TTCProveIt,TheArtofMathemathicalArgument052615 Page1of41. WhatareProofsandHowdowedoThem2. TheRootofProofABriefLookatGeometry

Questions:1. Wheredoesodd/evennumbersdefinitionscomesfrom?

TTCProveIt,TheArtofMathemathicalArgument052615 Page2of4Lecture1:WhatAreProofsandHowdowedoThem:WhatareProofsandWhyareTheyImportant:Aproofisalogicalargumentthatestablishesthetruthofastatementortheorem(statementthathasbeenproven).

Proofsteachyouwhysomethingistrue.Makesyouaprecisemathematician.Bestproofsare,hopefully,shortandelegant,&sometimesgiveyouinsightintowhythetheoremistrue.Earlymatheducationisbasedoncalculations.Thingsaregiventoyouandyouhavetoacceptthem,mostlybyfaith.Postcalculuscourses(Analysis,LinealAlgebra,Topology)areweightedonproofs,toemphasizeprecisemathematicalreasoning.SOMETHEOREMS:PythagoreanTheorem: FermatLastTheorem:foreachinteger 2,therearenopositiveintegersolutions, , suchthat Why.didtheyalwayscallittheoremifitwasunprovenuntilrecently?CONJECTURES:Unsolvedproblems.Ifsolved,becometheorems.CollatzConjecture:Takeanyinteger 0.Ifeven,divideby2,ifoddmultiplyby3andadd1.Itclaims,willeventuallygetto1.No found.AKAcounterexample 3 1 conjecture.COUNTEREXAMPLE:ExamplethatshowsaconjectureisfalseHOWDOYOUGETGOODATPROOFS:1. Understandtheproblem.Trysomenumbersonit2. Doyoureallybelieveit,andcanyouproveit?3. Haveyouprovensimilarthingsinthepast?4. Practiceandbecreative.Playingcantriggeraproof5. Ifyoususpecttheconjectureisfalse,getcounterexampleFirstProofs:Letssupposeyoumultiply2oddnumbers.Istheresultevenorodd?Youcantryseveralexamples:7x9=63,5x7=35,3x9=27.Itseemstheresultisalwaysodd,butbecareful,EXAMPLESDONTPROVEITWILLALWAYSBETRUE.Maybeatsomegiantnumberitbreaks.Ithashappened.So,howdoyouproveforsureoddtimesoddwillalwaysbeodd?Well,1styouneedtodefineobjectivelyandprecisely:(1)Whatsanoddnumber(2)whatsandevennumberEvenNumbers:Andevennumberisalwaystheproductof2timesanotherinteger,(e.g.2k).wherekisanotherinteger(oddoreven,negativeorpositive).Remember,theconceptofodd/evenappliesonlytointegers.Example:8=2x4,10=2x5,6=2x(3).Anintegerisevenifitisoftheform ,whereisaninteger.

OddNumbers:Anintegerisoddifitisoftheform ,whereisaninteger.Someauthorsusetheform .

Inotherwords,andoddnumberisanevennumber(),plus1.Forexample,7isanoddnumberbecauseitisoftheform:7 2 3 1.9 2 5 1LetstrytoprovefirstthefollowingTheorem:P1TheproductoftwoEvenintegersiseven.(1) Beginwithtwoevenintegers,2and2(2) Multiplythem:2 2equals4(3) 4equals22(4) 2isaninteger,multipliedby2,getsanevenLetsnowtrytoprovethefollowingTheorem:P2TheproductoftwoOddintegersisanOddinteger.1. Beginwithtwooddintegers,2 1and2 12. Multiply:2 12 1 4 2 2 13. Regroup:22 14. Notethat22 isaneveninteger,plus

1,makesitanoddinteger;thatistheproof.DirectProofs:Youbeginwithanargument,directlymakesomemath/algebra,andarriveataconclusion.

.LikebothproofsaboveEXAMPLE1:Considerthesetofpositiveintegers 1,2,3,4 andthesetoftheirsquares 1, 4, 9, 16 ,whichsetisbigger?Youmightargue,intuitively,that becausethereseemstobeonlyoneelementbetween1and2(1stand2nditemsof ),but3elementsbetween1and4(1standA2nditemsof );however,bothsetsaresamesizeBbecauseyoucanmatcheachelementinwithanelementin.1 1, 2 4, 3 9, 4 1 andsoon,forever.6Canyouproveit?EXAMPLE2:ConsidertheFibonaccinumbers(FN) 1,1,2,3,5,8,13,21,34 .Whatsthe1000thFN?,Youcangeneratethelistonebyone,butyoucanalsocomeupwithaformulathatpopsoutanyFN.Doesthatformulaexist,anddoesitworkalways,nomatterhowlargethenumber?Canyouproveit?Problem:P3ProvethatEVEN+ODD=ODD1. AddoddNo(2s+1)andeven(2k):2 1 22. Distribute:2 2 13. Regroup:2 +14. 2 isaneven,plus1,isanodd.

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