An Investigation of Continued Fractions

Transcription

1 Ivetigtio o Cotiued rtio Kriti Ptto Ohio Norther Uiverity d Ohio 8 Emil: k-ptto@ou.edu dvior: Dr. dr hroeder -hroeder@ou.edu d Dr. Rihrd Dquil rdquil@mukigum.edu btrt: The tudy o otiued rtio h produed my iteretig d eitig reult i umber theory d other relted ield o mthemti. Cotiued rtio hve bee tudied or eturie by my mou mthemtii uh Wlli Euler Gu Lgrge Rmuj Cuhy d Khihi. oetio betwee otiued rtio d the iboi equee be reveled by emiig utiol prmeter o vriou rtiol utio. Thi work mke ue o eitig reult oerig otiued rtio d Mthemti to eplore the reltiohip betwee otiued rtio d rtiol utio. Cotiued rtio re oe o my wy o repreetig rel umber. The tul term otiued rtio h bee roud or yer ie it w irt ee i Joh Wlli rithmeti Iiitorum i 6. otiued rtio i equee o iteger tht repreet rel umber. Thee equee hve trog impt i umber theory d hve bee tudied by my mthemtii uh Euler Gu Lgrge Rmuj Cuhy d Khihi. Not diiult to udertd d iumerble i their ue otiued rtio provide mthemtii with dieret wy to epre the umber they work with dy i d dy out.

2 The iite imple otiued rtio repreettio o rel umber h the orm: where... Ο re o - egtive iteger d... >. bbrevited wy o writig thi otiued rtio i: [ ]. It i ot diiult to obti the otiued rtio orm o rtiol umber; the proe very loely reemble tht o Eulid lgorithm or idig the gretet ommo divior. The proe begi by idig the gretet iteger prt o the rtiol umber d the determiig the rtio remider p. To otiue the rtio the q remiig rtio i repled by q p. The proe i repeted with thi ew rtiol q umber d otiue util the remiig rtio itel i o the orm p emple o thi proe i how below: Z.

3 o the otiued rtio repreettio o i: or []. 9 Uig Mthemti I took bkwrd pproh to lookig t otiued rtio o rtiol utio. Rther th trtig with the rtiol utio themelve I iput otiued rtio epio o the orm [] [] [] [ ] to ee the type o geertig rtio tht were produed. Wht I oud w quite iteretig: Eh redued orm o the otiued rtio [... ]i o the orm : - or ll where ll o the grph o the otiued rtio hve poit o ourrey o the lie y tht i o the orm :.

4 Uig power erie epio whoe Tylor oeiiet re the poit o ourrey d it ojugte epliit ormul or the equee be geerted. metioed the iitil ivetigtio beg by lookig t the otiued rtio rom bkwrd perpetive. y iputtig the otiued rtio epio I w ble to lyze the geertig rtio tht were produed rom thee epio. Whe the iboi equee rie i the oeiiet o d the ott i both the umertor d deomitor o thee geertig rtio. ptter: Cloe ipetio how the iboi equee pperig i the ollowig o elemet i the epio. where d i the umber There pper to be imilr ptter ourrig i ll o the ueive rtio well. Ivetigtio olude tht thee geertig rtio re o the me orm thoe with the iboi equee oly with dieret yet drmtilly imilr geertig equee. The rtio ll oti equee uh tht where d - - where i the ott ued i the otiued rtio epio d i the umber o term i the epio. - - I the iboi e the geertig rtio were o the orm -. proo by mthemtil idutio how tht thi i ideed the e.

5 Theorem: Proo : or umber o elemet i the otiued rtio equee where [... ] y Idutio. Let [ ] -. ume - - Wt Thereore by the priiple o mthemtil idutio [... ] where umber o or ll - - term i the otiued rtio epio. - - d i the. QED The geerl e i delt with i the me mer the e where. With proo by mthemtil idutio we ee tht or y vlue o the geertig rtio will be o the orm where the equee - - with iitil vlue.

6 Theorem: or umber o elemet i the otiued rtio equee - [... ] - - where Proo :y Idutio. Let [ ] -. ume - - Wt Thereore by the priiple o mthemtil idutio [... ] where or ll QED. Lookig t the plotted geertig rtio poit o ourrey or ll o the geertig rtio o the lie y or eh vlue o pper. The poit o ourrey ot oly be determied uig the grph but lytilly well. gi we will look t the e whe iboi d the ue imilr tehique to how tht thi hold true or ll vlue o. 6

7 7. whe ourrey i the poit o d thu - - : ourrey to id the poit o The ] [... ] [ ] [ The deie the proeedig rtio uh : ]. [ We begi by lettig ± Ο Λ The egtive olutio to the equtio i ot poibility it i ppret tht there i ot poit o ourrey there. We olve or the ourrey poit the me wy i the geerl olutio.. ourrey i d the poit o - - ourrey : To id the poit o ]... [ ]... [ ] [ d ] [ Let ± Ο

8 rom thi we olude eh otiued rtio epio o the orm [...] h poit o ourrey or ll i i whih i the poit o iteretio or the lie y d ll o the grph o the utio d i o the orm. o r thee otiued rtio hve bee delt with eluively with reurive equee. However it i poible to deie epliit rule or the equee tht lo relte to the poit o ourrey oud bove by uig power erie epio or utio. The ollowig i the reurive equee we hve bee diuig o r: d. Thi give equee with the ollowig term: d d ddd d dd d d d.. To id epliit rule or the equee ue the erie epio o the utio: or d The uig the eod epio olve or to obti wht we will ll the geertig utio. 8

9 9 d d d d d d d d d d Now we hve the geertig utio or the equee. We will gi begi with the iboi e. Remember tht the iboi equee i the equee o the orm: - - where. or thi equee we deied d d.. the ojugte o i where - - o d

10 y uig prtil rtio to deompoe the geertig utio we will be ble to epliitly ee the role tht the poit o ourrey ply i the equee itel. The - - hve we o d We ow equte the oeiiet to get epliit ormul or : Thi i the epliit ormul or the equee geerted with or the iboi equee. We ow ee how the poit o ourrey i thi e or the Golde

11 Rtio relted to the origil reurive equee. Oe gi we will repet the tep ued to geerte the epliit orm o the iboi equee to geerte the epliit ormul or ll previouly deied equee. I geerl we deied the equee to be: - - or ll... o i geerl we hve i d d the. Uig the me erie epio o the utio beore or ll equee the geertig utio will be:. y emiig the root o the deomitor we oe gi ee tht ± whih w the olutio diued previouly whe idig the poit o ourrey. We the tor the deomitor how below: where gi by deompoig the geertig utio we epliitly ee the role the poit o ourrey ply i the equee itel:

12 The - - o we hve d We ow equte the oeiiet to give epliit ormul or :

13 Thi i the epliit ormul or y equee - - where d ow we epliitly ee how the poit o ourrey relte to the origil equee d the geertig rtio or ll vlue o. Cotiued rtio re ot diiult to udertd but they do produe iteretig reult whe looked t more iteely. Cotiued rtio epio o the orm [ ] were ivetigted. The reult oud iluded ptter i the otiued rtio itel uig reurive ormul to id the oeiiet the eitee o poit o ourrey o the grph o ll o the epio d the epliit ormul or the reurive ormul produed by thee otiued rtio epio. Reeree Kott Dr. Ro Itrodutio to the Cotiued rtio July. Otober. Old C.D. Cotiued rtio. The Mthemtil oitio o meri. 96 Nith Pritig: U. Roe Keeth H. Direte Mthemti d it pplitio. th editio oto: MGrw Hill. pp. -7.