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B0R0M 2r = 1Ht8lP ` , S 0Aformul3" u > Z"v @8,$D  0 > V ? S &Api0  b A s 2AL @piv s  B Z'v @PP,$ 0 > 6 O Z*v @P(P,$D 0 p*TANER YNDER SER0LER0   R  `.v2 @BP,$D  0 >  S  fvd @8  U  `3v2 @BP,$D  0 >  V  fvd @8 X  fvd @PP \ c ZA 43dflagsdotcom_turke_2faws0<3 ] # l8vd @0P >  ^ c ZA 43dflagsdotcom_turke_2faws`<c _ <;p   l~ 2'$  # l$Av @?" ?  M< >   N?" A   N?"}A`   N?"]A@  N?"=A   N?"A  N?"A  N?"A  N?"A  N?"A  N?"A  N?"A  N?"A  N?"A  N?"Ac  N?"`AC  N?"@A#  N?" A  N?" A    `v @?"?H  0޽h ? ̙33y___PPT10Y+D=' = @B +!    @*y 0 { (   C 8  y D   <xKT P4- B0R0M DA0RE 00NE 0Z0LEN E^KENAR GEN0N ALANINDAN YARARLANILARAK, B0R0M DA0REN0N ALANININ HESAPLANARAK r 2 'YE BOLUNEREK SAYISININ BULUNMASI. B0R0M 2r =1Hn7b   s 2AL @pi@`   S 0Aformul4%p   <pPr B5- TM BU 0^LEMLERDEN SONRA SAYISININ BULUNAB0LMES0 00N GENEL 0FADE YAZMAK 0STERSEK,FAVOR0 SER0M0ZDE...&om!Mh   8AL @?piV `   S 0Aformul5    <@ 0`  bE^0TL00 ORTAYA IKAR - HATA N0SYAN MSTESNADIR. ,20F   <̀;  NOT: YUKARIDAK0 SER0LERDE vs. 0LE GSTER0LEN SAYILAR 0^LEM BEL0RL0 B0R NOKTADA KES0LMEK 0STEND00NDE 0FADEN0N SON OLARAK ARPILACAI KATSAYILARDIR. 0^LEM KES0LD00 NOKTADAN NCE SER0LER0N ALTINDA VER0LEN KATSAYILAR D0KKATE ALINMAMALIDIR'=2\   c ,A?x4 @    Z{v @8,$D  0 >    Zv @PP,$ 0 >      `0v2 @BP,$D  0 >     fvd @8 !   fvd @PP "  c ZA 43dflagsdotcom_turke_2faws0<3  #   `кv2 @BP,$D  0 >  $   fvd @8 %  # lLvd @0P >  &  c ZA 43dflagsdotcom_turke_2faws`<c6 c  Z4ëv @P(P,$D 0 p*TANER YNDER SER0LER0  e  # ltǫv @?" ?  M< >  f  N?" A  g  N?"}A`  h  N?"]A@ i  N?"=A  j  N?"A k  N?"A l  N?"A m  N?"A n  N?"A o  N?"A p  N?"A q  N?"A r  N?"A s  N?"Ac t  N?"`AC u  N?"@A# v  N?" A w  N?" A  x   `v @?"?H  0޽h ? ̙33y___PPT10Y+D=' = @B +   C;0'r$0 (  $8  r$  $ <ϫ  $KAYNAK GSTER0LEK YAZILI MAKALE VEYA 0NTERNET'TE L0NK AILARAK YAYIMLANAB0L0R. F0K0R VE SANAT ESERLER0N0 KORUMA KANUNU KAPSAMINDADIR.. TASD0K TAR0H0 03.^UBAT.1997 NO: 4085 2.NOTER MAN0SA / TRK0YE *D5 xf $ c 6Api_anim7  $ <o t ,$ 0 VBu seriye devam ettiiniz takdirde H,"3P   $ S &Api0 C ,$ 0 $ <a0p  LTFEN A^AIDAK0 HAREKETL0 KIRMIZI AN0MASYONU TAK0P ED0N0Z nA 3x  ` $ <xf t ,$ 0 :say1s1na ula_acaks1n1z. F3 $ Zlv @8,$D  0 >  $ Zpv @ ~P,$ 0 >   $  `xtv2 @BP,$D  0 >  $  fvd @8 $  fvd @PP $ c ZA 43dflagsdotcom_turke_2faws0<3  $  `yv2 @BP,$D  0 >  $  fvd @8 $ # l |vd @0P >   $ c ZA 43dflagsdotcom_turke_2faws`<c6 \$ ZHv @P(P,$D 0 p*TANER YNDER SER0LER0  ^$ # lPv @?" ?  M< >  _$ N?" A  `$ N?"}A`  a$ N?"]A@ b$ N?"=A  c$ N?"A d$ N?"A e$ N?"A f$ N?"A g$ N?"A h$ N?"A i$ N?"A j$ N?"A k$ N?"A l$ N?"Ac m$ N?"`AC n$ N?"@A# o$ N?" A p$ N?" A  q$  `v @?"?H $ 0޽h ? ̙33y___PPT10Y+D=' = @B +@  ?? B4`V?(  4R 4 s *i>8  4  4 <  6 Dap1 2r =X=1 olan dairenin iine izilen 4 , 8, 16 , 32 gen in birer kenarlar1n1 s1ras1yla, X1, X2, X3, X4 diye tan1mlarsak, Pisagor ba1nt1s1ndan faydalanarak, kenar uzunluklar1n1n, X1=0.707106781..n, X2=0.382643439..n, X3=0.0195090314..n, X4=0.098017136..n deerlerinde olduunu buluruz,tabii bu kenarlar1n birbirlerine aralar1nda bir oran olmas1 ka1n1lmazd1r ama bu oran, acaba nas1l bir orand1r. X / X1 den ba_layarak bulunan kenar uzunluklar1n1 birbirine blme i_lemleri s1ras1yla ......... X / X1 / X2 / X3 / X4..n _eklinde yap1l1rsa. #3 /"      N              4 B|0P BSIRA"  4 B$0 P P0LK KENAR" l  4 <0 P  4 B 0P XSONRAK0 KENAR"   4 BÇ+0pP R BULUNAN ORAN*   4 B|LJPp  X / X1= P 4 B ͇P p h2 r = 1(  4 BчP p Z /  4 Bև Pp | 0.707106781...n433 4 BۇPp U= 1.414213562...n( 4 B߇p  s X1/X2= ( 4 Bp  S0.707106781...n(33 4 B,p  Z /  4 B  p  p 0.382683439...n( 4 Bxp  U= 1.847759064...n( 4 Bx   s X2/X3= ( 4 BT  S0.382683439...n( 4 B  Z /  4 B   p 0.195090314...n( 4 B$   U= 1.961570562...n( 4 B   s X3/X4= ( 4 Bȡ  S0.195090314...n( 4 B  Z /  4 BD   p 0.098017136...n(  4 Bx   U= 1.990369453...n(4 $4 <䲓 0 t  >Oranlar1nda olduunu greceiz.( 6    +4 <ී` `4  Bu kere X / X1 ...n olarak nerilen serimizdeki ilk i_lemden ba_layarak, ( X=2r=1 ) VU   V ,4 S &Aff P o S .4 6 d P  Z...n& 2  54 Z Ɠv @8,$D  0 >  =4 <ϓ Dj0_lemler0 s1ras1yla yap1l1rsa ve nerilen seri doru ise, bir nceki i_lemde pisagor ba1nt1s1yla hesaplanan 4-8-16-32 genlerin kenar uzunluklar1n1 aynen bulmam1z gerekmektedir.>>>         >4 Zٓv @P(P,$ 0 >  ?4 Zݓv @2P,$D 0 I. j @4 <8Z,$D 0 J SAYISININ P0SAGOR TEOREM0YLE 0SPATI@&b(B(#@B(%  D4  `<v2 @BP,$D  0 >  E4  fvd @8 F4  fvd @PP G4 c ZA 43dflagsdotcom_turke_2faws0<3  H4  `v2 @BP,$D  0 >  I4  fvd @8 J4 # lPvd @0P >  K4 c ZA 43dflagsdotcom_turke_2faws`<c 4 # lPv @?" ?  M< >  4 N?" A  4 N?"}A`  4 N?"]A@ 4 N?"=A  4 N?"A 4 N?"A 4 N?"A 4 N?"A 4 N?"A 4 N?"A 4 N?"A 4 N?"A 4 N?"A 4 N?"Ac 4 N?"`AC 4 N?"@A# 4 N?" A 4 N?" A  4  `v @?" ?H 4 0޽h ? ̙33y___PPT10Y+D=' = @B +N  ,M$M]8`L(  8R  8 s * L8 p 8p 8 BhAqz8 pSER0LER" 8 BDq8 `ONDALIK DEER* l 8 <8 8 B`Iq 8 P0LK KENAR"  8 BMq 8 `ONDALIK DEER*  8 B@Rq0 XSONRAK0 KENAR"  8 BWqp8*b 3 8 BZqs8pb  = 1.414213562...n4( 8 B`q8 b >=>  8 Bpeq8 b h2 r = 1(  8 Bhjq3 80b 1.414213562...n =0   8 B`oq8b z0.707106781...n (33  #8 Btqsbp  = 1.847750064...nD( $8 Bp{qb  >=>  %8 B4qb  w0.707106781...n(33  '8 B,q3 b0 1.847750064...n =0  (8 Bqb S0.382683439...n( +8 Bqsp  = 1.961570562...nD( ,8 Bq  >=>  -8 Bq5  w0.382683439...n(  /8 B8q3 0 1.961570562...n =0  08 BĤq S0.195090314...n( 38 Bqsp  = 1.990369453...n4( 48 Bq  >=>  58 Bq  w0.195090314...n(  78 Bq3 0 1.990369453...n =0  88 Bqq S0.098017136...n(Z 8 S *Aff12r` "8 S 0A ff22+ 22@.f *8 S 6Aff32+ 2+ 2vrl 28 S <Aff42+ 2+ 2+ 2l <8 <4 R r =8 Bq4 B b P / >8 Bq4 b  P / ?8 Bq4   P / @8 Bq4   P / A8 <Pqcn   p Sonu olarak, daha nce Pisagor ba1nt1s1 ile bulunan kenar uzunluklar1n1, nerilen ve uygulanan yeni seri ile bulabildiimizden, ispat tart1_mas1z dorudur. Bu seriyi kullanarak bulunan 2n+1 kenarl1 okgenin, kenar uzunluklar1 toplam1n1 olan emberin evresini, apa blersek say1s1na giderek yak1nla_aca1m1z1 a_a1da grebiliriz.0q <         V C8 S &Api  Z8 Bgq  S0.707106781...n(33 [8 Bdq  =X  \8 Bq   _ 4 GEN  ]8 Bq   t= 2.828427124...n  ^8 B  S0.382683439...n( _8 BP  =X  `8 B   _ 8 GEN  a8 B   t= 3.061467512...n  b8 B  S0.195090314...n( c8 BP  =X  d8 B  B16 GEN  e8 BL  t= 3.121225024...n  f8 BS s S0.098017136...n( g8 BP S s =X  h8 B Ss B32 GEN  i8 BlSs t= 3.136548352...n  j8 B `  SRNEK k8 B  *  >=>  w8 Zv @8,$D  0 >  x8 <T00   _ ~ ,  "   }8 Zdv @PP,$ 0 >  ~8 c ZA 43dflagsdotcom_turke_2faws0`3 8 c ZA 43dflagsdotcom_turke_2faws``c  8  `v2 @BP,$D  0 >  8  fvd @8 8  fvd @PP 8 c ZA 43dflagsdotcom_turke_2faws0<3  8  `"v2 @BP,$D  0 >  8  fvd @8 8 # l%vd @0P >  8 c ZA 43dflagsdotcom_turke_2faws`<cj 8 <(Z,$D 0 J SAYISININ P0SAGOR TEOREM0YLE 0SPATI@&b(B(#@B(% 8 # l.v @?" ?  M< >  8 N?" A  8 N?"}A`  8 N?"]A@ 8 N?"=A  8 N?"A 8 N?"A 8 N?"A 8 N?"A 8 N?"A 8 N?"A 8 N?"A 8 N?"A 8 N?"A 8 N?"Ac 8 N?"`AC 8 N?"@A# 8 N?" A 8 N?" A  8  `v @?")?H 8 0޽h ? ̙33y___PPT10Y+D=' = @B +s;  ::.,` :(  ,R , s *x98  ,V , S &Api`  $  , <@`j@ d 1.Ondal1klar1 sonsuz olan say1s1 gnmze kadar ok dei_ik teoremlerle bulunmu_tur, hatta Japon Yasuma KANADA, 206.158.430.000 ondal1a kadar hesaplayarak bir dnya rekorunuda k1rm1_t1r,ancak say1s1n1 ispatlamak ve anlamak ayr1 bir konu olduundan, trigonometri , logaritma , integral , trev , limit vs. bilinmesi _artt1r,mevcut ispatlar1 7 s1n1f rencilerinin anlamas1da mmkn deildir. "               , <R`   D 3.Klasik yntemle,Yunanl1 Ar_imed, 'nin 3 ondal1k basama1n1; 1479'da Ulu Bey' in gk bilimcisi, Al-Ka_i 16 ondal1k basama1n1; 1579'da Frans1z F. Viete 9 ondal1k basama1n1 bulmu_, ayr1ca 2 / 'ye giden sonsuz bir seri yazm1_t1r, 1593'te Hollandal1 Adrian Van Roomen 15 ondal11; 1610'da Ludolph Van Coulen 35 ondal1k basama1 bulmak iin hayat1n1n byk bir k1sm1n1 harcam1_t1r.              V , S &Api&` 8  , <j`  D 4.Burada kulland11m yntemde klasik, yani dairenin kenar uzunluunun apa blnmesine dayanmaktad1r ama farkl1 olan, okgenlerin kenar uzunluklar1n1 bulmak iin uzun sre gerekmemektedir. Sadece ispattan tatmin olmak iin,pisagor teoreminden faydalanarak bulunan 4-8-16-32 genlerin kenar uzunluklar1n1,s1ras1yla birbirine blerek ortaya 1kan oranlar1n favori yeni serimizle bulunan oranlarla e_it olduunu gstermektedir, bu e_itlik 2n+1gen'e kadar devam edecei sphe gtrmediinden matematik evrelerince, bu serinin doruluu ve anla_1lma dzeyinin ORTA E0T0M seviyesinde olmas1ndan dolay1da kabul grecei muhakkakt1r.: "                       V , S &Api0  # , <P`  2.Oysa,^imdi nerilen yeni seri ve formlleri,eitim seviyesi c2=a2 + b2 ba1nt1s1n1 ( Pisagor teoremini ) ve 4 i_lemi bilen 7 s1n1f rencileri dahi say1s1n1n nas1l bulunduunu ve de ispat1n1 anlayabilirler. Ara_t1rmalar1m1za gre bu kadar basit anlat1ma veya ispata literatrde rastlanmam1_t1r. zH 8   |       V , S &ApiP ,, ZЯv @8,$D  0 > V @, S &ApifC B, Zv @PP,$ 0 >   C, Z̶v @ ~P,$D 0 G. 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L `  0A?L  `  0A?  `  0A?  `  0A?\ `  0A?\ `  0A? `  0A?( `  0A?(l `  0A?l `  0A? ,& `  0A?, p& `  0A?p & `  0A? & `  0A? <& `  0A?< & `  0A? & `  0A?  & `  0A? L & `  0A?L & `  0A? & `  0A? & `  0A? \& `  0A?\ & `  0A? & `  0A? (& `  0A?( l& `  0A?l & `  0A? ,F `  0A?, pF `  0A?p F `  0A? F `  0A? <F `  0A?< F `  0A? F `  0A?  F `  0A? L F `  0A?L F `  0A? F `  0A? F `  0A? \F `  0A?\ F `  0A? F `  0A? (F `  0A?( lF `  0A?l F `  0A?@ ,f `  0A?,@ pf `  0A?p@ f `  0A?@ f `  0A?@ <f `  0A?<@ f `  0A?@ f `  0A?@  f `  0A? @ L f ` ” 0A?L @ f ` Ô 0A? @ f ` Ĕ 0A? @ f ` Ŕ 0A?@ \f ` Ɣ 0A?\@ f ` ǔ 0A?@ f ` Ȕ 0A?@ (f ` ɔ 0A?(@ lf ` ʔ 0A?l@ f ` ˔ 0A?` , ` ̔ 0A?,` p ` ͔ 0A?p`  ` Δ 0A?`  ` ϔ 0A?` < ` Д 0A?<`  ` є 0A?`  ` Ҕ 0A?`  ` Ӕ 0A? ` L ` Ԕ 0A?L ` ` Ք 0A? ` ` ֔ 0A? `  ` ה 0A?` \ ` ؔ 0A?\`  ` ٔ 0A?`  ` ڔ 0A?` ( ` ۔ 0A?(` l ` ܔ 0A?l`  ` ݔ 0A? ,` ޔ 0A?, p` ߔ 0A?p `  0A? `  0A? <`  0A?< `  0A? `  0A?  `  0A? L `  0A?L `  0A? `  0A? `  0A? \`  0A?\ `  0A? `  0A? (`  0A?( l`  0A?l `  0A?,`  0A?,p`  0A?p`  0A?`  0A?<`  0A?<`  0A?`  0A? `  0A? L `  0A?L  `  0A?  `  0A? `  0A?\`  0A?\`  0A?`  0A?(`  0A?(l`  0A?l`  0A?,`  0A?,p`  0A?p`  0A?`  0A?<`  0A?<`  0A?`  0A? `   0A? L `   0A?L  `   0A?  `   0A? `   0A?\`  0A?\`  0A?`  0A?(`  0A?(l`  0A?ll  <$P A^AIDA 0FADE ED0LMEYE ALI^ILAN SER0LER,TARAFIMDAN YAPILMI^TIR,B0L0NEN VE KABUL ED0LEN 0SPATI YAPILMI^ 3.14159265358...n SAYISIYLA E^ OLUP,B0RE B0R DORUDUR,FANTAZ0 B0R ALI^MADIR, L0TERATRDE RASTLANMAMI^TIR.(2   <l%0 < A 8 8 8 8 = ( - ) + ( -------- ) + ( ------------- ) + ( ---------------- ) + ... n 3 (1x32)+3 [(1+2)x32]+3 [(1+2+3)x32]+34 23 V  S &Apif V  S &Apif_V  S &ApiP   < 1`   B 8 8 8 8 8 8 = --------- + --------- + --------- + --------- + --------- + ---------- + ... n (2x1)-1 (2x3)-1 (2x5)-1 (2x7)-1 (2x9)-1 (2x11)-1L! 23 U !D  H:c0 | 2 2 2 2 2 2} 2}  }V  S &ApifP    <@?0 `  y C 8 8 8 8 8 8 = ------ + ------ + -------- + -------- + ------- + ---------- + ... n (2)-1 (6)-1 (10)-1 (14)-1 (18)-1 (22)-1L 23 L .  HH 4 @  f2 2 2 2 2 2g 2g  g  <L P 0 l D 8 8 8 8 8 8 8 8 8 = ----- + ----- + ----- + ----- + ----- + ----- + ----- + ----- + ------ + ... n 3 35 99 195 323 483 675 899 11558 233 J   HN?    YUKARIDAK0 0^LEMLER0N SONUCU,A^AIDAK0 SAYISAL SER0YE G0TMEKTED0RCC C ! Z\Yv @PP,$D  0 B R " Z]v @X0P,$D 0 BTANER YNDER 0N D0ER ALI^MALARI"" " & Z@av @8,$D  0 >  ( c ZA 43dflagsdotcom_turke_2faws0`3  )  `ev2 @BP,$D  0 >  *  fvd @8 +  fvd @PP , c ZA 43dflagsdotcom_turke_2faws0<3 - c ZA 43dflagsdotcom_turke_2faws``c  .  `hv2 @BP,$D  0 >  /  fvd @8 0 # llvd @0P >  1 c ZA 43dflagsdotcom_turke_2faws`<c m # lpv @?" ?  M< >  n N?" A  o N?"}A`  p N?"]A@ q N?"=A  r N?"A s N?"A t N?"A u N?"A v N?"A w N?"A x N?"A y N?"A z N?"A { N?"Ac | N?"`AC } N?"@A# ~ N?" A  N?" A    `v @?"i?@H  0޽h ? ̙33y___PPT10Y+D=' = @B +.  --.}(0 m-(  (R ( s *//,8  }( ( < `Pf   1947' DE MAN0SA'DA DODUM,8'NC0 SINIFA KADAR REN0M GRDM. LAST0K BAY0L00 YAPMAKTAYIM,SAYILARA KAR^I OLAN 0LG0MDEN DOLAYI AMATR B0R MATEMAT0K0 OLDUM. x 10 : 9 VE BUNUN KARES0 9.876543210..n 0LE KAR^ILA^TIM, BU SAYININ DOAL GZELL00NDEN ETK0LEND0M. x 10 : 9 = 3.14269680529..n '0N SAYISI OLAB0LECE0N0 SANDIM.ST DZEYDE MATEMAT0K B0LMED00MDEN HATAMI HEMEN GREMED0M AMA B0RKA YIL SONRA 0RRASYONEL VE TRANSANDANTAL ( STN- A^KIN ) SAYILAR KAVRAMINI REN0NCE BU SAYININ NEDEN SAYISI OLAMIYACAINI ANLADIM. SAYISININ B0L0NEN AIKLANMALARININ YANINDA, TR0GONOMETR0 GEREKT0RMEYECEK OK DAHA BAS0T B0R 0ZAHININ OLAB0LECE0NE 0NANIYORDUM.ALI^MALARIM SONUNDA 2000 YILDIR GZDEN KAAN VE 0SPAT BLMNDE ANLATILAN OKGENLER0N KENARLARI ARASINDAK0 ORANLARI FARKETT0M. SONULARI = 3.14159265358...n OLAN NCEK0 TM NERME VE 0SPATLARIN HEPS0 DORUDUR, BAS0T B0R GSTERMEYLE MATEMAT0K DNYASINA KAZANDIRDIIM BU TEOREM G0B0 KOLAY ANLA^ILAN B0R YOL VARKEN, GEM0^TEN BU GNE KADAR MATEMAT0K0LER0N ONLARCA ZOR YOLDAN,HALA SAYISINI BULMA VE 0SPATLAMA ABALARINI HAYRETLE 0ZLEMEKTEY0M, NERD00M YEN0 FAVOR0 TEOREM0N TRK0YEDE VE DNYADA 8 YILLIK RET0M0N SON DNEM0NDE DERSLERE KONMASI 00N M0LL0 E0T0M BAKANLIININ YETERL0 0LG0Y0 GSTERMES0N0 ARZU ETMEKTEY0M. TANER YNDER n  3 !" 6    P     , *`#     /<Z ( S *A kok85Pl ( <A ?kok8@[V ( S &ApiPV ( S &Api&pV  ( S &ApijsV  ( S &ApiPV  ( S &Api \ ( S ,A taner;  ( B P KENAN EVREN SANAY0 S0TES0 1299 ADA NO: 39 MAN0SA / TRK0YE TEL : 0 ( 236 ) 233 33 33 $XW*  ( Zv @8,$D  0 >  ( Ztv @PP,$D  0 B "  ( Z4v @ ~P,$D 0 \Z GEM0^    #( c ZA 43dflagsdotcom_turke_2faws0`3  $(  `tv2 @BP,$D  0 >  %(  fvd @8 &(  fvd @PP '( c ZA 43dflagsdotcom_turke_2faws0<3 (( c ZA 43dflagsdotcom_turke_2faws``c  )(  `v2 @BP,$D  0 >  *(  fvd @8 +( # lvd @0P >  ,( c ZA 43dflagsdotcom_turke_2faws`<c i( # l`v @?" ?  M< >  j( N?" A  k( N?"}A`  l( N?"]A@ m( N?"=A  n( N?"A o( N?"A p( N?"A q( N?"A r( N?"A s( N?"A t( N?"A u( N?"A v( N?"A w( N?"Ac x( N?"`AC y( N?"@A# z( N?" A {( N?" A  |(  `v @?"?` B ( s *޽h ? ̙33y___PPT10Y+D=' = @B +<  c[;™`ӎ(  x  B?"8  ™ nT  # `  0A?, `  0A?,p `  0A?p `  0A? `   0A?< `   0A?< `   0A? `   0A? `   0A? L `  0A?L `  0A? `  0A?  `  0A?\ `  0A?\ `  0A? `  0A?( `  0A?(l `  0A?l `  0A?,@`  0A?,p@`  0A?p@`  0A?@`  0A?<@`  0A?<@`  0A?@`  0A? @`  0A? L @`   0A?L  @` ! 0A?  @` " 0A? @` # 0A?\@` $ 0A?\@` % 0A?@` & 0A?(@` ' 0A?(l@` ( 0A?l@` ) 0A?:,`` * 0A?,:p`` + 0A?p:`` , 0A?:`` - 0A?:<`` . 0A?<:`` / 0A?:`` 0 0A?: `` 1 0A? :L `` 2 0A?L : `` 3 0A? : `` 4 0A? :`` 5 0A?:\`` 6 0A?\:`` 7 0A?:`` 8 0A?:(`` 9 0A?(:l`` : 0A?l:`` ; 0A?Z,` < 0A?,Zp` = 0A?pZ` > 0A?Z` ? 0A?Z<` @ 0A?<Z` A 0A?Z` B 0A?Z ` C 0A? ZL ` D 0A?L Z ` E 0A? Z ` F 0A? Z` G 0A?Z\` H 0A?\Z` I 0A?Z` J 0A?Z(` K 0A?(Zl` L 0A?lZ` M 0A?z,` N 0A?,zp` O 0A?pz` P 0A?z` Q 0A?z<` R 0A?<z` S 0A?z` T 0A?z ` U 0A? zL ` V 0A?L z ` W 0A? z ` X 0A? z` Y 0A?z\` Z 0A?\z` [ 0A?z` \ 0A?z(` ] 0A?(zl` ^ 0A?lz` _ 0A?,` ` 0A?,p` a 0A?p` b 0A?` c 0A?<` d 0A?<` e 0A?` f 0A? ` g 0A? L ` h 0A?L  ` i 0A?  ` j 0A? ` k 0A?\` l 0A?\` m 0A?` n 0A?(` o 0A?(l` p 0A?l` q 0A?,` r 0A?,p` s 0A?p` t 0A?` u 0A?<` v 0A?<` w 0A?` x 0A? ` y 0A? L ` z 0A?L  ` { 0A?  ` | 0A? ` } 0A?\` ~ 0A?\`  0A?`  0A?(`  0A?(l`  0A?l`  0A?, `  0A?,p `  0A?p `  0A? `  0A?< `  0A?< `  0A? `  0A? `  0A? L `  0A?L  `  0A?  `  0A?  `  0A?\ `  0A?\ `  0A? `  0A?( `  0A?(l `  0A?l `  0A? ,& `  0A?, p& `  0A?p & `  0A? & `  0A? <& `  0A?< & `  0A? & `  0A?  & `  0A? L & `  0A?L & `  0A? & `  0A? & `  0A? \& `  0A?\ & `  0A? & `  0A? (& `  0A?( l& `  0A?l & `  0A? ,F `  0A?, pF `  0A?p F `  0A? F `  0A? <F `  0A?< F `  0A? F `  0A?  F `  0A? L F `  0A?L F `  0A? F `  0A? F `  0A? \F `  0A?\ F `  0A? F `  0A? (F `  0A?( lF `  0A?l F `  0A?@ ,f `  0A?,@ pf `  0A?p@ f `  0A?@ f `  0A?@ <f `  0A?<@ f `  0A?@ f `  0A?@  f `  0A? @ L f ` ˜ 0A?L @ f ` Ø 0A? @ f ` Ę 0A? @ f ` Ř 0A?@ \f ` Ƙ 0A?\@ f ` ǘ 0A?@ f ` Ș 0A?@ (f ` ɘ 0A?(@ lf ` ʘ 0A?l@ f ` ˘ 0A?` , ` ̘ 0A?,` p ` ͘ 0A?p`  ` Θ 0A?`  ` Ϙ 0A?` < ` И 0A?<`  ` ј 0A?`  ` Ҙ 0A?`  ` Ә 0A? ` L ` Ԙ 0A?L ` ` ՘ 0A? ` ` ֘ 0A? `  ` ט 0A?` \ ` ؘ 0A?\`  ` ٘ 0A?`  ` ژ 0A?` ( ` ۘ 0A?(` l ` ܘ 0A?l`  ` ݘ 0A? ,` ޘ 0A?, p` ߘ 0A?p `  0A? `  0A? <`  0A?< `  0A? `  0A?  `  0A? L `  0A?L `  0A? `  0A? `  0A? \`  0A?\ `  0A? `  0A? (`  0A?( l`  0A?l `  0A?,`  0A?,p`  0A?p`  0A?`  0A?<`  0A?<`  0A?`  0A? `  0A? L `  0A?L  `  0A?  `  0A? `  0A?\`  0A?\`  0A?`  0A?(`  0A?(l`  0A?l`  0A?,`  0A?,p`  0A?p`  0A?`  0A?<`  0A?<`  0A?`  0A? `   0A? L `   0A?L  `   0A?  `   0A? `   0A?\`  0A?\`  0A?`  0A?(`  0A?(l`  0A?l|  6$;60 Matematik blmmz taraf1ndan deerlendirilmi_ ve web sitemizde yay1mlanmas1 uygun bulunmu_tur. Prof. Dr. Necdet B0LD0K Manisa Celal Bayar niversitesi Matematik Blm Ba_kan1 Telefon : ( 0 236 ) 241 21 53 - 188 MAN0SA e-mail : necdet.bildik@bayar.edu.tr L ZU 0  BD@  ]ORDER   B|D  b FIRST EDGE  l  <   BhI  a NEXT EDGE    B`N@ n FOUND RATIO(  Z0Sv @PP,$ 0 > &  Z@Wv @ ~P,$D 0 `ONAY *   Z[v @8$,$D  0 >  ! c ZA 43dflagsdotcom_turke_2faws0`3  "  `av2 @BP,$D  0 >  #  fvd @8 $  fvd @PP % c ZA 43dflagsdotcom_turke_2faws0<3  &  `ev2 @xi,$D 0 >  '  fvd @8,$D 0 ( c ZA 43dflagsdotcom_turke_2faws``c ) # lDjvd @0P > l 2 <   c ZA 43dflagsdotcom_turke_2faws`<c  3 rmv @?" ?  M< >   T?" A   T?"}A`   T?"]A@  T?"=A   T?"A  T?"A  T?"A  T?"A  T?"A  T?"A  T?"A  T?"A  T?"A  T?"Ac  T?"`AC  T?"@A#  T?" A  T?" A    fv @?" ?   c .A cbu! A  U 0H  0޽h ? ̙33y___PPT10Y+D=' = @B +!   !!/H0  (  HM 8  H/ H <ud U 4 DIFFERENT WAYS OF CALCULATING WHICH'S PROOF IS BASED ON PYTHAGORAS THEOREM &VT Tb H s 2AL @pi `  H <{j k 1- BEING THE NUMBER OF EDGES, TO CALCULATE FROM STARTING WITH A SQUARE INSCRIBED IN A CIRCLE AND CONTINUING BY FORMING 3, 6, 12, 24...n GONS AND DIVING THE SUM OF THE EDGES OF THE LAST POLYGON, BY DIAMETER (2r = 1) @ d  H s 4AL @n+12b  H s 2AL @pi`0V  H S &Api0  @`  H S 0Aformul1(0   H <D  x 2- 3 x BEING THE NUMBER OF EDGES, TO CALCULATE FROM STARTING WITH A EQUILATERAL TRIANGLE INSCRIBED IN A CIRCLE AND CONTINUING BY FORMING 3,6,12,...n GONS AND DIVING THE SUM OF THE EDGES OF THE LAST POLYGON, BY DIAMETER (2r = 1) @ d H s 4AL @n+1VP`b H s 2AL @pim:` H S 0A formul2  g H <l   g 3-TO FIND USING THE AREA OF A SQUARE INSCRIBED IN A CIRCLE AND DIVIDING IT BY r 2 ( 2r = 1 ) ThW g` H S 0Aformul3P   H ZLv @8,$D  0 >  "H Zؙv @P(P,$ 0 >  %H Zv @X0P,$D 0 YSERIES TANER YNDER  'H c TA.3dflagsdotcom_uk_2fawm``e  (H  `|v2 @BP,$D  0 >   )H  `Lv2 @BP,$D  0 >  *H  fvd @8 +H  fvd @8 .H  fvd @PP /H c TA.3dflagsdotcom_uk_2fawm0?5 0H # l0vd @0P >  2H c TA.3dflagsdotcom_uk_2fawm`?e qH # l@v @?" ?  M< >  rH N?" A  sH N?"}A`  tH N?"]A@ uH N?"=A  vH N?"A wH N?"A xH N?"A yH N?"A zH N?"A {H N?"A |H N?"A }H N?"A ~H N?"A H N?"Ac H N?"`AC H N?"@A# H N?" A H N?" A  H  `v @?" ? H H 0޽h ? ̙33y___PPT10Y+D=' = @B +  *nL@5(  L} L < 4-TO FIND USING AREA OF AN EQUILATERAL TRIANGLE INSCRIBED IN A CIRCLE BY CALCULATING THE AREA IF THE CIRCLE AND DIVING IT BY r 2 . (2r = 1)H X L S 0Aformul4%p0Y L <B s 5-AFTER ALL THESE CALCULATIONS, IF WE WISH TO WRITE A GENERAL FORMULA FOR WE FIND THE FOLLOWING EQUALITY. BtpsZ L s 2AL @pigX  L S 0Aformul5   L <   4IF THERE ARE ANY SYNTATIC ERRORS, PLEASE DO REPORT. 0534  L <) e c NOTE : IN THE ABOVE FORMULAS, BY , WE MEAN THAT IF THE OPERATION IS TO BE CUT AT A CERTAIN STEP, THE RESULT WILL BE MULTIPLIED WITH THAT COEFFICIENT ( THE NUMBER OF EDGES OF THE LAST POLYGON ) AND THE OTHER PREVIOUS COEFFICIENTS ARE TO BE DISCARDED$T  L c ,A?x4[  ^ Z L s 2AL @pi S L Zv @8,$D  0 >  L  `v2 @BP,$D  0 >  L  fvd @8 $L  `|v2 @BP,$D  0 >  %L  fvd @8   =L ZLv @X0P,$ 0 YSERIES TANER YNDER | >L c TA.3dflagsdotcom_uk_2fawm``e ?L  fvd @PP| @L c TA.3dflagsdotcom_uk_2fawm?05 AL # lvd @0P > | BL c TA.3dflagsdotcom_uk_2fawm?`e ZL 3 r|v @?"?  M< >  [L T?"A  \L T?"A}`  ]L T?"A]@ ^L T?"A=  _L T?"A `L T?"A aL T?"A bL T?"A cL T?"A dL T?"A eL T?"A fL T?"A gL T?"A hL T?"Ac iL T?"A`C jL T?"A@# kL T?"A  lL T?"A  mL  fv @?"?, H L 0޽h ? ̙33y___PPT10Y+D=' = @B +  'iT j(  T28  iTf T c 6Api_anim@  T S &Api@ J ,$ 0 T <4Pp  PLEASE FOLLOW THE RED ANIMATION J!3    T <^  THE ABOVE FORMULAS ARE ALL MY CREATIONS AND COMPLETE OR PARTIAL REPRODUCTION, WITH PERMISSION OF THE AUTHOR IS FREE.THIS FORMULAS CAN USE IN SCIENTIFIC EDUCATIONS WITHOUT PERMISSION OF AUTHOR. THIS WORK IS PATENTED ( DATE 03/02/1997 NR.4085 MANiSA 2Nd PATENT OFFICE-TURKEY ) ,. T <h   gIf you continue you will reach , 3 T Zv @8,$D  0 >   T  `@v2 @BP,$D  0 >  T  fvd @8   T  `v2 @BP,$D  0 >  !T  fvd @8 8T Z4v @P(P,$ 0 >  9T Zv @X0P,$ 0 YSERIES TANER YNDER  :T c TA.3dflagsdotcom_uk_2fawm``e ;T  fvd @PP   >T c TA.3dflagsdotcom_uk_2fawm`?e UT # lLv @?" ?  M< >  VT N?" A  WT N?"}A`  XT N?"]A@ YT N?"=A  ZT N?"A [T N?"A \T N?"A ]T N?"A ^T N?"A _T N?"A `T N?"A aT N?"A bT N?"A cT N?"Ac dT N?"`AC eT N?"@A# fT N?" A gT N?" A  hT  `v @?"L?#H T 0޽h ? ̙33y___PPT10Y+D=' = @B + F  1E)EG Y D(  XaD8   Y X </    and if the suggested series if correct, then we should obtain the same lengths of edges that we computed by Pythagoras theorem.>>> 66   Y X <K   Let X1, X2, X3, X4 be the lengths of edges of a 4, 8, 16, 32-gon, respectively, which are inscribed in a circle of 2r =X=1 diameter. Then using Pythagoras' theorem we find the values X1=0.707106781..n, X2=0.382643439..n, X3=0.0195090314..n, X4=0.098017136..n of course,there sould be ratio between these lengths of edges but what kind of ratio can this be? starting from X/X1, if we divide succesively the lengths of the edges in the following manner, X / X1 / X2 / X3 / X4, we get the ratios. d  d^3     Q   X B  X Zv @P(P,$ 0 >  X B|` hSERIES" X B`3 X FIRST EDGE   X B ` W NEXT EDGE   X BC`@ _ DECIMAL VALUE  X Z@v @ ~P,$D 0 I.  X HL ,$D 0 aPROOF OF THE SERIES FOR B " X H  L ,$D 0 n USING PYTHAGORAS' THEOREM$B 0 X <  @ Z,$D  0  @ b((B(B  X  ` v2 @BP,$D  0 >  X  fvd @8 X  fvd @PP X c TA.3dflagsdotcom_uk_2fawm0?5,$ 0 X c TA.3dflagsdotcom_uk_2fawm``e,$ 0  X  ` v2 @BP,$D  0 >  X  fvd @8 X # lL vd @0P >  X c TA.3dflagsdotcom_uk_2fawm`?e,$ 0  Y 3 r  v @?" ?  M< >   Y T?" A   Y T?"}A`  Y T?"]A@ Y T?"=A  Y T?"A Y T?"A Y T?"A Y T?"A Y T?"A Y T?"A Y T?"A Y T?"A Y T?"A Y T?"Ac Y T?"`AC Y T?"@A# Y T?" A Y T?" A  Y  fv @?"l?CH X 0޽h ? ̙33y___PPT10Y+D=' = @B +S  RRd\`[R(  \#R8  \ \ Bt$ z hSERIES" \ B)  _ DECIMAL VALUE l \ < \ BH.   X FIRST EDGE   \ B2 3 0 _ DECIMAL VALUE   \ B7  W NEXT EDGE  l #\ <4  : (\ <C   7As a consclusion, because the corresponding values computed are the same, the correctness of this series for is proven. Below, we can see that using the series, the sum of lengths of edges 4 8 16 32 of the 2n+1 gon divided by the diameter of the circle approaches to > 8      V )\ S &Api @  *\ BL;   S0.707106781...n(33 +\ BT   =X  ,\ BY    s 4 GON ( -\ B]    s= 2.828427124...n  .\ Bb   S0.382683439...n( /\ Bf   =X  0\ Bj  `  s 8 GON ( 1\ Bo    s= 3.061467512...n  2\ Bt   S0.195090314...n( 3\ BXy   =X  4\ B}  ` f16 GON  5\ BT   s= 3.121225024...n  6\ B$ @ ` S0.098017136...n( 7\ B  @ ` =X  8\ B`  @0` f32 GON  9\ B @` s= 3.136548352...n  :\ B  `  TSAMPLE ;\ B    >=> V C\ S &Api} p  L\ Z, v @8,$D  0 >  Z\ B   ]ORDER  [\ B   b FIRST EDGE  l \\ <  ]\ B   a NEXT EDGE   ^\ B @ n FOUND RATIO( _\ Zl v @P(P,$ 0 >  b\ Z v @ ~P,$ 0 >  {\ Bȿ 8*b 3 |\ B s8pb  = 1.414213562...n4( }\ BD 8 b >=>  ~\ B 8 b h2 r = 1(  \ B 3 80b 1.414213562...n =0  \ B 8b  0.707106781& n (330  \ BP sbp  = 1.847750064...nD( \ Bl b  >=>  \ B, b  w0.707106781...n(33  \ B 3 b0 1.847750064...n =0  \ BD b S0.382683439...n( \ B4 sp  = 1.961570562...nD( \ B   >=>  \ B 5  w0.382683439...n(  \ B 3 0 1.961570562...n =0  \ B  S0.195090314...n( \ B@ sp  = 1.990369453...n4( \ B   >=>  \ B(   w0.195090314...n(  \ B" 3 0 1.990369453...n =0  \ B(  S0.098017136...n(Z \ S *Aff12r` \ S 0A ff22+ 22@.f \ S 6Aff32+ 2+ 2vrl \ S <Aff42+ 2+ 2+ 2 \ Bh. 4 B b P / \ BT1 4 b  P / \ B5 4   P / \ B2 4   P /  \  `|< v2 @BP,$D  0 >  \  fvd @8 \ # l$A vd @0P >  \ c TA.3dflagsdotcom_uk_2fawm`?e,$ 0  \  `,D v2 @BP,$D  0 >  \  fvd @8 \  fvd @PP \ c TA.3dflagsdotcom_uk_2fawm0?5,$ 0l \ <0 Pl \ <0 P \ HH L ,$D 0 aPROOF OF THE SERIES FOR B " \ HL  L ,$D 0 n USING PYTHAGORAS' THEOREM$B 0 \ <dR  @ Z,$D  0  @ b((B(B \ # lPX v @?" ?  M< >  \ N?" A  \ N?"}A`  \ N?"]A@ \ N?"=A  \ N?"A \ N?"A \ N?"A \ N?"A \ N?"A \ N?"A \ N?"A \ N?"A \ N?"A \ N?"Ac \ N?"`AC \ N?"@A# \ N?" A \ N?" A  \  `v @?"?cH \ 0޽h ? ̙33y___PPT10Y+D=' = @B +t5  44p4wl` 4(  l38  wl l Za v @8,$D  0 >   l Bc   ]ORDER   l Bg   b FIRST EDGE  l l <  l Bl   a NEXT EDGE   l Bq @ n FOUND RATIO( l Zdv v @P(P,$ 0 >  l Ztz v @ ~P,$D 0 I. N l <~ `Z,$D 0 " THE H0STORY OF LB(@B(b(B( l < `  e  2. However, the formulas and series suggested above are easily understood by 7th grade students who only know and four mathematical operation. There is not any other calculation as simple as those that we have suggested. BNbjb B  b BbB/b B @$b$(B(dW  /BX l <ĕ `   f 3. Archimedes found the third decimal place of , Ulug Bey s astronomer Al-Kasi carried it to 16 decimal places in 1479, in 1579 F. Viete figured out 9 decimals of and wrote an infinite series equals 2 / , Adrian Van Roomen found 15 decimal place of in 1593, and Ludolph Van Coulen had spent most of his life when finally in 1610 he carried to 35 decimal places.  b B/b B (bBLbB b $$(b( ,B,0b0448b88b8  *l  fvd @8 +l  fvd @PP ,l c TA.3dflagsdotcom_uk_2fawm0?5,$ 0  -l  `< v2 @BP,$D  0 >  .l  fvd @8 /l # l vd @0P >  0l c TA.3dflagsdotcom_uk_2fawm`?e,$ 0 cl # l$ v @?" ?  M< >  dl N?" A  el N?"}A`  fl N?"]A@ gl N?"=A  hl N?"A il N?"A jl N?"A kl N?"A ll N?"A ml N?"A nl N?"A ol N?"A pl N?"A ql N?"Ac rl N?"`AC sl N?"@A# tl N?" A ul N?" A  vl  `v @?"?H l 0޽h ? ̙33y___PPT10Y+D=' = @B +  `T ?(  8   nT  # `  0A?, `  0A?,p `  0A?p `  0A? `   0A?< `   0A?< `   0A? `   0A? `   0A? L `  0A?L `  0A? `  0A?  `  0A?\ `  0A?\ `  0A? `  0A?( `  0A?(l `  0A?l `  0A?,@`  0A?,p@`  0A?p@`  0A?@`  0A?<@`  0A?<@`  0A?@`  0A? @`  0A? 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IT IS JUST AWORK OF FANTASY WHICH HAS NOT BEEN FOUND IN THE LITERATURE.(2  A H4 D p  =THE CALCULATIONS ABOVE GIVE US THE FOLLOWING RESULTING SERIES>> > C B8   ]ORDER  D Bp   b FIRST EDGE  l E <  F B   a NEXT EDGE   G B @ n FOUND RATIO( H Z v @PP,$ 0 > J I Z v @ ~P,$D 0 :OTHER WORKS OF TANER YNDER S  L < 0 < A 8 8 8 8 = ( - ) + ( -------- ) + ( ------------- ) + ( ---------------- ) + ... n 3 (1x32)+3 [(1+2)x32]+3 [(1+2+3)x32]+34 23 V M S &Apif V N S &Apif_V O S &ApiP  P < `   B 8 8 8 8 8 8 = --------- + --------- + --------- + --------- + --------- + ---------- + ... n (2x1)-1 (2x3)-1 (2x5)-1 (2x7)-1 (2x9)-1 (2x11)-1L! 23 U !D Q HT c0 | 2 2 2 2 2 2} 2}  }V R S &ApifP   S < 0 `  y C 8 8 8 8 8 8 = ------ + ------ + -------- + -------- + ------- + ---------- + ... n (2)-1 (6)-1 (10)-1 (14)-1 (18)-1 (22)-1L 23 L . T H  4 @  f2 2 2 2 2 2g 2g  g U <  P 0 l D 8 8 8 8 8 8 8 8 8 = ----- + ----- + ----- + ----- + ----- + ----- + ----- + ----- + ------ + ... n 3 35 99 195 323 483 675 899 11558 233  W Z v @8$,$D  0 >  Y  fvd @8 Z  fvd @8 [ c ZA 43dflagsdotcom_turke_2faws0`3  \  ` v2 @BP,$D  0 >  ]  fvd @8 ^  fvd @PP ` c ZA 43dflagsdotcom_turke_2faws``c  a  `xv2 @BP,$D  0 >  b  fvd @8 c # l<vd @0P >  e c TA.3dflagsdotcom_uk_2fawm`?e,$ 0  c TA.3dflagsdotcom_uk_2fawm0?5,$ 0  # lv @?" ?  M< >   N?" A   N?"}A`   N?"]A@  N?"=A   N?"A  N?"A  N?"A  N?"A  N?"A  N?"A  N?"A  N?"A  N?"A  N?"Ac  N?"`AC  N?"@A#  N?" A  N?" A    `v @?"?H  0޽h ? ̙33y___PPT10Y+D=' = @B +4  44P4{p@3(  pW38  xp p Zv @8,$D  0 >  p B  ]ORDER  p B@  b FIRST EDGE  l p <  p BT   a NEXT EDGE   p B%@ n FOUND RATIO( p Z*v @PP,$ 0 > ( p Z+v @ ~P,$D 0 bABAUT THE AUTHOR\ p S ,A taner;  p B2  ` |KENAN EVREN SANAY0 S0TES0 1299 ADA NO: 39 MAN0SA / TURKEY TEL & FAX : 00 90 236 233 33 33,[Zx*  * p <9   I was born in 1947, in Manisa, Turkey. I couldn t go to school after the 8th grade. I am a tire dealer now. Since I am interested in numbers I became an amateur mathematician. When I sawx x 10 : 9 and its square I was fascinated by the nature of the numbers. I had thought x 10 : 9 = 3.14269680529..n could be it self but after I few years when I learnt advanced math and irrational and transcendental numbers I understood why this can t be However I believed that could be proven by some other easier ways without trigonometry. I realized something that had been ignored or missed for 2000 years: the ratio between the polygons edge-lengths. As shown before, the calculations are all correct. While there is such an easy way to understand and prove , why do mathematicians still try to prove or calculate through difficult methods? I would like to teach to children and young mathematicians by my theory in school books. 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