By Claudi Alsina, Roger B. Nelsen
Theorems and their proofs lie on the center of arithmetic. In talking 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'. fascinating Proofs offers a set of exceptional proofs in hassle-free arithmetic which are tremendously stylish, jam-packed with ingenuity, and succinct. by way of a shocking argument or a robust visible illustration, the proofs during this assortment will invite readers to benefit from the fantastic thing about arithmetic, and to advance the power to create proofs themselves. The authors contemplate proofs from subject matters equivalent to geometry, quantity conception, inequalities, aircraft tilings, origami and polyhedra. Secondary university and collage academics can use this ebook to introduce their scholars to mathematical splendor. greater than a hundred thirty workouts for the reader (with options) also are incorporated.
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Theorems and their proofs lie on the middle of arithmetic. In talking of the basically aesthetic features 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'. fascinating Proofs offers a suite of exceptional proofs in hassle-free arithmetic which are exceedingly based, filled with ingenuity, and succinct.
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Extra info for Charming Proofs: A Journey into Elegant Mathematics (Dolciani Mathematical Expositions)
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.
10. 7. 1C1=n/n g is increasing and bounded above and hence converges. 1 C 1=n/n g. a C b/=2. ] simplifying . 11. Let 0 < a < b. a Cpb/=2. a C b/=2. ✐ ✐ ✐ ✐ ✐ ✐ “MABK014-02” — 2010/7/10 — 13:30 — page 32 — #14 ✐ 32 ✐ CHAPTER 2. 7). 6)? 7) yields 2:71824 < e < 2:71837. 6). 8 An infinite series for e We now show that e can be expressed as another limit, and consequently as an infinite series. This limit is usually established using Taylor polynomials. We prove it using only integration by parts [Chamberland, 1999] to evaluate the sequence of partial sums of the series.
17. We again continue to the final case, wherein the tiling begins with n dominos and a square, which can be completed in f0 D 1 way. Summing yields the desired result. See [Benjamin and Quinn, 2003] for many more such Fibonacci identities and combinatorial proofs. Identities involving powers of Fibonacci numbers such as squares and cubes can be illustrated nicely with two- or three-dimensional pictures. 15. For n 1, F12 C F22 C C Fn2 D Fn FnC1 . Proof. 18 [Bicknell and Hoggatt, 1972].
Charming Proofs: A Journey into Elegant Mathematics (Dolciani Mathematical Expositions) by Claudi Alsina, Roger B. Nelsen