Using Origami to Find Rational Approximations of Irrational Roots

Transcription

1 Usng Ogam to Fnd Ratonal Appomatons of Iatonal Roots Jeemy Lee Amest Regonal Hg Sool Hudson Rve Undegaduate Matemats Confeene Wllams College Apl 6 t, 0

2 Ogam Instutons

3 Foldng pape allows us to fnd atonal appomatons of te squae oot of two.

4 Sde Unt fts Dagonal One wt emande. Ogam Instutons

5 Sde Unt fts Dagonal One wt emande. Ogam Instutons Remande fts sde unt twe wt emande spae

6 Sde Unt fts Dagonal One wt emande. Ogam Instutons Remande fts sde unt twe wt emande spae Remande fts Remande unt twe wt Remande left ove.

7 Summay of Ogam Eese As we eep dong te eese, you always get te numbe two afte te fst step. Te net emande wll ft nto pevous emande twe agan.

8 Summay of Ogam Eese All tese appomatons of te squae oot of two ome fom te ontnued fatons epesentaton of te squae oot of two.

9 Contnued Fatons Metod a 0 a a a

10 Contnued Fatons Metod a a a

11 Contnued Fatons Metod a a

12 Contnued Fatons Metod a

13 Contnued Fatons Metod Tee wll be twos ontnung tougout te faton afte one.

14 How to Use Contnued Fatons to Buld Appomatons 7 7

15 Ratonal Appomatons Of te Squae Root of Two

16 Denotng Pats of te Squae Root of Two Contnued Faton Let Bg Bo equal Wat does te bg bo onvege to?

17 Algeba tat Contnued Fatons Conveges to te Squae Root of Two 8 0 Wy does te answe ave to be? Assumng tat ts ontnued faton onveges, tat s wat onveges to. Teefoe te ontnued faton equals te squae oot of. ) (

18 Te Babylonan Metod n n n S S Let s ty t wt te squae oot of Have 0 equal, ou fst appomate.

19 Algeba tat Babylonan Metod Conveges to te Squae Root of Two Te eason wy s postve s beause f te Babylonan Metod onveges, te esult s wat t wll onvege to. If s negatve ten te esult s te negatve squae oot of two. Tee s a wondeful blog enty tat s foused on te Babylonan Metod and ow t wos wt te squae oot of two. ttp://jonalosbaez.wodpess.om/0/ /0/babylon-and-te-squae-oot-of-/

20 + / + / Wat ae te eplt fomulas fo ea te numeato and te denomnato of all atonal appomatons of te squae oot of two? Tee s almost a ommon ato between onseutve numeatos of te atonal appomatons. Tee s almost a ommon ato between onseutve denomnatos of te atonal appomatons. Te ato s aound + sqt() fo bot of tese elatonsps. Maybe, and ae te sums of two geomet sequenes wt a ommon ato of + sqt()

21 Ratonal Appomatons Numeato

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23 Ratonal Appomatons Denomnato 0 ) ( ) ( ) ( ) ( ) (7 ) ( ) ( 0 ) ( ) (

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25 Conneton between Contnued Fatons and Babylonan Metod Wat appens wen te ontnued faton appomaton s put nto te Babylonan Metod? ) ( ) ( ) ( ) ( Ts s te Babylonan Metod by tang a ontnued faton appomaton and aveage t wt twe ts epoal. Te esult s te ontnued faton appomaton of te squae oot of two, t s just te numbe of steps of te ontnued fatons appomaton s doubled fo te Babylonan Metod. You only dd one step of te Babylonan Metod.

26 Conluson We dd an ogam epement tat fnds te same atonal appomatons as te ontnued fatons metod. We found tat te ontnued faton and Babylonan Metods podued te same atonal appomatons fo te squae oot of two. Te Blog Enty ttp://jonalosbaez.wodpess.om/0//0/babylon-and-te-squae-oot-of-/

27 Conluson We dd an ogam epement tat fnds te same atonal appomatons as te ontnued fatons metod. We found tat te ontnued faton and Babylonan Metods podued te same atonal appomatons fo te squae oot of two. Could we do sometng smla wt Fbona Numbes and te Golden Rato w as te squae oot of fve n t? Te Blog Enty ttp://jonalosbaez.wodpess.om/0//0/babylon-and-te-squae-oot-of-/