Two of th most bautiful ad sigificat umbrs i mathmatics ar π ad. π (approximatly qual to 3.459) rprsts th ratio of th circumfrc of a circl to its diamtr. (approximatly qual to.788) is th bas of th atural logarithms. Both π ad ar idtifid by mathmaticias as trascdtal umbrs. Trascdtal umbrs ar umbrs that caot b xprssd as th root (i.., solutio) of ay algbraic quatio with ratioal cofficits. This mas that π caot xactly satisfy quatios as follows: π = 5 π 4 8π + 4π 6 = 0 Not that ths quatios ivolv itgrs (tchically, itgrs ar ratioal umbrs) with powrs of π. You ca rplac π with ad com to th sam coclusio: caot xactly satisfy ths quatios. Th mathmaticia s pattrs lik thos of th paitr s or th pot s must b bautiful. Th idas, lik th colors or th words, must fit togthr i a harmoious way. Bauty is th first tst. Thr is o prmat plac i th world for ugly mathmatics. G. H. Hardy (877-947) Although th umbrs π ad ar ot solutios of algbraic quatios, ths umbrs ca b xprssd as ifiit cotiud fractios or as th limit of a ifiit sris. Not th bautiful pattrs that ufold i th followig xampls: π= 3 + (Ifiit cotiud fractio) 3 5 7 9 π = + + + + + (Ifiit sris) 6 3 4 = + + + + + (Ifiit sris)!! 3! 4! I 88, Grma mathmaticia Frdiad vo Lidma (85-939) provd that π is trascdtal, puttig a d to arly 500 yars of cojctur. By his proof h showd that π trascds th powr of algbra to display it i its totality; i.., π caot b xprssd i ay fiit sris of arithmtical or algbraic opratios. This trascdig of th fiit is why th word trascdtal is usd to dscrib a umbr lik π. Th dcimal xpasio of π ad vr d, or has ayo dtctd a ordrly pattr i thir arragmt. Hc, ths umbrs ar also irratioal; i.., thy caot b writt as th ratio of two itgrs. 3 I gral,! = 3 4. Not, amazigly, that th frqucy rat of th dcimal digits i th dcimal xpasio of π is qual (i.., th digit 0 occurs 0%, occurs 0%, occurs 0%, tc.). 3 Othr irratioal umbrs ar, 3, 5, tc. Copyright 009 of 5
With computrs, w hav b abl to stimat π to arly o trillio digits. Usig a fixd-siz fot, w could ot writ th dcimal xpasio of π o a pic of papr strtchig from o d of th uivrs to th othr! Th proof that π is a irratioal umbr is a challg to follow, but it ca b do. Provig that is a irratioal umbr is slightly asir (of cours, asy is i th ys of th bholdr!). To rady ourslvs for th proof that is irratioal, lt s first prov that = + + + + +. To do!! 3! 4! this, w must first prov that = + + + + +. W ot th followig: 4 8 6 = + = + + 4 4 = + + + 4 8 8 = + + + + 4 8 6 6 W ca cotiu this procss ad ifiitum. Thrfor, + + + + + + = 4 8 6 3 QED. Do you s this? W ow rarrag what w wat to prov, = + + + + +, as follows:!! 3! 4! + + + + =!! 3! 4! W ow wat to compar ths two quatios: + + + + =!! 3! 4! with + + + + + + = 4 8 6 3 Th first two trms ar qual. W ow compar th third ad subsqut trms i th two ifiit sris: 3! 3 6 4 4! 4 3 4 8 Copyright 009 of 5
5! 543 0 6 ad ifiitum It is clar that th sum of th ( ) sris is lss that. Thrfor, th sum of th sris must b lss tha 3. Wh w add th trms i th sris, w gt a valu of =.788884 QED. Mathmaticias mak us of symbols to writ a shorthad vrsio of a ifiit sris. Lt s ow rviw ths symbols. rprsts th sum of all th trms i a ifiit sris (Not: Σ is th capital Grk lttr sigma). rprsts th sum of all th trms i a ifiit sris from = to (approachig). rprsts ay positiv itgr.! is factorial, th product of all th itgrs from to or, as w hav alrady otd,! = 3 4. W ca show that 0! =. W ot that! = 3 4 ( - ) = ( - ) 4 3. Usig this aalysis, w obsrv: 6! = 6 5 4 3 = 6 5! =70 5! = 5 4 3 = 5 4! =0 4! = 4 3 = 4 3! =4 3! = 3 = 3! =6! = =! =! = 0! = Basd upo th pattr idicatd, w ca coclud: Sic 0! =, th 0! =. QED. Hc, = + + + + + = + + + + +. Usig th abov symbols ad otig th!! 3! 4! 0!!! 3! 4! bautiful pattr, w ca ow writ a compact dfiitio of : = 0! W ca also algbraically play with ths symbols ad writ i a varity of ways: = + + + + +!! 3! 4! = + =!! = + + ( + ) =! ( + )! Sic = + + + + +, th = 0!!! 3! 4!! ( ) Copyright 009 3 of 5
Copyright 009 4 of 5 Th formula for compoud itrst is coctd to i a wodrous way. Th xprssio for calculatig compoud itrst is + m whr is th capital, is th itrst rat, ad is th umbr of yars that m th moy is goig to b dpositd. Not, if th itrst rat is 5%, th m = 0. From this simpl formula, w x ca grat aothr o: +, which, as (quivalt to cotiuous compoudig), is qual to x! Usig symbols, w gt: x x x x x As, + = + + + + = or, usig limit otatio from th calculus,!! 3! x x lim + =. Wh x =, w gt = = + + + + +, our formula for! By rplacig x with (-x) ad!! 3! 4! obsrvig th (-x) raisd to a odd powr rsults i a gativ umbr ad (-x) raisd to a v powr rsults i a positiv umbr, w ca driv th ifiit sris for -x : x x x x x x x x = + + + + + = + +!! 3! 4!!! 3! 4! x x If x =, th = = + + 0.36787944 =!! 3! 4!.788884 W ar fially rady to prov that is irratioal. To do this, w bas our argumt upo th ifiit sris rprstd by th rciprocal of or. Hr is our startig poit (otig that = ): 0! = + + 0!!! 3! 4! W cosidr what happs with th diffrc of coscutiv partial sums of this sris. W lt P(k) rprst th k th partial sum of this ifiit sris. By doig so, w immdiatly ot th followig: P( k) P( k ) =± k! Lt s illustrat this truth for k =,, 3, 4, ad 5: k P(k) P(k ) P(k) P(k ) P() = 0 P(0) = 0 = P() = P() = 0 3 P(3) = 3 P() = 0 = = 3 6 ± k!! +! 3!
4 P(4) = 3 P(3) = 3 = + 8 3 8 3 4 4! 5 P(5) = P(4) = 3 3 = 30 8 30 8 0 5! W ow compar ach pair of coscutiv partial sums usig commo domiators. W ot: 0 ) P() = < < = P(0) 0 ) P () = < < = P ( ) 3) P( 3) = < < 3 = P( ) 6 6 4) P( 3) = 8 < < 9 = P( 4) 4 4 44 45 5) P5 ( ) = < < = P4 ( ) 0 0 64 65 6) P5 ( ) = < < = P6 ( ) 70 70 ad ifiitum Th immdiat cssity for cocludig that (ad ) is irratioal should b vidt from this dmostratio. Th followig aalysis xplicats why this coclusio is justifid. Not th boudary coditios o. Each succssiv boudary coditio boxs btw two ratioal umbrs. Not also that ach pair of boudig umrators diffrs by, ad th domiators ar factorials (!, 3!, 4!, 5!, tc.) or m! Lt s ispct ths comparisos by lookig at th third boudary coditio. It tlls us that if is ratioal its domiator caot b a divisor of 6, bcaus th it could b writt for som itgr, ad thr is o 6 such itgr gratr tha ad lss tha 3. Similarly, th fourth boudary coditio provs that th domiator of caot b a divisor of 4, ad th fifth provs that it caot b a divisor of 0, ad ifiitum. It is clar that, by cotiuig with this logic, th domiator of caot b a divisor of ay m! (for m =,, 3, 4, ). Not carfully that vry itgr k is a divisor of m! (for all m k). Sic is boudd by ths succssful partial sums, th umrator of caot b a itgr. Hc, QED. (ad thrfor ) caot b a ratioal umbr. Copyright 009 5 of 5