐ ... 13. We conclude this section with a theorem representing a cube as a double sum of integers. 10. i C j 1/ D n3 . Proof. We represent the double sum as a collection of unit cubes and compute the volume of a rectangular box composed of two copies of the collection. 14. 14. i C j 1/ fit into a rectangular box with base n2 and height 2n, hence computing the volume of the box in two ways yields 2S D 2n3 , or S D n3 . 3 There are infinitely many primes Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons.

16. n C 2/-board). Summing yields the desired result. 14. For n 0, f0 C f2 C f4 C C f2n D f2nC1 . Proof. 2nC1/-board are there? By definition, f2nC1 . For a second way to count, we condition on the location of the leftmost square. 2n C 1/-board begins on the left with a square, it can be completed in f2n ways. If the tiling begins on the left with a domino and then a square, it can be completed in f2n 2 ways. If the tiling begins on the left with two dominos and then a square, it can be completed in f2n 4 ways.

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Charming Proofs: A Journey into Elegant Mathematics (Dolciani Mathematical Expositions) by Claudi Alsina, Roger B. Nelsen


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