V/O The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.

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Th of Oz: From Fractons to Formulas Th of Oz: From Fractons to Formulas Davd Hllo, I m Davd Brannan. Flms hav oftn bn usd as cratv tools wth whch to tach mathmatcs.

1 Th of Oz: From Fractons to Formulas Th of Oz: From Fractons to Formulas Davd Hllo, I m Davd Brannan. Flms hav oftn bn usd as cratv tools wth whch to tach mathmatcs. On of th bst known xampls s th 1939 classc Th of Oz, partcularly th famous scn whr th Scarcrow gts hs brans and rcts Pythagoras s Thorm albt ncorrctly! V/O Th sum of th squar roots of any two sds of an soscls trangl s qual to th squar root of th rmanng sd. Of cours th Scarcrow should hav rfrrd to a rght-angld trangl not an soscls trangl, and to squars not squar roots, so that: Th squar of th hypotnus of a rght-angl trangl s qual to th sum of th squars of th othr two sds. In ths podcast w r gong to tak you on a mathmatcal journy calld th of OH- ZED, or OH-ZEE f you ar Amrcan. It s a parody of th of Oz, and s basd on an Opn Unvrsty rado programm; but nstad of a Scarcrow, talkng Tn-Man and cowardly Lon, our ncountrs som rathr challngng mathmatcal concpts! V/O Intgrs, fractons, quatons Oh My; Intgrs, fractons, quatons Oh My; Intgrs, fractons, quatons Oh My Phl Wth th hlp of, w ll mt fractons and complx numbrs ncludng a rathr colourful charactr I bfor a clmactc fnal nvolvng on of th most fascnatng quatons of all tm Eulr s Magcal Formula. Audo Clp Toto, I hav a flng w ar not n Kansas anymor. Oh dar! I v sad that alrady. Who ar ths tny craturs? Hllo? Hllo? Who ar you? Fracton W ar th fractons. Fractons? My, arn t you small? Fracton Wll, that s bcaus w ar propr fractons. W ar lss than a whol. Isn t somthng lss than a hol stll a hol? You can t hav half a hol you know? Fracton

2 Not th mpty sort of hol, th full sort, unty, compltnss, onnss, that sort of whol. wh whol, w whol. Oh, ar thr mpropr fractons? Fracton Of cours! Ovr thr! Look! What, th fracton standng on ts had? Fracton That s rght. Any of us bcom mpropr, f w turn ourslvs upsd down. Of cours, and thr ar so many of you. Davd So thr w just hard how mt th fractons. But what ar fractons? And what s th dffrnc btwn mpropr and propr fractons? Phl Wll, Davd, w can xplan fractons by dscrbng a numbr ln. Frst, w mark som pont on th ln that w call zro. Thn n on drcton, to th rght, w mark succssv qually-spacd ponts as 1, 2, 3, and so on, just lk a rulr. And n th oppost drcton w mark succssv qually-spacd ponts as mnus 1, mnus 2, mnus 3, and so on. That gvs us all th whol numbrs, or th ntgrs. Thn w start agan. But ths tm, w mark qually-spacd ponts n th postv drcton at dstancs of on-half from ach othr. From zro th frst s calld on ovr two, th scond, two ovr two, th thrd, thr ovr two, and so on. In th oppost drcton w mark succssv ponts as mnus 1 ovr 2, mnus 2 ovr 2, and so on. Ths gvs us all th ponts on th ln that rprsnt fractons wth dnomnator, that s th numbr on th bottom, 2. Som of thm ar rpats of th ntgrs of cours, for xampl 2 ovr 2 s th sam as 1. Thn w rpat ths procss for ach whol numbr, q, say, whr th dstanc btwn succssv ponts s 1 ovr q nstad of a half. Th ponts that w gt thn rprsnt all th fractons, p ovr q. If a fracton s p ovr q and q s bggr than p, thn w say that th fracton s a propr fracton; and p ovr q s lss than 1, othrws w call t an mpropr fracton. Davd And s that all th numbrs thr ar? Phl No: thr ar lots mor numbrs that somhow l btwn ths fractons. Th fractons ar oftn calld ratonal numbrs, bcaus thy ar ratos of whol numbrs. Th othr numbrs ar thn calld rratonal numbrs - not bcaus thy r llogcal, but bcaus thy ar not ratos of whol numbrs. You can approxmat ths rratonal numbrs by a procss of takng lmts of ratonal numbrs. Takn togthr all ths numbrs ar calld ral numbrs. All ral numbrs can b wrttn as dcmal numbrs, whch ar whol numbrs plus so many tnths, plus so many hundrdths, plus so many thousandths, and so on, possbly for vr and vr - that s an xampl of a lmt. Lt s just look at th tnths, hundrdths and so on, th propr fracton bt. If w wrt th multpls of th tnths, hundrdths and so on, w gt a whol squnc of numbrs btwn 0 and 9 n succsson. If th orgnal numbr s a fracton, thn ths numbrs vntually bcom a squnc of batchs rpatd, w say that th dcmal xpanson s rcurrng. If th orgnal numbr sn t a fracton but on of th othr ral numbrs, thn th squnc sn t a st of rpatd batchs, w say th dcmal xpanson s non-rcurrng.

3 Davd Now, t looks lk s about to mt a 300 yar-old wtch wth a famous curv calld th Wtch of Agns so lt s s what happns. Audo Clp I don t want to dsturb anybody. Fracton Wll, you wll crtanly dsturb hr. You man that angry old lady ovr thr wth th flowng bll shapd cloak? Fracton Sh s th Wtch of Angs, a famous curv. Almost 300 yars old. Vrsra You ll rgrt ths. What s sh so angry about? Fracton Sh s msundrstood, or rathr mstranslatd. Sh s Italan, and hr nam s Vrsra, that mans turnng. That s not a bad nam for a curv. Fracton No, but sh s bn translatd nto Englsh as a vrsra, whch s wtch. Whch s whch? Oh, I s what you man, whch s wtch! Fracton Rght, and sh s bn prtty unplasant about t vr snc. I m sur f I m nc to hr sh ll hlp m. Ah, Vrsra? Vrsra Vrsra, not a vrsra! Can you hlp m plas? Vrsra Probably, but that dosn t man that I m gong to. What do you want? I want to gt back to whr I cam from plas. Vrsra Back to your orgn you man?

4 Plas. Vrsra Wll I m no us to you. I don t pass through th orgn. Oh, do you go anywhr that s usful? Vrsra Wll, I approach th x-axs, th orgn s on th x-axs you know. Is t? Wll ys, I suppos t s. So f I follow you to th x-axs Vrsra I sad I approachd t, not that I got thr. I am asymptotc, I don t gt thr untl nfnty. So do I hav to go to nfnty thn? Vrsra Ys! Why? Vrsra It spols th story f you don t. What s n nfnty? Vrsra Wll, th nds of thngs, nfnt thngs that s. And th! Th? Fractons Th? Vrsra Th of OH-ZED! Shouldn t that b th of oh, no I suppos t shouldn t, t would spol th story. Vrsra You ar larnng. You should hav sn th, and prhaps h wll hlp you gt back to th orgn. (7 46) Won t you com wth m thn? Vrsra How can I? W don t start from th sam plac. So whr do you start from thn?

5 Vrsra From mnus nfnty of cours. Mnus nfnty? Oh dar, ths s all trrbly confusng. Vrsra You ll probably fnd m along th way. Or jonng m you man? Vrsra No, of cours not! Gttng closr and closr th furthr you gt along th x-axs. What do you thnk asymptotc mans? I rally don t know. Vrsra I don t know what ducaton s comng to ths days! Gt on your way! Whch way? Fracton That way! Ho ho! Davd Can you xplan nfnty and mnus nfnty? Phl Wll, f you start at any pont on th numbr ln and walk n th postv drcton - that s, to th rght- kpng walkng so that you pass vry pont on th ln, thn w say that you hav gon to nfnty. And f you walk n th oppost, th ngatv, drcton, kpng walkng so that you pass vry pont, thn w say that you hav gon to mnus nfnty. Actually ths lnks wth th da of srs, somtms calld nfnt srs. If you hav a lst of nfntly many numbrs, thn you can try to add thm all up: so, you wrt th frst on plus th scond, plus th thrd, thn plus th fourth, and so on for vr. Ths xprsson s calld an nfnt srs, or just a srs for short. Somtms th valus of th succssv sums gt closr and closr to a partcular numbr, n fact ndfntly clos as you go on. Thn w say that th srs convrgs and th sum of th srs s that numbr. For xampl, f you start wth th lst a ½, a ¼, a 1/8 and so on, and thn add thm up ths way, thn what you gt s a ½, ¾, 7/8, and so on, Ths numbrs convrg to 1, so th sum of th nfnt srs ½+1/4+1/8 and so on s 1. Othrws w say that th srs dvrgs. Davd Actually, th Wtch of Agns curv has got nothng to do wth wtchs, as you hard, but t s a vry good nam for a curv. Anothr thng, what dd th wtch man by Asymptotc? Phl Ah, ths s whr thngs bcom two dmnsonal! You look at a plan, not just a ln. So, w tak a horzontal numbr ln n th plan that w call th x-axs, and a vrtcal numbr ln that w call th y-axs; wth th two axs mtng at zro on ach numbr ln w ll call that pont of ntrscton th orgn. Suppos nxt that you hav a curv n th plan that mts ach vrtcal ln just onc and that gos nfntly far to th rght. Now suppos

6 that th vrtcal dstanc btwn th curv and th x-axs gts closr to zro, n fact as clos to zro as you plas, th furthr and furthr you go to th rght, thn w say that th curv s asymptotc to th x-axs. Davd OK, shall w gt back to s sarch for nfnty? Audo Clp Oh my, what a long way ths s. And oh no, thr s a fork n th road, whch way do I go? You ar absolutly at rght angls. Howdy! Who s that? Why thr s a strang lttl man standng a lttl way up ths branch? Ah, hllo? Who ar you? I m. I bg your pardon, I don t man to b rud but shouldn t that b I m m? Not at all, I m, magnary, you know? Ths s th magnary axs. Do I want th magnary axs? I m lookng for nfnty. I nd to fnd th. I was followng th x-axs. All axs s lad to nfnty vntually. But t s dramatcally mor convnnt f you stay on th othr road - th ral axs. Wat! Whr hav you gon? I m ovr hr! Back whr you cam from. Oh what dd you do thn? I squar myslf. You know. squar quals -1, I m ral now. Wrn t you ral bfor thn? Not at all. I was magnary. But I can t magn what an magnary numbr could b. It s qut asy. Thnk of mnus on lk I am now. I don t hav any problms wth that.

7 Now I v takn my squar root I m back ovr hr! s th squar root of -1! Oh, shouldn t that b I am th squar root of no, I suppos t shouldn t. So I should tak th ral axs thn that way? That wll crtanly tak you to nfnty, but I warn you, t s a long way! Oh dar. Hy, nvr mnd, I com along wth you. Bsds, I can ask th to hlp m too! What wth? I m too smpl, I gt gnord. Popl don t blv n m, I want to b mor complx. A complx numbr? But arn t you alrady? Oh, nvr mnd. I xpct t would spol th story. Don t you nd to squar yourslf agan thn? Oh yah, yah, sur. Thr! Lt s go! Davd You r lstnng to of OH-ZED: From Fractons to Formulas from Th Opn Unvrsty. W just hard mtng an magnary axs on hr rout to nfnty. Can you xplan I and th squar root of -1 nxt? Phl Wll, that brngs us back to th complx numbrs that mt arlr. W look agan at th plan, wth th x- and y-axs n t. Any pont n th plan can b dscrbd by sayng how far along th drcton of th x-axs t s from th orgn, a dstanc x say; and how far along th drcton of th y-axs from th orgn t s, a dstanc y say. Thn th numbrs x, y ar just th coordnats of th pont. Now hr s th cunnng bt! Th ponts on th x-axs corrspond to ral numbrs, so lt s calld th ponts on th y-axs magnary numbrs. Thn th pont 1 on th y-axs w ll dnot by th symbol I, whr stands for magnary. So th pont 2 on th y-axs s 2, and so on. Thn w ll dnot th pont n th plan wth coordnats x and y by th sngl xprsson x plus y. Thn w ll call x plus y a complx numbr. And so w don t los all conncton wth ralty, lt s agr that complx numbrs hav to satsfy all th usual ruls of arthmtc that s, th ruls for addng, subtractng, multplyng and dvdng numbrs. And ths works fn, just so long as w add th xtra assumpton that multplyng th magnary numbr by tslf gvs mnus 1; that s, I squard quals mnus 1 as you hard. Davd W should fnd out how s gttng on n hr qust to gt to nfnty. Audo Clp

8 Hoh, w v bn walkng for ags now, t nvr sms to gt any nar! That s th problm wth nfnty you know. (Vrsra s laughtr) Vrsra Not as asy as you thought t would b ay? But thr s a way! Wll, how? I m not sur a wtch s advc can b trustd. Vrsra You spd up. Gt fastr and fastr rgularly. Look, n th last hour, you dd two mls. In th nxt on, do thr, and n th on aftr that, do four and so on. Wll I stll thnk t wll tak forvr. Vrsra Mayb, but t s somthng to thnk about. I ll b sng you agan. I thnk I m just gong to go as fast as I can and s what happns. I ll rac you to that pl of junk ovr thr Oh don t, that s not th way to rfr to a dstngushd nfnt srs. Who ar you? What m? H! Not h,! Is vry body n ths country namd aftr a lttr? Oh, t s varabl, but most of us ar. Ltrally, I m th xponntal srs or functon. Tsk tsk tsk, vryon s so casual nowadays. But you can call m, most popl do. Wll, can you lp, r, hlp us? W ar tryng to gt to nfnty. But t sms to b takng rathr a long tm.

9 Wll, I m rathr good at gttng to nfnty. You v hard th xponntal growth I prsum? Ah huh. Ys, wll, I can gt to nfnty fastr than anythng ls you car to nam. So f w wnt along wth you, wll w gt thr quckr? Oh, quckr, crtanly. But t would stll tak forvr. Oh dar, what can w do? Lt m thnk about t, I m sur thr s a way. I suppos w chang how w gt fastr. How? Wll, now, w ar just spdng up on ml at a tm, prhaps w can do bttr than that. Couldn t w go two mls fastr n th nxt hour, thn four, thn ght, sxtn so on, I m sur thr s somthng thr. I can assur you that no good wll com out of t. Why s travllng wth you? H wants th to mak hm mor complx. But sn t h alrady? No, nvr mnd. I xpct t would spol th story. Dos th gv popl thngs thy want? So I m told. Thn would you mnd f I jond you. What do you want thn? To hav som valu! It s all vry wll bng nfnt, but I nvr sm to hav any valu. But surly oh, nvr mnd. / It would probably spol th story. Davd

10 Now, thr w mt th xponntal functon! What s that? And can you xplan th valu of? Phl Frst. W hav to us th shorthand trm n factoral to rprsnt th numbrs 1, 2, 3 and so on up to n all multpld togthr. So 2 factoral Is 1x2, 3 factoral s 1x2x3, that s 6, and so on. So, tak th nfnt srs 1, plus 1 ovr 1factoral, plus 1 ovr 2 factoral, plus 1 ovr 3 factoral, and so on ndfntly. It turns out that ths s a convrgnt nfnt srs whos sum s approxmatly and so on; and that s a non-rcurrng dcmal. And w call ths sum th numbr, also known as th Eulr s numbr. Eulr was a famous Swss mathmatcan who bcam th court mathmatcan of Cathrn th Grat n St Ptrsburg. Davd And th xponntal functon? Phl Rght: tak th nfnt srs 1, plus x ovr 1 factoral, plus x squard ovr 2 factoral, plus x cubd ovr 3 factoral, and so on ndfntly. Ths nfnt srs also convrgs, whatvr valu x happns to b; and ts sum s to th powr x. That s an amazng fact about th numbr. Davd Of cours w nd th and to com togthr wth for Eulr s Magcal Formula: = -1 Audo Clp Oh, w v bn usng th Wtch s mthod for ags, I m not sur that sh was tryng to hlp us at all. (Vasra s laughtr) Just lav m alon for a bt longr, I m sur I m gttng somwhr. Wll w ar not. Infnty s no narr than t was bfor. And I m gttng vry puffd up wth all ths spdng up. It s no bttr than f w wr just gong around n crcls. Spakng of gong round n crcls Why s that lon chasng ts own tal round and round? H wll war hmslf out? Hllo? Who ar you? Ar you an magnary lon? No, I m a ral lon. Thr ar lots of us lons around hr. Paralll lons, ntrsctng lons, straght lons, curv lons, th lon at nfnty Th lon atnfnty? Can h hlp us? W ar tryng to gt to nfnty just to s th you know. I m sur h ll b usful. But h s th lon at nfnty, you s, so h s not hr just now, but ths road s gong th rght way.

11 I s, you ar a ral lon, do you hav a nam? I m, you s, I m not a whol lon, just a lon sgmnt, not a vry long on thr, but a vry rpttv on, I just go round and round n crcls, round and round ths unqu damtr. I know t s rratonal, but I m vry trd of rcurrng ths. But surly oh, nvr mnd, ths s obvously th way th story gos. Do you thnk f I cam wth you to th, h could hlp m? To stop rcurrng? Hmm. Oh yah, do plas jon us. It s so nc to mt somon who s just not a sngl lttr. Wll, I m afrad that that sn t tru. It s my forgn ancstry. I m not p,, you know, t s, Grk. Davd What s th valu of thn? And ddn t w mt non-rcurrng bfor? Phl Wll, th paramtr of a crcl of damtr d s d, so th numbr s just th paramtr of a crcl of damtr 1. And th valu of s approxmatly and so on, a dcmal xpanson that nvr rpats tslf anothr on, that s non-rcurrng. And astonshngly f you put n plac of x n th srs for xponntal functon, that s to th powr, t turns out aftr a bt of work that th sum of that nfnt srs s xactly mnus 1. And that s Eulr s Magcal Formula = -1. You can rarrang that formula as +1= 0, a formula whch nvolvs th 5 most mportant numbrs n mathmatcs 0, 1,, and. A rmarkabl formula! Davd Now, thy r comng to th nd of thr journy whch mans w must b clos to mtng th and Eulr s Formula too? Audo Clp And just who do w hav hr? Plas sr, I m, and ths s, and. And just what do you all want wth m? Plas sr, f t s not too much troubl, I m trd of bng magnary, and I d lk to b a lttl mor complx.

12 Ah uh. And plas Mr, I want to hav som valu, t s not a good lf as an nfnt srs. And I m trd of rcurrng sr. And plas sr, I want to go hom. I want to gt back my orgn. Plas sr. Can you hlp us? Oh darng m, what a collcton of msdrctd non-problmatc popl w do hav hr. Oh ys, what a collcton. But a rathr spcal collcton mnd you. Thn you can hlp us? Oh ys oh ys. Com hr,. I want to b complx. But you ar complx. No I m not. I m wholly magnary. It s all a mattr of how you look at thngs. All you nd s a dffrnt labl. What labl would that b? I thnk that ths on should do ncly. 0+1? Ah, I m complx! I m complx! I m a complx numbr! Oh thank you, thank you, thank you vr so much! Nothng to t, ltrally. And now, lt s s about th rst of you. You,. Ys, sr. What s ths about not havng valu? Wll, sr, I m just an nfnt srs. Just?

13 1 plus 1 plus 1 ovr 2 factoral, plus 1 ovr 3 factoral, plus 1 ovr 4 factoral and so on Dfntly! Oh, phw, rathr ndfnt actually. I just go on and on forvr, on and on to nfnty. Wll, now you v got to nfnty. You v addd all your trms. And I thnk you mght fnd that you v got a valu ! Ha ha! Ha ha, bttr now? Oh ys, ys. But don t tll vryon about t. It s not as asy as that for all nfnt srs. Thr ar qut a fw dvrgnt ons around you know? ! I v got a valu, I v got a valu! Now, what was your problm agan? Oh, I m fd up wth rcurrng ovr and ovr, just gong round and round th sam crcls. And you thnk that s what you ar dong? Ys, vry , I gt back to whr I startd. Evry how many? (don t stop) I havn t hard you rcur yt! Enough! you can go on all day forvr and you wll nvr rcur. It s just a mattr of how you look at thngs. Plas sr.

14 Ys,. What about m sr, I want to gt back to th orgn. Ay, ys, I hadn t forgottn you. But ths s somthng that I thnk your frnds ar bttr plac to hlp m wth. How s that sr?, stop murmurng dgts to yourslf and go ovr thr and stand nxt to. Lk ths??, w ar gttng thr. Now whr s? s bhnd you All Oh no, h sn t. Stop ths nonsns., gt ovr thr and xponntat thm!?. W ar almost don now. It s just on mor thng I nd. Lt m s, I nd 1 of somthng. Ah, ys, Toto, you had nothng to do n ths story so far. I nd to add 1 but not yt, as soon as w do that, you ll b back to th orgn. How com sr? Wll, as Eulr, on of th gratst mathmatcans obsrvd that =-1, so So that +1=0 and I can go hom now! Indd my dar, and frankly I thnk you should. Davd Is t possbl to fnd out mor about th complx formulas that ncountrd? Phl At th Opn Unvrsty w hav a wd rang of maths courss that xplor th numbrs and das that ncountrd. Our ntroductory cours Y162, Startng wth Maths looks at

15 fractons and smpl formulas; whl th cours MU123, Dscovrng Mathmatcs looks at th us of functons as modls of ral-world phnomna, ncludng xponntal functon and othr topcs lk applcatons of basc algbra. Thn th scond lvl cours M208, Pur Mathmatcs dals wth ral and complx numbrs, as wll as lookng at functons, graphs and xponntal functons n gratr dpth, and th scond lvl cours MST209 Mathmatcal Modls uss complx numbrs to hlp solv dffrntal quatons. Fnally, th thrd lvl cours M337, Complx Analyss xplans complx numbrs n much mor dtal, gong all th way from Eulr s Magcal Formula to topcs lk complx functons appld to flud flow and vn fractals lk Jula sts and th Mandlbrot st. Davd On last thng what happnd to th wtch? Phl I thnk th has th last word. Audo Clp Wll, that s just about that xcpt that I thnk I can har somon n th background. Vrsra? Com on out! Vrsra Oh, I v got hr too lat, thy v gon! Oh rally my dar, you should try a mor sdulous study of convrgnt. Vrsra Wll, now I m hr, at nfnty at last, what happns nxt? Wll, I thnk ths could b th bgnnng of a bautful frndshp, or was that a dffrnt mov?

Lucas Test is based on Euler s theorem which states that if n is any integer and a is coprime to n, then a φ(n) 1modn.

Lucas Test is based on Euler s theorem which states that if n is any integer and a is coprime to n, then a φ(n) 1modn. Modul 10 Addtonal Topcs 10.1 Lctur 1 Prambl: Dtrmnng whthr a gvn ntgr s prm or compost s known as prmalty tstng. Thr ar prmalty tsts whch mrly tll us whthr a gvn ntgr s prm or not, wthout gvng us th factors