The Elementary Arithmetic Operators of Continued Fraction
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1 Americ-Eursi Jourl of Scietific Reserch 0 (5: 5-63, 05 ISSN IDOSI Pulictios, 05 DOI: 0.589/idosi.ejsr The Elemetry Arithmetic Opertors of Cotiued Frctio S. Mugssi d F. Mistiri Deprtmet of Mthemtics, Uiversity of Beghzi, Beghzi, Liy Astrct: The fiite d ifiite simple cotiued frctios re cosidered. The dditio, sutrctio d equlity of two simple cotiued frctios re preseted. The pplictios of cotiued frctios re lso studied. Key words: Simple cotiued frctio Negtive rtiol umer Multiplictive iverse INTRODUCTION is clled cotiued frctio. The vlues 0,, d 0,,... c e either rel or complex d their umers The cotiued frctios hve ee studied i c e either fiite or ifiite. mthemticl (Diophtie d Pell's equtios d physicl (ger rtio [-4]. Our simple cotiued frctios Defiitio : A fiite simple cotiued frctio is tht we study re remiiscet of vrious other cotiued cotiued frctio (defiitio i which for ll, frctios prolems. However they re very differet from tht is: ours. Oe is lytic of cotiued frctios [3-8], which hs ee discussed for the rel d complex vlues. 0 + Aother study is the cotiued frctios for the iteger + vlues [, 4, 6]. + Curretly, cotiued frctios hve my prcticl + uses i mthemtics. Foristce, we c express y umer, rtiol or irrtiol, s fiite or where is positive iteger for ll, 0 c e y ifiitecotiued frctio expressio. We c lso solve iteger umer. The ove frctio is sometime y Diophtie cogruece thtis y equivlece of represeted y [ ;,,..., ] for fiite simple cotiued the form x (mod m. I other words, i most rel- frctio d [ 00;, ] for ifiite simple cotiued world pplictios of mthemtics, cotiued frctios frctio. I this pper we will use the symol (S.C.F. for re rrely themost prcticl wy to solve give set of the simple cotiued frctio. prolems s deciml pproximtios re much more useful (d computers c work with decimls t much is fiite complex vlued cotiued + fster rte. However, some iterestig oservtios c + still e mde usig cotiued frctios. Nmely, i this 3i pper, we will e explorig how cotiued frctios c frctio. e used to dd d sutrct the umers d c. We strt with some defiitios d theorems tht is fiite S.C.F [3,6,7,9,7] we used to defied the dditio d the sutrctio of 6 + two simple cotiued frctios Defiitio : A expressio of the form. 7 0 is ifiite S.C.F [4;,3,3, ] Correspodig Author: S. Mugssi, Deprtmet of Mthemtics, Uiversity of Beghzi, Beghzi, Liy. 5
2 Am-Eurs. J. Sci. Res., 0 (5: 5-63, 05 Theorem : A umer is rtiol if d oly if it c e expressed s fiite S.C.F. [4]. For exmple [;,,3,] Remrk : 0 +, d the we use the sme techiques s i theorem for. To expd egtive rtiol umer (,, > 0 ito S.C.F. we tke the gretest iteger umer for the first term of S.C.F. tht is, To expd rtiol umer, ( > > 0 ito S.C.F. we write 0 where 0 < 0 +. We write, 0 + d the we use the sme techiques s i theorem to get the remiig terms for. Tht is, if [ ;,..., ] the [ 0 ;,..., ]. Exmple : ( [0,, 3] ( 5 + [-6,,, ] Lemm : Ay fiite S.C.F. [ 0;,..., ] c e lso represeted y [ 0;,...,,] [5]. Defiitio 3: The S.C.F. [ ;,..., ] c e defied y 0 K ( K ( [ ;,..., ], or [ 0;,..., ] K( K( where, 5
3 K ( K ( K ( K ( 0 0 K ( + K ( K ( + + K ( K ( K ( + K ( K ( K ( + K ( i 0 i i 0 i 0 i i i I geerl Am-Eurs. J. Sci. Res., 0 (5: 5-63, 05 i Ki( j i+ ( j Ki ( j + Ki ( j, i,,...,, j 0,,..., K ( 0 K ( i j 0 j Lemm : If C j 0 i, [c 0; c,...,c j, c j, c j+,...c ] for some j with 0 < j <, the we c replce cj y cj c j + c j+ d delete c j, c j+ from the simple cotiued frctio expsio, without chgig the vlue of S.C.F. Lemm 3: If c j c j+ 0 i [ c0; c,..., cj, cj, cj+, cj+,..., c ], for some j with 0 < j < the we c delete c j, c j+ from the simple cotiued frctio expsio, without chgig the vlue of S.C.F.. Defiitio 4: Let [ ;,..., ], [ ;,..., ] e two simple cotiued frctios, the, i i [ ;,..., ] [ ;,..., ] if for i 0,,,...,. [ ;,...,, ] [ ;,...,,,]. 0 0 [ ;...,,,0,,,..., ] [ ;...,, +,,..., ]. 0 j j j+ j+ 0 j j j+ j+ Theorem : Let d [ ;,..., ] e two S.C.F. with i 0 for ll i,,...m d, j 0 for ll j,,... d [ ;,..., ] [ ;,..., ] the [ 0;,..., m] 0 0 m 0 If, let >, the [ ;,..., ] > [ ;,..., ]. 0 m 0 If i ifor ll i0,,,...k, k< mi (m, let >, with k + mi (m, k+ k+ the [ ;,..., ] > [ ;,..., ] if k is odd 0 m 0 d [ ;,..., ] < [ ;,..., ] if k is eve. 0 m 0 If i ifor ll i 0,,,...k. k mi (m, letk m, the [ ;,..., ] > [ ;,..., ] if k is odd 0 m 0 0 m 0 d [ ;,..., ] < [ ;,..., ] if k is eve. 53
4 Am-Eurs. J. Sci. Res., 0 (5: 5-63, 05 For exmples [3;, ] > [;, 4], sice 0 > 0. [3;,, 3]> [3;7], sice 0 0 d > i. [;, 3] < [;, 3, 4], sice i ifor ll i 0,, d k m (eve. [;, 3, 4] > [;, 3, 4, 5], sice i ifor ll i 0,,, 3 d k m 3 (odd. Defiitio 5: Let [ ;,..., ] d [ ;,..., ] e two S.C.F., we defie dditio y 0 m 0 m ( If m the [ ;,..., ] + [ ;,..., ] [ c ; c,..., c ] (5 where, c K( 0 + K( 0 K( K( c + K( + K( ( + + ( + c ( + ( + c[ ( + + ( + ] K( K( + K( K( K( K( c[ K( K( + K( K( ] K3( K3( K0( c [ K( K3( + K( K3( ] K( c c3 [ K( K3( + K( K3( ] K( c K3( K3( K( c Ki( Ki( Ki 3( c [ Ki ( Ki( + Ki ( Ki( ] Ki ( c, i odd [ Ki ( Ki( + Ki ( Ki( ] Ki ( c Ki( Ki( Ki ( c ci [ Ki ( Ki( + Ki ( Ki( ] Ki ( c Ki( Ki( Ki 3( c, i eve Ki( Ki( Ki ( c [ Ki ( Ki( + Ki ( Ki( ] Ki ( c for i,,. The lst term c of the resultig S.C.F. is to e expded gi s S.C.F. if ecessry d ot to e treted s the gretest iteger umer s the precedig terms hve ee treted. ( If m the Suppose tht m< the [ ;,..., ] + [ ;,...,,,..., ] [ c ; c,..., c, c,..., c ] 0 m 0 m m+ 0 m m+ (5 where c j c jfor j 0,,,, m, d c s we did for cse m, while j cj cj, j m d 54
5 c j, j m Km( Kj( K j 3( c [ Km ( Kj( + Kj ( Km( ] Kj ( c, j odd [ Km ( Kj( + Kj ( Km( ] K j ( c Km( Kj( Kj ( c [ Km ( Kj( + K j ( Km( ] K j ( c Km( Kj( Kj 3( c, j eve Km( Kj( Kj ( c [ Km ( K j( + K j ( Km( ] K j ( c forj m+, m +,,. c c ( + + ( + c ( + ( + c[ ( + + ( + ] 5(3 + + (5 + (3 + (5 + [5(3 + + (5 + ] 5(7 + ( [;,5,, ], (7( [35 + ] Am-Eurs. J. Sci. Res., 0 (5: 5-63, 05 Also, the lst term of the resultig S.C.F. is to e treted s metioed erlier. Remrk : If the umer of the S.C.F. term s for [ 0;,..., ] d [ 0;,..., ] re, the it is ot ecessry for the umer of terms of the resultig S.C.F. [ 0;,..., ] + [ 0;,..., ] to e. If [ 0] d [ 0;,..., ] re two S.C.F. the [ 0] + [ 0;,..., ] [c0; 0,..., ] where c 0 is give i equtio (5. Exmple : Fid [;,5] + [4;3,]. Solutio: Let [;, 5] [ 0;, ] d [4;3, ] [0, ], we get m. From equtio (5 we hve [;,5] + [4;3,] [ 0;, ] + [ 0;, [c 0;c,c ], where c is the lst term d the [;,5] + [4;3,] [5;,,,5,,] Exmple 3: Fid [7;7] + [8;,,3]. Solutio: Let [7;7] [ ; ] d [8;,, 3] [,,, ], we get m, 3 d m < From equtio (5 we hve, 55
6 [7;7] + [8;,,3] [ ; ] + [ ;,, ] where c is the lst term d 3 [ c ; c, c, c ] 0 3 c0 c c c [ K0( K( + K( K( ] K0( c K( K( K ( c c c, K( K( K0( c [ K0( K( + K( K( ] K( c [ ( + + ] ( + 0 ( + [( + + ] c ( K( K3( K0( c [ K0( K3( + K( K( ] K( c c3 c3, [ K ( K ( + K ( K ( ] K ( c K ( K ( K ( c ( [( ( 3+ ] c [( ( + ]( cc + ( + + c ( [( (3 + ]0 [( ( 3 + ] 7 ( ( [( ( 3 + ] [;,,] Am-Eurs. J. Sci. Res., 0 (5: 5-63, 05 Therefore [7;7] + [8;,,3] [35;0, 0,,, ] [35;,, ]. Exmple 4: Fid [4] + [;, ]. Solutio: Let [4] [ 0] d [;, ] [ 0;, ] we get m 0,, m<. From remrk (ii we hve, [4] + [;, ] [ 0] + [ 0,, ] [c0, ] where c Therefore [4] + [,, ] [5,, ]. Corollry : If c is o-zero iteger the Exmple 5: Fid [;, ]. 3 c[ 0,,, 3,...] [ c0,, c,,...] which is ot ecessry S.C.F.. c c Solutio: From corollry we hve [;,] [ (;, (] [ ;, 4]. 56
7 Am-Eurs. J. Sci. Res., 0 (5: 5-63, 05 Remrk 3: Note tht the multiplictio of S.C.F. y o-zero iteger see i exmple 5. does ot ecessrily led to S.C.F. s we hve Defiitio 6: Let [ ;,..., ] e S.C.F., the we c defie dditive iverses y 0 [ ;,..., ] [ ; ] 0 0 where, d K( K ( K ( [ d ; d,..., d ] therefore, l (to e treted s S.C.F. [ ;,..., ] [ ; ] [ d ; d, d,..., d ] l Defiitio 7: If [ 0;,..., m] d [ 0;,..., ] re two S.C.F. the we defie Sutrctio [ 0;,..., m] -[ 0;,..., ] y the dditio, [ 0;,..., m] + [ d0; d,..., dl] where [ d0; d,..., d l ] give y defiitio 6. Tht is; If ml the [ ;,..., ] + [ d ; d,..., d ] [ c ; c,..., c ] 0 m 0 m 0 m (7 where c,c,...,c s give i equtio5. 0 m If m l, l<m the [ ;,...,,,..., ] + [ d ; d,..., d ] [ c ; c,..., c, c,..., c ] 0 l l+ m 0 l 0 l l+ m (7 where c0, c,..., cm s give i equtio 5. The motivtios of our defiitios d the lytic prove re pulished i [5]. Exmple 6: Fid [;,,3,5,3,,,4,9] + [;,8,4,,,3,5,3,,] d [;,8,4,,,3,5,3,,]. Solutio: To fid [;,,3,5,3,,,4,9] + [;,8,4,,,3,5,3,,]. Let [;,,3,5,3,,,4,9] [ ;,,,,,,,, ], m d [;,8,4,,,3,5,3,,] [ ;,,,,,,,,, ], 0, m. From equtio (5 we hve [ ;,,,,,,,, ] + [ ;,,,,,,,,, ] [ c ; c, c, c, c, c, c, c, c, c, c ]
8 Am-Eurs. J. Sci. Res., 0 (5: 5-63, 05 where c is the lst term d 0 c0 c c c K( K( + K( K( c c K( K( c[ K( K( + K( K( ] (9 + 8(5 58 (9(5 45 K ( ( ( [ ( ( ( ( ] ( c3 c3 3 K3 K0 c K K3 + K K3 K c [ K( K3( + K( K3( ] K( c K3( K3( K( c K3( K3( K( K3( + K( K3( K3( K3( K( c 7( (33(7 + (7(37 (7(37 9 [ K3( K4( + K3( K4( ] K( c K4( K4( K( c c4 c4 K4( K4( K( c [ K3( K4( + K3( K4( ] K3( c [(37 (83 + (47(90] (83(90( 6 3 (83(90(4 [973](3 687 K5( K5( K( c [ K4( K5( + K4( K5( ] K3( c c5 c5 [ K4( K5( + K4( K5( ] K4( c K5( K5( K3( c (87(0(4 [(8(0 + (07(87]( (44869(00 [34440](3 970 [ K5( K6( + K5( K6( ] K4( c K6( K6( K3( c c6 c6 K6( K6( K4( c [ K5( K6( + K5( K6( ] K5( c [(377(395 + (443(55](0 (377(443(3 (670(43 [7580]( K7( K7( K4( c [ K6( K7( + K6( K7( ] K5( c c7 c7 [ K6( K7( + K6( K7( ] K6( c K7( K7( K5( c (04(335(43 [(04(08 + (335(48](33 (36674(09 [430735]( [ K ( ( ( ( ] ( ( ( ( c8 c8 7 K8 + K7 K8 K6 c K8 K8 K5 c K8( K8( K6( c [ K7( K7( + K7( K8( ] K7( c [(454(664 + (7448(857](09 (454(7448(4 (338368(606 [663377]( K9( K9( K6( c [ K8( K9( + K8( K9( ] K7( c c9 c9 [ K7( K9( + K8( K9( ] K8( c K9( K9( K7( c (490(73(85 [(440( (73(73](4 (940860(09 [750]( [ K8( K0( + K9( K9( ] K8( c K9( K0( K7 ( c c0 K9( K0( K8( c [ K8( K0( + K9( K9( ] K9( c [(490( (490(73](09 (490 (4 ( (85 [ ]( [;5] 5 58
9 To check, we hve [;,,3,5,3,,,4,9] + [;8,4,,,3,5,3,,] d K6( [3;3, 3, 3, 3,,, 5] 3 + K ( Let [;8, 4,,, 3, 5, 3,, ] [ ;,,,,,,,,, ], we get: 0, [;,8,4,,,3,5,3,,] [ ; ], 0 0 Am-Eurs. J. Sci. Res., 0 (5: 5-63, 05 therefore, [,,, 3, 5, 3,,, 4, 9] + [;8, 4,,, 3, 5, 3,, ] [;0,, 3, 3, 3, 3,, 0, 0,, 5] [3;3, 3, 3, 3,, 0, 0,, 5] [3;3, 3, 3, 3,,, 5] which is true. To fid [;,8,4,,,3,5,3,,]. From defiitio 6 we hve where, K0( 0 K ( K ( K ( Sice 9 [;,8, 4,,,3,5, 3,,] 0 +, (y defiitio K0( d K ( 490, 0 K( 7979 K ( ( Therefore, K0( K ( K ( [9;4,,, 3, 5, 3,, ]. Therefore [;,8,4,,,3,5,3,,] [-;9, 4,,, 3, 5, 3,, ]. Theorem 3: Let 0 e irrtiol umer d defie the sequece 0,,,... recursively y for k 0,,,.The is the vlue of ifiite S.C.F. [ 0;, ]. k k, k+ k k 59
10 Am-Eurs. J. Sci. Res., 0 (5: 5-63, 05 For exmple 6 [ ;,,,,,...] [;,4,,4...] [;,4] We c use the sme opertios of fiite S.C.F. for ifiite S.C.F.. Exmple 7: Fid [ ;] + [;, ] Solutio: Let [;] [;,,,,...] [ ;,,,,...] d [;,] [;,,,,...] [ ;,,,,...] From equtio (5 we hve [ ;,,,,...] + [ ;,,,,...] [ c ; c, c, c, c,...] where, c c ( + + ( + c ( + ( + c[ ( + + ( + ] (3 + (5 (3(5 6 5 K3( K3( c3 K( K3( + K( K3( K3( K3( K( c (4 5(4 + 3( ( [ K3( K4( + K3( K4( ] K( c K4( K4( K( c c4 K4( K4( K( c [ K3( K4( + K3( K4( ] K3( c [(( + (8(9] (9(( (9((7 [364]( K5( K5( K( c [ K4( K5( + K4( K5( ] K3( c c5 [ K4( K5( + K4( K5( ] K4( c K5( K5( K3( c (70(5(7 [(9(5 + ((70](6 (05( [050](
11 [ K5( K6( + K5( K6( ] K4( c K6( K6( K3( c c6 K6( K6( K4( c [ K5( K6( + K5( K6( ] K5( c [(70(4 + (30(69]( (69(4( (699(7 [7940]( K7( K7( K4( c [ K6( K7( + K6( K7( ] K5( c c7 [ K6( K7( + K6( K7( ] K6( c K7( K7( K5( c (408(56(7 [(69(56 + (4(408](6 (69(7 [848]( [ K7( K8( + K7( K8( ] K6( c K8( K8( K5( c c8 K8( K8( K6( c [ K7( K7( + K7( K8( ] K7( c [(408(53 + ((985](7 (985(53(8 (50705(39 [7744]( Therefore [;] + [;, ] [;0,, 6, 0, 0,, 4, 0, ] [3;6, 0, 0,, 4, 0, ] [3;6,, 4, 0, ] Exmple 8: Fid Solutio: [;] + [;,,, 4] Am-Eurs. J. Sci. Res., 0 (5: 5-63, 05 Let [;] [ ; ] d [ ] [ ;,,,,...] 0 ;,,,4] [;,,, 4,,,, 4, From (5 we hve [ ; ] + [ ;,,,,...] [ c ; c, c, c, c,...] where c0 c c c K( + K( + c K ( 4 K 3( c c 3 [ K3( + K( ] K3( c ( ( (3 6 [ K4( + K3( ] K4( c c 4 K4( K( c [ K4( + K3( ] K3( c [(4 + (9(] ((4 ((4(7 [3](
12 Am-Eurs. J. Sci. Res., 0 (5: 5-63, 05 K5( K( c [ K5( + K4( ] K3( c c5 [ K5( + K4( ] K4( c K5( K3( c ((7(7 [(7 + ((](6 (39(7 (34(8 4 4 [ K6( + K5( ] K4( c K c 6( 6 K3( c K6( K4( c [ K6( + K5( ] K5( c [(3 + (0(](7 ((3(8 (6(39 (7( K7( K4( c [ K7( + K6( ] K5( c c7 [ K7( + K6( ] K6( c K 7( K5( c ((48(39 [(48 + (3(](34 (0(7 (96(8 4 Therefore [;] + [;,,,4] [3;0,, 6,, 4, 0,, ] [4;6,, 6, ]. There re my pplictios of cotiued frctios: comie cotiued frctios with the cocepts of golde rtio d Fiocci umers, Pell equtio d clcultio of fudmetl uits i qudrtic fields, reductio of qudrtic forms d clcultio of clss umers of imgiry qudrtic field.there is plest coectio etwee Cheyshev polyomils, the Pell equtio d cotiued frctios, the ltter two eig uderstood to tke plce i rel qudrtic fuctio fields rther th the clssicl cse of rel qudrtic umer fields [, 5]. The very ice elemetry pplictio of simple cotiued frctios is Gosper's ttig verge prolem which is, if sell plyer's (3-digit rouded ttig verge is 0.334, wht's the smllest umer of t-ts tht plyer could hve? (Bttig verge is computed s umer of hits. The solutio proceeds y otig tht rouded verge of ( t ts correspods to ctul umer i the rge (3335,3345, fidig the cotiued frctios for these vlues yields 667 d 669. This implies tht the 'simplest' umer withi the rge [0;,,666] [0;,,94,,,3] is 69 [7]. [0,,,95] This pper is the first prt for the opertios of the simple cotiued frctios. The secod prt will defie the multiplictio, multiplictive iverse d the powers of the simple cotiued frctios. CONCLUSION re much more useful (d computers c work with decimls t much fster rte. However, some iterestig We c lso solve y Diophtie cogruece tht oservtios c still e mde usig cotiued is y equivlece of the form x (mod m. I other frctios.nmely, i this pper, we will e explorig how words, i most rel-world pplictios of mthemtics, cotiued frctios c e used to dd d sutrct the cotiued frctios re rrely the most prcticl wy to umers d c. solve give set of prolems s deciml pproximtios 6
13 Am-Eurs. J. Sci. Res., 0 (5: 5-63, 05 REFERENCES 5. Mugssi, S. d F. Mistiri, 04. Simple Cotiued Frctios, I press.. Breu, E., 003. Pell's Equtio, Spriger. 6. Rose, K.H., 005. Elemetry umer theory d its. Euler, L. d D.B. Joh, 988. Itroductio to pplictios, Addiso-Wesley Pu. Co. Alysis of the Ifiite, New York U..: Spriger. 7. Vider, C., Jume B. Pelegrí d B. Lluís, 000. O the 3. Klie, M., 97. Mthemticl Thought from Aciet Cocept of Optimlity Itervl, UPF Ecoomics & to Moder times, New York: Oxford UP. Busiess Workig Pper, pp: Moore, C.G., 964. A Itroductio to Cotiued 8. Wll, H.S., 948. Alytic theory of cotiued Frctios, The Ntiol Coucil of Techers of frctios, New York: D. V Nostrd Co. Mthemtics: Wshigto, D.C. 63
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