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PPrrooooff aanndd tthhee AArrtt ooff MMaatthheemmaattiiccss PPPPPPrrrrrrooooooooooooffffff aaaaaannnnnndddddd tttttthhhhhheeeeee AAAAAArrrrrrtttttt ooooooffffff MMMMMMaaaaaatttttthhhhhheeeeeemmmmmmaaaaaattttttiiiiiiccccccssssss PPrrooooff aanndd tthhee AArrtt ooff MMaatthheemmaattiiccss PPPPPPrrrrrrooooooooooooffffff aaaaaannnnnndddddd tttttthhhhhheeeeee AAAAAArrrrrrtttttt ooooooffffff MMMMMMaaaaaatttttthhhhhheeeeeemmmmmmaaaaaattttttiiiiiiccccccssssss Joel David Hamkins JJJoooeeelll DDDaaavvviiiddd HHHaaammmkkkiiinnnsss The MIT Press Cambridge, Massachusetts TTThhheee MMMIIITTT PPPrrreeessssss London, England CCCaaammmbbbrrriiidddgggeee,,, MMMaaassssssaaaccchhhuuussseeettttttsss LLLooonnndddooonnn,,, EEEnnnggglllaaannnddd © 2020 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or chanical means (including photocopying, recording, or information storage and retrieval) me- pweirtmhoiusstion in writing from the publisher. This book was set LATE Xand TikZ by the using author. Library of Congress Cataloging-in-Publication Data is available. ISBN: 978-0-262-53979-1 To my students—may all their theorems be true, proved by elegant arguments that flow effortlessly from hypothesis to conclusion, while revealing fantastical mathematical beauty. TTTooo mmmyyy ssstttuuudddeeennntttsss———mmmaaayyy aaallllll ttthhheeeiiirrr ttthhheeeooorrreeemmmsss bbbeee tttrrruuueee,,, ppprrrooovvveeeddd bbbyyy eeellleeegggaaannnttt aaarrrggguuummmeeennntttsss ttthhhaaattt flflflooowww eeeffffffooorrrtttllleeesssssslllyyy fffrrrooommm hhhyyypppooottthhheeesssiiisss tttooo cccooonnncccllluuusssiiiooonnn,,, wwwhhhiiillleee rrreeevvveeeaaallliiinnnggg fffaaannntttaaassstttiiicccaaalll mmmaaattthhheeemmmaaatttiiicccaaalll bbbeeeaaauuutttyyy... CCoonntteennttss CCCCCCoooooonnnnnntttttteeeeeennnnnnttttttssssss Preface xiii A Note to the Instructor xvii PPPrrreeefffaaaccceee xxxiiiiiiiii A Note to the Student xxi AAA NNNooottteee tttooo ttthhheee IIInnnssstttrrruuuccctttooorrr xxxvvviiiiii About the Author xxv AAA NNNooottteee tttooo ttthhheee SSStttuuudddeeennnttt xxxxxxiii 11 AAAAA bbb CCooolluuuaatttss tttsshhhiieeecc aaAAAll uuuBBttteehhhggoooiirrrnn nniinngg xxxxxxvvv1 √√ 1.1 The number 2 is irrational 2 111111 AAAAAA CCCCCCllllllaaaaaassssssssssssiiiiiiccccccaaaaaallllll BBBBBBeeeeeeggggggiiiiiinnnnnnnnnnnniiiiiinnnnnngggggg 111 1.2 Lowest terms 4 111...111 TTThhheee nnnuuummmbbbeeerrr 222 iiisss iiirrrrrraaatttiiiooonnnaaalll 222 1.3 A geometric p√√√√√√roof 5 111...222 LLLooowwweeesssttt ttteeerrrmmmsss 444 1.4 Generalizations to other roots 6 111...333 AAA gggeeeooommmeeetttrrriiiccc ppprrroooooofff 555 Mathematical Habits 7 111...444 GGGeeennneeerrraaallliiizzzaaatttiiiooonnnsss tttooo ooottthhheeerrr rrrooooootttsss 666 Exercises 8 MMMaaattthhheeemmmaaatttiiicccaaalll HHHaaabbbiiitttsss 777 22 MMuullttiippEEEllxxxeeeee PPrrrcccrriiioossseeeoosssffs s 9888 2.1 n2 n is even 10 222222 MMMMMMuuuuuullllllttttttiiiiiipppppplllllleeeeee−− PPPPPPrrrrrrooooooooooooffffffssssss 999 2.2 O222ne theorem, seven proofs 10 222...111 nnn nnn iiisss eeevvveeennn 111000 2.3 Different proofs suggest different generalizations 12 222...222 OOOnnn−−−−−−eee ttthhheeeooorrreeemmm,,, ssseeevvveeennn ppprrroooooofffsss 111000 Mathematical Habits 13 222...333 DDDiiiffffffeeerrreeennnttt ppprrroooooofffsss sssuuuggggggeeesssttt dddiiiffffffeeerrreeennnttt gggeeennneeerrraaallliiizzzaaatttiiiooonnnsss 111222 Exercises 14 MMMaaattthhheeemmmaaatttiiicccaaalll HHHaaabbbiiitttsss 111333 Credits 14 EEExxxeeerrrccciiissseeesss 111444 33 NNuummbbCCCeerrrrreee TTdddiiihhtttsssee oorryy 11115444 3.1 Prime numbers 15 333333 NNNNNNuuuuuummmmmmbbbbbbeeeeeerrrrrr TTTTTThhhhhheeeeeeoooooorrrrrryyyyyy 111555 3.2 The fundamental theorem of arithmetic 16 333...111 PPPrrriiimmmeee nnnuuummmbbbeeerrrsss 111555 3.3 Euclidean division algorithm 19 333...222 TTThhheee fffuuunnndddaaammmeeennntttaaalll ttthhheeeooorrreeemmm ooofff aaarrriiittthhhmmmeeetttiiiccc 111666 3.4 Fundamental theorem of arithmetic, uniqueness 21 333...333 EEEuuucccllliiidddeeeaaannn dddiiivvviiisssiiiooonnn aaalllgggooorrriiittthhhmmm 111999 3.5 Infinitely many primes 21 333...444 FFFuuunnndddaaammmeeennntttaaalll ttthhheeeooorrreeemmm ooofff aaarrriiittthhhmmmeeetttiiiccc,,, uuunnniiiqqquuueeennneeessssss 222111 Mathematical Habits 24 333...555 IIInnnfififinnniiittteeelllyyy mmmaaannnyyy ppprrriiimmmeeesss 222111 Exercises 25 MMMaaattthhheeemmmaaatttiiicccaaalll HHHaaabbbiiitttsss 222444 EEExxxeeerrrccciiissseeesss 222555 viii Contents vvviiiiiiiii CCCooonnnttteeennntttsss 44 MMaatthheemmaattiiccaall IInndduuccttiioonn 27 4.1 The least-number principle 27 444444 MMMMMMaaaaaatttttthhhhhheeeeeemmmmmmaaaaaattttttiiiiiiccccccaaaaaallllll IIIIIInnnnnndddddduuuuuuccccccttttttiiiiiioooooonnnnnn 222777 4.2 Common induction 28 444...111 TTThhheee llleeeaaasssttt---nnnuuummmbbbeeerrr ppprrriiinnnccciiipppllleee 222777 4.3 Several proofs using induction 29 444...222 CCCooommmmmmooonnn iiinnnddduuuccctttiiiooonnn 222888 4.4 Proving the induction principle 32 444...333 SSSeeevvveeerrraaalll ppprrroooooofffsss uuusssiiinnnggg iiinnnddduuuccctttiiiooonnn 222999 4.5 Strong induction 33 444...444 PPPrrrooovvviiinnnggg ttthhheee iiinnnddduuuccctttiiiooonnn ppprrriiinnnccciiipppllleee 333222 4.6 Buckets of Fish via nested induction 34 444...555 SSStttrrrooonnnggg iiinnnddduuuccctttiiiooonnn 333333 4.7 Every number is interesting 37 444...666 BBBuuuccckkkeeetttsss ooofff FFFiiissshhh vvviiiaaa nnneeesssttteeeddd iiinnnddduuuccctttiiiooonnn 333444 Mathematical Habits 37 444...777 EEEvvveeerrryyy nnnuuummmbbbeeerrr iiisss iiinnnttteeerrreeessstttiiinnnggg 333777 Exercises 38 MMMaaattthhheeemmmaaatttiiicccaaalll HHHaaabbbiiitttsss 333777 Credits 39 EEExxxeeerrrccciiissseeesss 333888 55 DDiissccrreeCCCtteerrreee MMdddiiitttaassstt hheemmaattiiccss 33349991 5.1 More pointed at than pointing 41 555555 DDDDDDiiiiiissssssccccccrrrrrreeeeeetttttteeeeee MMMMMMaaaaaatttttthhhhhheeeeeemmmmmmaaaaaattttttiiiiiiccccccssssss 444111 5.2 Chocolate bar problem 43 555...111 MMMooorrreee pppoooiiinnnttteeeddd aaattt ttthhhaaannn pppoooiiinnntttiiinnnggg 444111 5.3 Tiling problems 44 555...222 CCChhhooocccooolllaaattteee bbbaaarrr ppprrrooobbbllleeemmm 444333 5.4 Escape! 47 555...333 TTTiiillliiinnnggg ppprrrooobbbllleeemmmsss 444444 5.5 Representing integers as a sum 49 555...444 EEEssscccaaapppeee!!! 444777 5.6 Permutations and combinations 50 555...555 RRReeeppprrreeessseeennntttiiinnnggg iiinnnttteeegggeeerrrsss aaasss aaa sssuuummm 444999 5.7 The pigeon-hole principle 52 555...666 PPPeeerrrmmmuuutttaaatttiiiooonnnsss aaannnddd cccooommmbbbiiinnnaaatttiiiooonnnsss 555000 5.8 The zigzag theorem 53 555...777 TTThhheee pppiiigggeeeooonnn---hhhooollleee ppprrriiinnnccciiipppllleee 555222 Mathematical Habits 55 555...888 TTThhheee zzziiigggzzzaaaggg ttthhheeeooorrreeemmm 555333 Exercises 55 MMMaaattthhheeemmmaaatttiiicccaaalll HHHaaabbbiiitttsss 555555 Credits 56 EEExxxeeerrrccciiissseeesss 555555 66 PPrrooooffssCCC wwrrreeeiidddtthhiiitttoosss uutt WWoorrddss 55557666 6.1 A geometric sum 57 666666 PPPPPPrrrrrrooooooooooooffffffssssss wwwwwwiiiiiitttttthhhhhhoooooouuuuuutttttt WWWWWWoooooorrrrrrddddddssssss 555777 6.2 Binomial square 58 666...111 AAA gggeeeooommmeeetttrrriiiccc sssuuummm 555777 6.3 Criticism of the “without words” aspect 58 666...222 BBBiiinnnooommmiiiaaalll sssqqquuuaaarrreee 555888 6.4 Triangular choices 59 666...333 CCCrrriiitttiiiccciiisssmmm ooofff ttthhheee “““wwwiiittthhhooouuuttt wwwooorrrdddsss””” aaassspppeeecccttt 555888 6.5 Further identities 60 666...444 TTTrrriiiaaannnggguuulllaaarrr ccchhhoooiiiccceeesss 555999 6.6 Sum of odd numbers 60 666...555 FFFuuurrrttthhheeerrr iiidddeeennntttiiitttiiieeesss 666000 6.7 A Fibonacci identity 61 666...666 SSSuuummm ooofff ooodddddd nnnuuummmbbbeeerrrsss 666000 6.8 A sum of cubes 61 666...777 AAA FFFiiibbbooonnnaaacccccciii iiidddeeennntttiiitttyyy 666111 6.9 Another infinite series 62 666...888 AAA sssuuummm ooofff cccuuubbbeeesss 666111 6.10 Area of a circle 62 666...999 AAAnnnooottthhheeerrr iiinnnfififinnniiittteee ssseeerrriiieeesss 666222 666...111000 AAArrreeeaaa ooofff aaa ccciiirrrcccllleee 666222