I : Constructions " & Compass stature of has basic - PowerPoint PPT Presentation

Applications " Ruler Geometric I : Constructions " & Compass

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(d) Mp another similar Ingle argument . Comdt text . Th DNA fold of constantine about this what say we can numbers ? constricting then Fact : if is > O Important a Ta is so . our Q . degree infinite numbers constructible are Cor : ( Vz ) E ⑥ ( %) E ( THE GQ E PI Consider - - -

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la , .ae ) , Ibi , but , Cayce ) , @ , .dz ) kegthem If in are ( a.aybi.bz , c. end . .dz ) ) for ⑥ E F E IR ( say Fh : F ' Sonu AB , c , and D constructible relation to lay ) Then ay extension of quadratic F lives in some . x. yet ( re ) ) that ( ie , exists ZEF so ( sketch ) Ef LCGD ) LL AB ) of and intersection hey ) case is 1 . : LCA , B ) ABEFZ , Then The gratin for Note

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⇒ (3) CCA .is/ncC4D ) if Cx , y ) same idea E TBT useful ? Hew is this austmtibk numbers . coustmtibte if Tim x. you IX. g) is " : @ 3=-2 [ ① Ca ) implies constable Cos is a n EIN for some . constructible HI Cao ) constable is . 2 is

. " , lxk-i.gr ) , ko ) ) a sequence { 10,9 , 4,0 ) , Hi , y . ) , hail Hence we cealtrufibk in the so that term sequence is each in the squirm preceding terms to relative some . s . but of to constable relative some Hence ( Xi , y ; ) is ( Xinyi ) is . . , ( Xi . . , you , ) Sinn 199,407 , IX. , Yi ) ; . coordmh canstmtihk in an when point , relink to know - > Xia , Yin ) ④ ( KH . , we . , Xii , Yi - D) c. 2 : ⑥ ( Xi , Yy [ ④ ( x , ,yy , Xi , yi ) . . , Xia , Tia . . .

g th degree along th tower ferula ' Have us - - EQ ( 14,9 , , E ④ ( kyyyxayr ) E . - yxk.ik.in ) . ) ④ ( Xi ⑧ E , y - : Q ) = 21 ( ⑥ ( x , ,y . , . " Yak ) get . ,Xk we . . . we get formula By the degree again Q ] net [ Ak ) - 2 for " some - : . DA

triseotnbk . all angles Cos are Not unit create a can If I we Prep says . ) length guilty Ingle . I Internal angles 60 are is equivalent to couslnetibil.ly angle trisecting this so angle triple However the . ) af cos 120 . fermin says fee µ f¥ cos 1307=4 cos ' la ) On - 3 cos O c. ska ) salish So .

. ] =3 [ Ql cos 200 ) t 2 " : Q get So we . constrict ble . cos 1207 isn 't Hence can 't duplicate cube . Cos a you is equivalent to aeustuetig Vz . duplication of cube PI - 2 - x3 irra ( Vz ) , so has V2 - But " =3 t 2 [ atra ) : - Dappy

circle . Cos you can 't square a constructible : if you have PI ' r Equitant te , , ft is coustutibk ) lie constructible then is rift . so , then [ ④ ( fu ) : Q ) for some " 2 If algebraic , be fries fit to . This n . But algebraic is and is * so it transcendental , so → ← DH .

ioastuetihilitg of regular If keen on are you 137 . you'll find consult pg gens , n - th ( Gauss ) If odd pain , then you is an p . get * i - goniff regular p . constrict p a can - Fermat t > 0 primes for sewn . or I 't 1=5 , 22 't 1=17 , 2%1 ' - 3 2 C. g. , - - th - = -