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8-1-1947
Continued fractionsLawrence Edgar YanceyAtlanta University
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Recommended CitationYancey, Lawrence Edgar, "Continued fractions" (1947). ETD Collection for AUC Robert W. Woodruff Library. Paper 683.
CONTI1~UED FRACTIONS
A. THESIS
SUBMITTED TO THJ~ FACULTY OF ATLANTA UN~IVERSITY
IN PARTIAL FUFILU~NT~ OF TH~ REQUIREMENTS FOR
TI~ DEGREE. OF MASTER OF SCIENCE
LAWRENCE EDGAR YANCEY
DEPARTMENT OF MATHEMATICS
ATLANTA, GEORGIA
AUGUST 1947
ii
T~ABL~ OF CoN’r~ETs
ahaptar Page.
i..~1_Historical Backgro.un~i. . . . . . . . . . . . . . . . . . . . a 1.i~cte~at. of Applic.atiori. • • • • • a a a a • • • • . . . . . . . aPurpose of this. I~reatise.................... 4
11. F AIVIE~1ciTAL CONCEPTS. AM) FOBMUI~A~... a ~.. ........ 5-
Corive~rg~e~iits. a a • • a • a a a a a a a a • a a a • a a a a a a . a a a a e SThe. Ide.ritit~ p~q~1 =
a~ •ea...a.aaa.. ••••• a • •a a.. •.a.a....a 11
&q~ivaIeat CantInu~e.d Ftractioris... a sa a a a a a a 15:
III.~2Q~
Prope.rtie.s of~21~ve~rgence. of a Simple. Coritinue~d Frao.tion5 30
Applications, to. Nunthe~rs..... a a a a ea. a... .. a. 31App1i~ationa to. Rational Numbers........... 33Applicatioas. to. Irrational Kuathe.rs......... 36¶~ranscendenta1. Nurnbers...................., 41A~pp-1ications: to. Iii.ophantine. ~QUatLOfls.. . a . 44
IV. G~RALCQI~4TINtjED,FRACTIOI~Saaaaeeaaaa.a.aa.a.a.a 49,Properties. of Coziverge.ri~ts......a......a.... SQOanve.rgerice. of. an. Infinite. Contirm6d.
~rac~tiozi. a • a s • a a- a • a a . • • a a a a a a a a a a a a a a •- a a 53Conve.raio.n of S~riaa inte~ Qontinu~d~
Frac.tions..aa. a a a... a a .a. a a a a a a a.aaaa a a a
Coiiversion. c- a Qc-n.tinued. Product- into aContinu.e.d., Fraction... a a a a a a a a a a a a a a a a a • a
BIBLIRAPI~Y.. a a... aa.~.. a... as a a a a a.... a • a a a.. •.a •a 5~5
£
LNTRODUCTIQN
Conu~d: fractiena. is ~. area. of isthematins that has
heerL urLusually Iacted by mathematicians and has been almost
a~1ete]~ ig~nored by; ~rierican writers.. Uofortunately, the
articles on. co t.inued fractions by American. autkiars,, with~ the
exception of one, have been articles dealing with. advanced
topics in ~ ~f: continued fractions.. American. sources
on the fun~ atals of continued, fractions are both. inadequate
arid: scattered.. The. theor~ of a-dfrneiasional continued fractions
baa been almost c~omp1etel~y’ igp~o.red and: the theory. of tea di
mensianal. general continued: fractions is abviau.sly quite inconi
plate.. This is not.. e new b such. of matE-ties for many of the
ftindamentala were established: in. -the 18th. century.. it has long
~e~j i- d. as an. important h such. of aiathe tics but littlE
has been done toward. the development of the theory.. The mast
complete and: comprehensive work~ on the sub~Ject~ is ‘~ie Leh~,
Won.. defiL K~tten ~ by 0.. Parron... Unfortuniately~ this book
is Un tamable at the present time... With. the exception of the
hook by Parran and: translations of the same,, other writers have
confid: themselves. to very narrow sections of the field ~j
usuaIl~r dIscuss very particnI-ar spacial cases of the general
continued fraction.. Saveral American authors hive included the
algebrai treatment of continued, fractions in. text ho ok~ on.
algebra, but mast of these hooks eoritaiii ftindamarital errors,
inadequate proofs and: bad: notation.. There is. rio hook pu1~lisbed,
I
aby~ an. AmericarL author, that treats both. the alg~bra arii
f~mctiari theory ~f: continued. fractions. There has been. &
teridaxiçy to se~pa~ ate. g~exiaral tmn.ued fractions into special
cases and. igaore. the g~ne.ral case.
The theory of continued fractions is. & very powerful
mathematical tool auL as. such should. be irlors f~.illy deve loped.
The possibilities for de.~telopwant in this field. are almest
infinite. The. applications of continued fi~aetiana are bath
numeraus arid. important. I1Lu~ch cf the theory of numbers is de
r±ved by the use of continued. fr~ac.tions. (ontiriued~ fractions
are. used frequently in. astronomy. They are. used te~ compute the
Uma of an expected eclipse, the number of days in a year and
many other related. problems. I~z fact any real xliirnbe.r, c~mmea-~
curable or in mmen~ahIe, may be computed. ta any- de~rae. of
accuracy by the. use of continued fracti~na. They may also be
used te determine the closest apprximation of a g~ive.xi number
with pre-datarmineci iirrrtt~ on the order of the size of the de -
nominator of the fractia. One of the ac1vantag~a of using~ a
continued fraction. to compute a g~ivea number is the fact that
&ter computing the g~ivea number to the desired. deg~es of
accuracy, we may compute upper and. lawer l~ rnfts for the error
between. our approximation an the. given. number. Gftea whari~ we
have, an indaterm~ nate. equation. at systeni. of equations, we way
obtain a solution or solutions. of the kind. desired. by imposing
outsida conditions on the equation QL~ 5y-5~fl~ of equatiena. If
the solutions that we desire must be rational, integral or
positive integral solutions, the equations at this kid. are
a
called flhi~pharitiiie aquations.. 1~&aziy fliaphantin~ eçuatioxia may~
he solved by: the u~ae of continued. fractions • Continued
fractions. play a profound. part in function theory. S iltje.s
i~rsed thew~ te obtain a complete slution. of a famous moment
problant. it. was while workin~ with continued fractions that
Steiltjes darive.d~ the famous Steilt3es integral. flurin~ the
18th century, Lmbart,. a GermarL mathemetician~ wed them. to
show that I, ~ and an~y rational power of e are I rational.
Continued fractions have also been used q~ite ext siveiy in.
the. theory of dofinite itegrals. Lime. Ball Talephcne Company
uses tbffllL oftezi in the. theory of current decay in electrical
trarisathsian lin~~. They are alaa~ vary closely related to
matrices: it two re.spec.ts4 i continued fraction may tie thought
of as a framaw~ork within. which a mathematical s-y tea of rational
functions may: be viewed. Since a. matrix may: be. thought of as
the same. tbing, they are clearly related. Thzi a system. is
placed within the framework of a continued fraction.,. often. we
may ebtaizi prope.rties of the. system.~ not easily obtained by other
metliade. M~Ch. of. the. advanced theory of matrices may be obtain
ad by the use. of cQntinued fractions. ry infinite continued
fraction may be. represented by an. infinite product of niatricea
and every finite. continued fraction may be represented by a
finite prodnat of matrices. M~uch of the ad~tanced theory of
~acabians may he obtained front then. These and. other well
kno~ applications of continued, fractions illustrate the in
p ortarice of the exploration and goneralizati a of the theory
of continued, fractions..
4:
In. this thesis, I shall discuss the. fundamaatals of
both~ the. algebra ansi the funetLan. theory of continued. fractions.
I shall also briofly compare. can.temporary aatation~ in. order to
facilitate ease. in. reading other articles on. the subj~ect~
Where. it is possihle, I shall discuss the. ~nest general. kind of
continued. fraction. and. let the special cases coma froni the
general case. Eawever, the. racist important special cases will
be. discussed. in sufficient detail that one. may easily use it
as. a backgraunLi for ad.vanaad work~. Màziy of the imp ortaat app1i~
cations of continued fractions will be discussed. in great d~
tail. Unless. a continued. fraction. is. designated as finite,, it
will be. understood to be infinite. Canciusions: will be. drawn.
s.t the end of e.acli proof and customary symbols will be. used
‘~the.a it is possihie..
CHA]PT~R ii
FM~ITAL~ CONC~PTS A~D FORMULAE
The. awat~ ~enerai c~atinued fi~ac~tiari is. ~i. expressiozi
of the farar
(a.ai ae ± ~i
ha ±
± a4
Ca; a~J
so — a —
where a1 Ci~ ~, ~, a, ~—~,n,—--4 ani = I, a, —,n, -—1may~ he a~y quantities. whatever~.. Such a fraction rnay~ he
pariorli~, rwn-periodic, finite or infinite .~ General contim—
e~ fractions, are tisuaily div~id~d ifltOE tza cIasses~ Cantinued~
fracrions of the form~
(a.21 a~ 1- a1
b~ +a~
LTh~ ~ —
an~aa~aaaaa ——
5
are Ca continuad frac~t-ions of the fist class. Continue;d
fractions of the form~
Ca.a)~
b3 —
are. caUe~ continued fractions of the. säconi class. Relative
i~ little. has een clans conc&rziinW continued fractions of the
sacond class. The fractLQn ± a~/ is called: the. ith. partial
Quetj5.nt ai.~ the; Lth~ c~wie.r~t~ fraction. of ~ continued
fraction. If the. number a &omporiant fractions is. finite,
the continued fraction is finite. If the nunthe~ of con orient
fractions is finite, the continued fraction is infinite.
When the. componexi.t fractions recur,, the. e tinued fraction is
said: to be. a p~j~ Q~ a recmrin~ continued fraction.
When all of the; component fractions recur, the continued:
fraction.. is said to be; a pure recui~rin~ continued. fraction...
When aon~ of the; partial. q~uotiants. recurr and. others do not.
recur~, the ontinued fraction i~ called e; ~xed recurring
continued fraction. If none of the; partial ~zatie.n~. recur
ordarbj, the continued, fraction. i~ said: is be nan~-periodic.
7
Le~t. u~. dezmte & piete c~oatinued. f~aatiexi ~y: ~ so that~
(a.4) F a0 ja1
% ±
and~ lat
= ~i ± a~
ha±a
Saa afl5_en
Ca.~6~ Qa:ha±aa
n_a_a__nSenan
eSaana
and so on~
The c~omp]ste coritiriued fractioa, F,. is ~afled the first
complets qjiotient of the. tinued. fraction El; 43LI
The.. c~ontiriued. fraction. ~ is the. second. complete. q~tiotisnt.
arid. is. the. third. complete. quotieat. ~quati~n. Ca.41 is no
less gei~aral when. eack partial Quotient has. one for its
nuwerator. Any anue.d. ma~j be written witb~. one as the
~of each partial qaotient. The niethod used.. will be
discussed. under a ivale.at continued. fractions, later.
There are several variations of notatin.. Th~~a.u~haut
this discourse,, the English notation will be. us~j. The.
En~lis~ notation f~ ~ a. Lraction is
—
so ± — ___ ___ ___ ---- ___. —-L 64 aa&j
The Continental notation. far suck a fraction le.
a. a..~za~ a0 ±_. ±_2 ±___ ___
hi
perhaps the better
freq~uaritly~ use.dL by
a a___ ± . ±___ xi ±--11;;4a4J
ha hawm~t notation. of all because. it is so
a ~ of sums. Or~ d~iff ness’. Most
the ~ng1ish. notation.
Converg~rLts
If we forrn~ fraro~ a continued. f~aetion
a3____ ___ ___ ___
This is..
that is.
afl
notatian. of the. two • A notation
C.a.tinental writers is
(a.a)~ a ±___
ThIS i~. probahly the.
e.asiiy~ confused. with
American. writers use
a
the suceassioti of finite aorLtinuad fraetiona
__ __ __ _ a~a. ~a..± ___ ,a ± ___ ___ ,a ± __ ___ ___a 0 bI±ba
a ± ~ ____ ___ a4 ~——---, an~± e~aauata ti~a
pin. the forms 0 , ______ , a 4
then ___ ía saii ta be the n.t~ con.vergen.t., ta ~ continued
fractioni,. ~ If the coritintzed frSctiQnI,. F, is. finite and has
exact1~ n. bra,. then p~/q~ ía a icuaI~y- the walue of F~ Let
a1 a1 aa
a2~ a: the:iiF ___ p~ ~ ja1 p1a1 C
F 1b~1~ ± a~ b~ ±
~ 1ba±aa~±ai~a a0 ±a0aa±a~b~ Paa ~iha±aa — a±~a
p050~~±a~ ±a~h~
~ ~za~b~ ±a~a~±aj~
pabaLao~ ~ ±a2~aa hatp~a~b1 ±a~ and
~rofQi~epa=h~pi±aap.. ~_b ±a~a(bi) ja~C1)
rnhut ~ ~ “~ = i~ = h2~ ± a~ Assume
that these forms hald. fart the nth~ r~zLt. That is t~ say
ca.ia) p-11 = %P11-~1 ± a~prL_a and
Ca.n). ~ ~ ± ~
The (ri~ i. 1)~st erni~vergent diff~ra from~ the 11tb~ c~onverg~at on1y~
is having ~ ± a~÷1M~÷1 i~ the plane of i~ The (ri ÷ IJ.st
a~an~xar~e11t is
(a~2J _____
aj~÷1
Pa÷i ‘~n~a—i ± J~ ±___ - ____________________
hut p~ = ± a~p ± ~q~_2~
± ~ri~j~ p111
~
±
±
therefore
(2~.~i3~ p114~1 — ~ an~
(a.~i4~ q~1 h~÷1q~ ±
if we substitute (11 -~ 11 for~ r~ is eq~uetiozis (a.~1[i and Ca.u),
11
wa ge~t ecjuatioris. (a.ia) and (a..14) W~ see from. equations
(2.13~ a~i C2b14~ that the law that we as&wied to ha true for
the rita coavarg~nt also holds for- the Cia. + I)st eon~targ~rit.
But the 1awd~e& hold for- the second3i conver~rit and so it must
haM for- e~tar~ cvergezit.~ rt should ha noticed. that in using
these reiationa, the fraction p/q~ must not he riuaed. Thus
in~ the continued fraction. I -t- 2 ~ ___ —----
the th~ird convergent must be taken ta he 6~4 and riot 3/~
Actually ~ and ~~ ~ f~jQ~ of. ~ indspendant
~ariaba a0, 5r aa~ ——-—--.~ a~ -~ b1, b~ h3, ~
and c~nriot be resolved. into factors.
The Ldentity’ - p~q~ =
Q~ ~ moat impor~an.t j~jftftj~ is the relation
ship hatwaei~ the co~onanta a~ two c~nsed.itive con3lergents.
This identity also. gives the difference hatweerL two cansactrtive
con.verg~1rLta. ~M. differen~ is important iri~ detarm~n ~ ng~ the
coz-verg~nca or- d verg~n .. of a continued fractiozi. One member
of the idaILtIty~ resembles a deiinitioa of a detsrmina~.
E~om. equations (a.2ü) and (a.il) we get
Pl~~ p0~1 . Ca~h~ j a~)C1i
- p1q~ bi(aoh~1a ~ a0a~ ± a1h~ - Ca~h1 j a1)~(b~b~ ± a2)
- p~q~ a~~h~b ± %a~b~ ±~ -(abs. ±
± 1- a~a~
- P~1q ± aa~ ± a~$~ - a~b~ ~ a0a~b~
~ - a~aa
= -
~ D~OW~ have - i~ci1~ ±a~andPacii~~~ P1~~- alaa
~ssum~ that tba Ia~ h~Ld~ fc~r~ tha ~omponexits of: tha ath~. ar~
Ca - hat eon r~ta. That ~s to say~
C~1~5~ pu_i C—hi~(~ :~~aa~_~
t~ain~ aquationa (a.ioi,. Ca.nI Ca~13~, an~ (a.14) wa gat
- p~clu_÷~ = ±~ (%~:.~ ±
± ±
± ±
+ au_a Pu_ 1qu_~ ~
± a ±
+
= h~&u_+IPrIcu_~L~ ± ±
÷~~ -
~ a1I÷j%P~j...jqu_1 a÷~u_~qu_
-
Ca~i~I ~rL4-j% ~r&l-i = -
± zi~-1~~&-a ~
÷
eq~uatians (a~1Q~ an& Ca~u~ ~ ge~t~
— ______________
&stituting tbea~ v~aiue~s far andL in~ (z.i€i wa g~t
÷~ri-1
± ~ Q~ - __________
÷ __________________
~ 7-I
- ~~SLi-i = -
± br~i(± p~q~ F b~pq1~ p~c~ ± hci~)
÷ a~1C±~ ~ ~
p~Q~ — - —
- ~ ~
± -
Ca.a2i P~11~1q~ ~ = C-iiC± 11 -
But - L—l~ (± li a1a2a~a~
trom~ equatioa (a~i~i and. putting this valu for~ p~q~1 -
irL ecjuatiori (z.i7i we get
- = (—Il (± 11 (_11 Z1 h1( ±11~alaaaaa4-~~-~akI
Simplifying the shove ecjuatio~ we get
(aiaI p~&i = (-li,~±~
if we place (xi i- ii fo~ ~ in eq~uatian (~L~l5); we get
ni-i- C—U (± U
‘this. is the same as ~iatioa (a.lBi ~ Tharefo if the l~:
halda f~or the. components ef the (xi - list and. nt~ coxivargerits,
it b~alda fai~ the coponents ef ~ .~.
varrits.~ It d.oea jiald for the poneutsaf the first and
seccid conivergants so~ it must haLi far the components of any
~Q. ecutive convergen.ts. ~4uatioxi Ca.IE. may be written
- (—li(j l(—l1\aaaaa4~an
Combining similar~ powers we get
~ ~ri—i - ~~3:.iS ~ 1)~ -a
&jui~ta1e~rLt Q~mtinued Fi~actiotis
When. two~ c~ont nued frac.ti oxis, F arid. Ye,. are so related
that every nver~en.t of E~ ~s equal to. the ~anvar~eat~. of F’
of the same order, the~j are said to he equivalent. ~~ra
tinued. fractions may also have a~ (m~.n)~ equivalence. ~ey
have an (m~,ni ivalezice. when. the foi1owin~ equation. h~ild~.
1-a~t us caisider the t~o caatintie& fractions
a1 a~. a a a~~a.2ai F _ - ____ ___ ___
and
~ F’ = c1a1 CiC~2~ ______ -— ________
e1b1± c~ jc~b3 ± ± c~b~ ±
Equations Ca.ao) and Ca.aU may be. reWarded as. the g~.ne~ral
type of continued fraction. for every contimied fraction. of
the form=
a a a a.F—a ~i 2 3 ii
may be rLttan~ as
1 a a. ac2~.22~ F __ 1 1 a -~
We see frooi the definition of a ecntinue.d fraction. that we
have, written. F in the fallowinW fara te get equation. (a.221
F ~ —F
F
Let i~is dex~ate the ath. canvergeat o±~ F hy p~/q~ and~ the ritb~
eaavergerit o±~ F~ by~ P~fdi1~ The,n~
p1 a~ p~ a1a1 c1p~__ - ___ __ = ~
___ cleaaj?a _______________ — ~i~ahiha ± c2(h,~h ± a~J
Assurne~ that the. law~ holds far the nth c~anvergent~ Thea
(a~aa~ ec3~--------- -- -
=
ca..a~i q~ e1c~3
~ qL cieae3 -----eq
F~on~ aquatL~ns. (~.lO~) and (a.n); we see that
(a.zi~ p’ eh~p~ ±~ an&
Ca.2a~ c~ ±
Since p~ ani q4~ di fei~ from. and~ Q~ onIy~ ia having
ee a~ 1) •~. a Eli-i. z2*1 iL the place of c~b , we get from~
4-1
equations Ca..afland~ (2..2~
i~
______ —
d~Z1i.1
7~ ca~~
c.~ a± 1-1~n*1
C -L n~i-i zii-~
Mip1yin~ th~ numera-tor and de~iominator by
p-I_z1i~i_ ——
•~n~i-i
.rrHl —
____ —
~n1÷i~n-i± ~1~P~j~j± ni~%÷iP~-ac÷~q~1~ ÷1a~~1q~1~ 1_11~b~~1q~
~b1p~2~± afll..L~1± 1a~~11.1
~~
÷i,a÷~~1~—i ± %~j~P-’~~I ± a~p~17
— ± ± a~1ç1/
___ %,t%?nP~~~i± P~-~? ±~c~_1c~a~q~1 ±
(a.a~ and Ca.2a) in~ eqw:tion Ca~29i• w~ get
±± ~a~141)~ —
&tb~tituting the. values for p~2~ ,. ph ,. ani q~!~ given by
equations (a.2:. ,~ (a~24a, (a.a~.) ani (aba6~ in equatioxi (a.~sC~)
and. we get
Ca.~ai) ____
±
±~
TI1
p1EL4-i
a1
U~ing~ equations
Ca~3a) ___ =~-11
From equation~ (a.31)~ we. get
p.~ ~c4—-.--- -c~%1(b~P ± ap~1) and
~~-cC~c~ ± aa+ia~
U~iz the equalities given by equations (2.13) and (2:•14) •Wa
readily see. that
(2.32~ pt1 e~c~e3e4—--------~1p and
C2~33i q — e c~ c. e —-———e q1aa4
Since. equatiuna (a~aa) and (2~33) are the same as those
obtained, by rep1aein~ a by (a + 11: iii equations. (2..~a4~ and
(a.a€~, the law holds far~ the Cri. + l)at cmwergent if it halds
far’ the nt~h. eo.nver~ent.. 3u.t the law: d~as. hold for~ the first
and. second caziverge.nts~. Therefore it must hold f~r~ the third
and every su~cce.edin~ conv-argent. Froni.. equations’. (a.a4) and
(a.26) we. see that
p’ C.. C~ C C. ——-———— -C. PCaa~ __ i2~34 an ~ri
W~ conciud froni. eq~uatiori (a.a4~I that the ntb cQnver~e;nt of
F~ is equal to the rith~ convergent of F far every value of a.
This: mai~es F and F~ equivalenthy definition.. It should he
noticed that ~ (i 1,. a, 3,~ -~----,n, ------i inay~be any
q,uaritities whatever but c~ mu 3±. be I.. It should a-iso be
noticed that F~ may be obtained froni ? by multiplying aj~ b~
and by~ c~. The same type of proof might be used on. an
Cm~,n) equivalence. The only difference is the difference in
19
the- o•rde-rof the. convergenta ef the- twa fractions. A very
interesting special case- of the- equivalent Continued fraction
Fand F’ a~risee- when. c~ c — I • The-na—in— a
xl
a, a aaC— 1. ~ a ,C— ____ and:
a1 2~a~ 3 a~a~
a1a3—-a~1 aaa4a6~-~-a2
— a —--- a a a —~a24 a~ 2~3 ~
~quatioxi (a.ali now he.comes~
(a•3~ F” = 1~ 1. 1_ Ib1 ±a~h~ ±aab3 ±a1a3h~ ±
a1a3 a2a
_______________ I± a1a3 a~1~ ± a2a4-a~b~÷1
aa—a a-a——--a2n. 13 2ni-1
CHAPTER III
SIMPLE. CONTII’~UED FRACTIONS
The. type of continued fraction. of; niest practical
interest and. im ortanceis the. simple or regular continued.
fraction. Unless otherwise. stated, we shall re.far~ to a
simple continued fraction when we spe.ak~ of a continuad. fract
ion. in this chapter-. A simple. continued fraction is a fract—
ion of the. type.
(a.i) fb~÷ I I I Ib1+ ~* h3i~ ÷
where b1,. h~, b~, —-—---~- L b~,—------ are. positive. integers.
~34a4I. In. equation (3.1);, b~ is an integer- but may be
positive,, negative. or aero. The. sImple continued fraction is
a spe.cial case. of the cIassi general cm~tinuad fraction where.
~ I a~ ~ aa a3 =
and~ -,z1,----—)arepo.sitive.
inte~gers..
in a simple continued fraction,. b. is called, the. ith.
partial quotient. It. should be. noted that b~ is the re~ipro
cal of the. partial quotient as defined in chapter II. Sucii
a fraction may he finite.,, infinite,. non-pe.riodic~, pure
periadic. or mixed pe.riodic~ The. criteria for determining
whether a continued fraction is finite, infinite, non-periodic
pure. periodic or mixed. periodic is the. same. as that given for
the general continued fraction discussed in chapter Il. Let
2G~.
u~s denote. a complete. simple. continued fraction. by £ sa that
f — h: j- 1 1 1 1 -~
a ~ ~i- h~+ ÷
and let
(3..2L) ~ b ÷ I I I ----- 11 1 b2i-b3+b4÷
and
(3.31 Q’~bi-1 I I —~-- __a aa 4. S xi..
and sa an.. The. complete. continued fraction,. f,. is called the
first complete qjiotiaat of f.. ~]3 431j. The continued
fraction. is the second camp lete. quatiexit and is. the.
ti~Lrd complete quotient..
P±~ape;rtias of Can~argeats.
SI ce. the coxivergents of a simple continued fraction.
a~e abtained in the same. w~ay as these of a general cantiriusci
fraction, we. may obtain. recursion. formnlae~ .far~ the convergerits
of a simple. continued fraction. by modif~ing the. farraulae far
general continued fractions. From~ equations (a.ID:) and (a.ii1we. get
(3.4) ~ and
(3.51 ~n~n~—1 ~
f~ there. are~ ~o. mmn.u~ sigfls in simple caxitinued fractions
&tuationa (3.41 and (3.5) together with~ the. initial conditions
froirL
fa~rni
(2.6)
(a..7)~
It we.
(a.a)~
22
p b p b~1~ ÷10 0: ~j I 0 1 ~ c~Qmputac~~ 1
su~ce.ssiva c~oavergents to the. ao.ntinue.d~. fractioa f ia equ.atiozz
(a.iI.Since. is a pasiti~te. inte.ge.r~ far a ~. Cl~, it fQ11aW3
equations C3.4)~ an~ (3.51 that (zi 0, 1, 2, 3,
irreasin~ sequezicas of inte~rs.. That is to say
~ p ~ p ~ p3 c —---- - - ~ ~rii s~ P ~
q0< q1~ q2~~< q3c ------------- ~q~1_1~ ~
divide. Eioth~ meiiihers of ecjuatioa (3.41 by p~1 we ~et
__ ~b + _____ —b ÷ _____
Pa—i nt—i. ri~—1
and~ suhstitutinW (ri—il, C (z3~-31, --- ~, 2 far a
ía (a.ai we. have.
P i____ p ___
p1~ ÷ ~ +
pa b2~i- I ~÷ I I
p1 1 0
pa
2a3y~ suhstit~itinW these. ~ta1u~s s~ccessivei~ in e~uatiozi (a.~a)
(a.ai __ ___ __ 1.~ ~—a + ~
If wa divide batti tne~nib~rs of equat~ion Ca.51 by
(a.i~) ___ _____ _________
A~ai substituting (ti~-l)~ ,(n~-2i,. (n~.-a),. —---—---, a i~i
~uatiar~ (3.i0~ ws ge~t
____ = b~ + - I _____ + - I
______ —
cL~L~ —b+ I b+ L I
cia ~a a
an~ci substituting thase. ~raluas su ssiveiyE in~ aquatian. (3.10)
(a.iii ~b÷ 1 1 _
ci hi-b .i- i-bn~-I EL—S 1
Tha following proparty of succe.ssiva eonvar~arits is
~ important arid is ofte.~ oailad tha datarrlliriarLtal proparty
becau~sa it is similar to ens of tha dafinition.s of a dstsrniinant.
it is a relationship between two. consecutive c~o.nve.rgenta.
Wè may obtain. the. forrnnI~ of this pr-perty fur simple; eo:_
tinued fractions by placing t2ia necessary restrictions on~
the general formula. The. general formula is given by equation
(a.l&) and~ states thatri—i
- = C—i): (± U~ ~~ri
But we Iuia~t that a1a~a3a4-—--.--a I far- the. simple. cari-~
tinned. fractiori and. there. are only pusitive. signs mi the
simple contined fractiozi in the p-lace of ± signs. Therefore.a
(± U 1.. Makm~ these. shstitu.tioaa an. the. general for
mula we. get
(3.12) pci1 p~~_~%.= (i)~
Suppose that p~/q~ is not in its lo.we.s.t terms. Thia a~ans
that p~ ansi cj~ have. a factor in eQmmon • Bat if this is. true ,~
that factor ie. also~ a factor of the. left member of equatiozi
(3.32):. Since; it is a factor of the. left meather of e~uatioa
(a.iai,. it must also be a. factar of the. right member of (3.32).
Since; this is imp ossinie ~ p,&~,. as computed. front equations
(3.4): and. (3.5): must be in its lowest terms. if we. divide
bot& side.s of equation (3.12) by we get
(a.iai ___ ‘~-J~ C- 1):
This is important for it gives ~as~ the differeri e between two
cQnse~u.tive. cQrivergents fQr~ ariy ‘~taIu~ of n~
W~ may write tha ritb convergent of a simple. continued.
fraction as Lt~ &J.
___ — pa, + ___ ~o __ — p1 ~÷_~~ L~
fp~ p_~1a ___
~ SI
Wë nae~d only to remove. the. brackets to se~ that the. above
e~qtiatiori Ia true. Si~ibatitaiting the. values far eack bracket
as founci by equation (a.iai anci we. have
p n—iab÷ ~- 1 ~ __ __
SI CI~CL1
It is often useful to knew the. relationship between
twa. alternate ooavergazits. Uäing equations (3.41 and (a.5)with. a replaced by (ri-li we have.
- ~a-aSI ~ (b~p~ + - P~a(b~ri%~ i
-~ b1q~~ - b~pq~~1
=~
B~rL p1q~ (~ i)~ front equation (a.ia)
witk a raplacad by a. Thus~a—a
(3.15);
~ divide. botk side.a of equation (3.151 ~ we get
•°1-~~qiI~
-,-____
-4d
115-t~b+h1b~t~—115
41~á~4~--;
_______________—
+~—
~~M~pTI~~imT~m1b~rduioo~~tq
~J~ZI~OT1~J~p~nW~1IOO~Oi~M~TIT~IO~
~%t411,.~____
4~+t~Q=~2t’C)
~)~pu~~ET~)St1oT~flb~
IDOJA•I~&TI0~O~Up~td~tr~q~o~oT~i~p~rrp~oo~
01.~rr~A.TI~~~u~tiO~~~pu~~ip~~
ppo~ore~~.uo~~pu~‘~flTr~TioD~TO~t1M
~~ss~iro~~re-J~pJo~U’~J~JO~~AUO~~
‘1.u~J~Attot~ppoX~r~i~~~s~is~
~~•~~£Mot~So~~e•~ut~
~.t~11O~1.r~Ae~ptre~r~pu~p~tio~~ti0~
ppO~~~~~OS’~~[1~J~OA!1O~~
~ITIO~.O’rI~(~;[~~)~t1~
11-i:z~11-)
tr~i~ire~‘~~~-~q~imitu~v.1i~ae~
°~~~rr~ov~o~~
1rn141_~~q~mimJ~AUT~~~
~SS~~~oI~J~tIO~~UO~~OJd~~q~ti
S~~u~o~nb~tfl.~
~j~xw~—~~ss’e~kx~iurizx
weo~tz~sixt~oddo~t-;~~tw~tth~
iuo.x~;~~MOTTO~~IIl-rib~~b‘~L’~)tIOWLb~
1UO~~J~PtI~t~~V~T~iod~xe~bwe
~tw~i~T~nTY~-~qp~epmtp~r~)
÷t_lib____
-t~J(6I’~)
tib÷V+~%)tii~—____
{~ItZb-1--+~d~+~t~%-
____—~t_1~-r~t4tb~bt_~tXd4~—
+___
—t~d~%—~~j
~T~-~9t1~~O~~TD~Wtl~TV~t1b~S~
.~q~iiw~t~~~~tE~)~rieq~y
-~1tt-)
~(~‘E)~.~1~b
~t4~~!ZbtZb—T4UB___
~in-
(~‘~)~xorih~e~Xt1
‘SM~OTTOJ~
-~pJOpii~~~‘~ptv~~Tu~uI~OJ~pzzOt1~p~t1~JJ~~
4;11
~4~tT~~
p~o~d~nr~o~~~tr~q~~~‘~xo~rrp~oddo
rn0J~o~p~z;fl~T1O~D~Ot~M~;o~~~tp~o~dd~
pu~euop~~x;To~~xw~EUJJO~~1Ot~ppo~‘J70I1W~1b~
im~m~ro;x~i~ue~T1eA~~(~j~)~PU~
~OT~T~TD~1U0J~SMO~flO~~J~AIXO~p~~d~
~UTj~J~SOT~~T~T~S.T1OD1p~pmjtt~t~~
~~~r~i~tzooppo~~pu~;u~q~s~s~~
ttTVIO~3~‘t~M~~O~p~9tt~I~ATJO~D~t1~Zupje~
ic~q~~~PU~8~!Ufl~J~MOT~
~~pu~(g~)‘~g~)‘f~~)~o~ç~wnb~~~o~b~a
~U~T~~~ur~J1O~~~O
9SO~~~T1~S39~)tzoT~nb~.~q~~~T1~
4___
d
~t~~)~or~b~~
~~
_____~÷T1~~t_____
=+‘t4tL~+11
-)
~~ttcr~nb~iuo~z~~
~4~11b%________
=
~~~~)~T~o~ç~tlTYe~wo~tj~rw~
~
{1~~)~,uo~x~
~~IO~enb~~qt[~A~~iw~iw~~
~on’~te.rgence of a Simple Coritinueci actiaa
It should ha zzo.tice.~ that the definition, of the cons
vergenta of a continued fraction determines the. meaning
attached to the chain of operations in. a continued.. fractions..
If the continued fraction is: finite, it is. possible to. begin
with. the last partial quotient. and. al~ebraicly reduce it to
a rational fraction.. When. the. continued fraction is infinite,
this is not possible. Equation. (3.22Q~I states that
pa Cli
by taking the ahao.lute value of both.. members of (3..22)~ we get
p 1.Ca.a7 El. f ______
q
but e.cjuati.orL (~3.7) indicates that Liui Liui Q~~1 = ®
XI’— ED El— (0
The~.tfaxe
p. p.Ca.2a1 ___ f =0 or flf
as n. becomes indefinitely larg~. Equation. (3.22~: shows. that
the limit of the nth.. convergent as a increases indefinite.IT
is I to the c.antinue~i fraction. Equati.on.. (.2~fl shows
that the cliffereace between the nth. convergent. ani the cQntinued
f±~actiori. can. be made smaller than any previously chosen
pos~tive quantity for- ii sufficiently large.. A. simple c:ontiaued
fraction.. is. said to. converge when.. the; difference. bet-wee n the.
31
ath nv.e.rgent and: the. continued fraction can. be made. smaller
than. any pre~riously chosen positive quantity. Therefore it
fallows front e;quatio.a (3..22)~ that. every simple. cQntinuei
fraction c~on.verg~-s. Equation. C3..14)~ proves. the. existence of
Lint p. i~ for- p1/s is written as~ an. al e.rnating series with.a
each~ tern less than. the pre.cading~ te~.rat and. the. limit of the
nth. ternt approaches zare. as a approaches infinity. This type
of series is known. ta eQnve.r~e..
Appiicaticns te. Kumbers
Every real number, commensurable or incommensurable,
may be expressed. uniquely as a simple continued fraction.,
WhiciL may or may not be finite. LI;. 424J. A number is. said.
to be commensurable. when. it may be. expressed: in. terms of
integers. It should. be noticed that incommensurable numbers
include, both. irrational and. transce.nden.tal numbers.
Let K0 be. the number in. question and let. h~ be the.
greatest. irtager that d.oes~ not e~cee.d N0. Then. we may write.
(a.2~) N. = b• ÷ 1~ where N~ ~ 1, although N may be.N k. 1.1
no i~the..r integral nor commensurable. Again. le.t be the
greatest inte. that. does not exceed: N1. Then we have
K b.÷ I whereN~ landagain.N.maybe.nei.therN2 2.
integral nor comxae. nsui~able • Assume. that the. law holds for the.
nth~ case. Then
Ca.3cx~ N b~ 1- 1.zr xi
To~ get N~ we let b1 ~ th greatest integer that does
not exceed N~÷1 and. we. g~t
(3.32~ N — b~ i- i.. where N. 1n.i-i — ~-i~l N Z1~i% >
E~uatiea (3.3.1) may be derived from~ e.quatioxL (3.3Q) by re
placingxiby Cri~i-U • Sinethe helds. far and.
a = a, it holds for every positive xi.~ If we. repeat this
process (a ÷ 1) times and. substitute the. values of N~ Ci. = 1,
a, a, -—--—, xi) iaequattozi Ca..2~ we. get.
(3.32) N ~h i- I __ __ —-•---- 1 ___0- 0 hi- ba+ h÷ i-b1 xi a-i-i
Where. b0 is an. integer, positive, negative. or- zero,. and.
Ci I,. a, ~, —-, xi) is a positive. ixite~a.r-. Since equation.
(3.321 is valid. for~ every positive integral value. of a, we
may choose a in such. a way that the. dofinition of a. coxitinued.
fraction. is. satisfied.. Since N0 is any real number, every
real xiumhe~ may be. expressed. as a~ simple. continued fraction.
It. is interesting te realize. that (i = Q.~ 1, a, 3., —-, n)
is actually the. Ci -~- i)st com.ple.te. qjiatient..
It should also be noticed that. the method. of continued
fractions possesses the. two most important advantag~s that
any system. of numerical calculation. can have. It furnishes
a regular- series of rational ap~iroximations to- the quantity
33
to be evaluated,. which increase step. by step in complexity,
but also. ia exactnas-s. One, eommensurahle. approximation to
a numbex.’, commensurable or incommensurable., is said to; be.
m.ore. complex than another when. the denominator- of the
representative. fraction. is. greater in one. case than in the
other. The error~ committed by arresting the approximation
at any step can at once be estimated.. C~ontrary t~ popular
belief,, every simple continued, fraction, representing a real
number,, has. a. ‘value between ~ and. + w on. the axis of reals
and is not necessarily between. Q and 1- U).
The representation. of N0 is equation (3.32) as~ a. COE1~
tinued fraction is obviously unique for there can. only be one
greatest integer-, that does; net exceed K~. If is negative,,
b i.e negative. but the fractional part always. remains positive.
It N0 is a positive, proper fraction., b0 is ae.ro. If is
greater than. one., h~ is positive..
Applications to. Rational Numbers
If N~ be.c.owas an, integer for seine finite value of a,
the. coatinuad fraction. in equation. (3.321 terminates or~ is.
finite. This follows from equation. (3.30) and we see that if
i- 0 ,. equation (a.aa) now becomes
N b+ 1. __ 10 b1÷ ba+ b~1
Ci i,. a,, 3;~ ————, n) are. positive in.tege;rs:,, % is
a~ integer and n. is. finite. The value. of the right member of
eq~iatian. (3.331 is the nth~ c~o rge.nt by definitiea. Equations
(3.41 arid (3.51 together with. the initial conditions. that
= ~‘ % 1 and q~ ~ shei~ that p.and ~ for~ the right
member of aquatiozi (3.331, are finite rational. polyziomials i~
Ci Q~, I, 2, 3, ---—,ri). Since, all of the b~s are integers
p and are integers • Therefore N is the. quotient of two
integers and. as such is rational or~ commensurable.
is. a rational. number, it may be written as.
N0 ~ rn/n where in. and n are integers. To. convert m4n iate a
continued fracti.ari, we must first decide. if rn/n. is less than
or’ greater thaxi 1. If rn/n is greater than 1, we. divide flL by
ri and get
(3.a41 m/zi~b ÷r/n vthe.reb istheqactientarnir. is.C 1 0 1
the remainder • Then. we write. (3.341 as.
(3.351 rn/rib ÷ IG
Then we. treat nirj~ in the same. way and we get
(3.361 n/r ~b ÷rfr’ b i- I
We continue, this process until one of the. remainders becomes
Ze.PO:. The remainders are. integers and they form. a decreasing
sequence. of numbers that mast become zero: in a: finite number
of steps. Substituting these. values successively in (3.351
we. get:
(3.371 rn/n. b0 ~ I - 1 1tl+ba~
where b is an intege.r,. b~ (i 1.,,. 2., 3., ——---a,) are. positive
integers: and. n is finite • This: proves that any commensurable
number can. he represented by a finite. simple. continud fraet
ion. When ni/b. ~ 1 we- write
(3.38) mjh = 1 and~ convert. nhin into, a: continued fractn/a
ion. ~just as We. c.enve.rte.d ni/n- into, a continued fraction. The.
fact that m/h~ I. means that 0-. If. the. rational. number
in question is rn/n,. converting it into a simple., continued
fraction is assezztiaily the. seine, as finding the greatest coin—
nion. measure. between in and. n..
kfte.r a. rational. number ha-s been. converted into- a
continued fracti.~n, this may be. written with. an. even or an odd
number of partial quotients. From. equation. (3,~.37): we have
1 1 1.
where is an- integer and. b.(i = 1, a, a, -—--—, n> are pos
itive. integers. This may be. written as
Ca.aa~ n/n.b+1 1 1. 1i- b~ ÷ b3 + ÷ (b~ 1) ÷ 1
Than.
(a•4~> rn/~ib0+ I ~ I _1÷ ÷ ÷ (b~—l) + I
Since, equation (3.3?> has Cn i- 11 convergeats and equation.
Ca.4o~ has (ri i- 2~ canve.rgents,. whe.a one- equation has an even
number of partial quotients, the. other has an add number of
partial quotients. This is the. only exception. to the unique—
nasa of a finite continued fraction and this. arises because
b~—1..is not the. greatest integer that does not exceed
Therefore. if we. always take the. greatest integer that does
not exceed (1. 0,l,2,3,. —, xi), the. repre.se.at&tioxi
of a rational number as a continued fraction will, always be
unique..
Applications, to Irrational. Numbers
The. continued fraction which represents an irrational
number” is an infinite continued ~naction and. the. represe.xi-~
tation. is always unique. 3J. Let N0 be. any irrational
number. Then from. equation (3.32) we. have.
N — b0 ~- 1. 1 ,s~ 10 121+ h2~ ~ bz:L~ ~n~1
Assume that N~+l is infinity for S.Qma finite value. of a. Then
N b. i- 1 1 ~aa—°—— — 1..0 b ÷ b i- ÷ b
1 a xi
but this is the same: as. (3.33.1. The. value, of the right member
of C3.33) is p~/q~ by definition and p~ and. q,for the. right
member of (3.33) have been shown to he. integers. Therefore.
N where pnand~ q~ are integers and the. irrational
number , N0, is. the quotient. of two inte.ge.rs. Since, this is
a. contradiction, N cannot be a for: a. finite value. of a norni-i
canN be. aixrte.ger for a;,finite value of~a. ItNcan.only
he.. an integer far an infinite value of xi,. (3.32) be.come.s
(3.41) N — b~ ÷ 1 1 — — ____
0 0 ~1÷ b2÷
Equation (3.41) is an. infinite continued, fraction. Assume.
37
that tha irratienal numbs.r N can bs. writtan as.a
(3.421 ~ + 1 1 1b + b i- i- h +1 2.
an~
(3. .43)~ N i— I a__a — —aaa_ I —a__aQ~O b4i-b~÷
The~a
(a.441 b ÷.1_ i -~-b~ ÷ 1. ~--a __
b + ÷ h i- b’÷ ÷ b’1 a. Li
Le~t = b ÷ 1 ---—--- I1 1 b-i- i-b
1 II
and ~ b’ + _1 —--—-- I1 1 b’+ + ~
2. a
Then (a.~44)~ bacamas
(a.~1 b. ÷1 —b~’+ I~
S:ince~’ ~1an~iQ~ ~1, 1 arid 1 are bath IassthanI •~
1 1
than one • If two numbers are e.qua1,~ tha integral p~rts must
be~ equal and. the two. fractional parts must be eq~ia1. There—
far€bb~ arldQ~q. Then
Ca~4~61 b. ÷ 1 = b~ ÷ I1. 1
Lat ~ b i- 1 ------ -~ Ia a hi- i-hi-3 a
an~i Q~ h’ i- I~ 1.
h~+
Thea equation (3.461 becomes
as(a.4~7)• I -h~+ 1
H~re~ a~aib~b~ an~1c~ Q
Ass mis that~ ths iaw~ holda far~ tha at~ casa •
b + I -h’ 1- 1ri-i
1
and. ~ — -1- 1.zi~ b’ ÷
a+l~
Ia ~tiaa (3.45) ws have. b~ = ~ am. Q~ Q~
Ia the. Cmi + list case. we. have. Q~ =
(3.4~);. b~ ÷ I I —~h’ ÷1 1b ~-t-b + b’ ÷b~ ~*au-a ri+1 a÷2.
Le.tI~’ —b i-I‘n.+1~ al_i b ÷
and. Q’~ — - + 1. _aa_ —
a-LI 111.i ht ~au-a
Thari we. ha~ts frera equatiami (3.49)
(•3~5Q). b + 1 -tb’ i- 1_a El
na-i “flu-I
Ji4uatian. (3.&~ is the. same as equatiert (3.45) witk ri re.
placedhy (a ÷1) ac atthalawboi .foi~the. (au-list
c~ase~~ Since. it. holds far the. first and. secand case, it holds
fer evary case. • Thus we. get
1 1 __
1- •~~~1- %i- c11-
(3.~fl b.. b’ (i ~ i~a~—--~-~.-, ri~ -)
EQua~tiarI (3~5I) proves that equati~ris (3 ~42) azicl (3 ~43) are.
identical th~ pravin~ that the representation is. uniq~ua.
E~ve~ry~ periadIc eentinusd fraction. repre se rits a quad—
ratid irrational nunber. L~; 7J. Let us~ consider the general.
pe.riodIc~ cont.inusd~ fraction
(3.5a) w = b ÷ 1 i --- ___ ___ ___ -~0 h~i- %i~
_l ___ ___ 1 1c c. + c i- ÷ c i- c ÷r 1 r
where. the periodicity begins after the nUi partial quatiazzt.
Let
(a.sa) w~ c ÷ _i. ____ --------—-—
ci- ci- 1-c + ci-a r~ 0
se that
Ca.s~i We i-_I. I --------~--- ___ 1e i- c i- c i- W~1 a r
Then.p’V÷ p~
(~~) ~ r r—i
÷
where. p’/~’ is the rth. convergent of the. rigkt member of
eqwtian (3.53). Clearing the fraction. in equation (3 .55~ we
get
Ca.s&) + (q~— pr
Their
c3.5~7)
Since. the discriminant of equation. (3.56) is positive and
aot a perfect square., we may: write equation (3.57) as
Ca.ss) w p ~S
where. Y, R.,. and S are integers,. R. is positive and not a perfect
square. • WEe may also write
w— b ÷ 1 1 1 1+bi-Wa
WEp÷p(3.~a) w— __________
Wa ÷q-U n—i
wher p~iq11 is the. nth~ convergent of the. right member of
equation (3.521. If we multiply the numaratax~ and. denominator~
of the right member-of equation (3.601 byW~q -q ,.we.ge.txi a-1
(3.61) ~ = (Wp~ ~- —
2w — (q~1)
W pq~ .t- (p~1q — pq~~W —~
~aa aw. — (q~1)
This may be. written as
P’ ± ~fRT(3.621 _______________
St
where pt, RI and. 5’ are. integers and. R’ is positive but not
a perfect square • This proves that every mixed. simple con-~
tinned fraction. can be expressed. as a quadratic irrational
number of the form~ given by equation (3.6 L~. Equation (3.52)
is a mixed periodic simple continued fraction.
Transcend~xital Nbmbera
The. anCient Greeks.. kneW; of the. existence of irrational
numbers. They knei~ that the. diagonal of a unit. square was not
a rational nuthbe.r~ ar.ii they suspecte~i that W was. also not
rational. We know now that if is not. rational 1~trt~. we. also know
that Wand /~ are. two different kinds at nrnbars~ FQr
/2 is. a root a an. irreducible. algebraic. scjuatioa with~ rat-~
ianal coefficients. (x2 a ~ 0;) and.. I i.e not a root of such
an. equation.. The last statemant may be proved, by the. use. of
continued, fractions bat it is ne ither easy nor short. We may
use continued fractions to prove. the. existence. of transcanden.t
al numbers. L7; aj.Ta do this. we first prove. the. following lemma due to
Liau.viIle. If’ x is a root of an irreducible algebraic
equation. of degree n > 1 with rational coefficients., then.
there. exists a. positive number e ~ 2. such that. for all inte
gers p ,q ~ 0 the following inequality holds.
Ca.6~a) ____ —
q xi
The. inequality’ obviously holds for every positive number
~ I when. p, q are. positive integers such. that
pfq x ~ 1. so we. need only to consider the values of
p, q far which pfq — ~ I. Suppose that
i-%zGwharec.(iz0.~
1., a, --~—, ii) are. integral, c0 ~ 0, is the equation
4Z
that x s&tisfies. Since. f(x) is rredueibie, f(p/q) ~ C)
and ~l
fr~’~I J_%P ÷ ~ -
far the. numerator is an integer at least as great as one.
Since. f(x~) — 0 , we have by one form, of the. the.~rera of the
(3.65) f(p/q) = f(pfq) fCx0) = Cpfq~
where ~ is a properly cbose.n. number between p/q and Xe
~4uatioris (a.6~ and (3.65) give us
(3.66) Cp/q — x0)f’(~) I ~ I
Since. lies between piq arid and since p/q — x ~ 1,.
it follows that ~ I ~ i- I. Then
(3.67) ~ .~ fric0 n-i1 ICa—i)ci a—2 1÷Replacing by t- 1 we get.
(3.65) ( f (~) ( < nc f ÷ j•)ril -‘- j (n—I) c1( t x~, I ÷ii~
n—i
Then
(3.69) f’(4•) 1/c:
where 1/c is aqual to the right member of (a.6a). Suhstitu~ting
the value, of (3..6~) in equation (3.66) we get
p/q — ~ _ia
q
~ultiplying bath mambers ~: e.q~1~j~ (37Q) by c we. get
(a.7i) pfq~ a.n
thus proving the. lemma..
‘~tith the. aid~ of the. lemma, we. are now able. to. eoa-~
struct tra see. n.tal numbers. by the. use of continued fractions
Let
(3.72) xb+l 1 10 b ÷ b + ~ b. ÷
1 a n.pl~
and. let ____ he the. k~tK convergent of the right member of
aquatio.n (3.72.. Suppose. that the. positive. integers b~ Ci = 1.,
aLa,-.--
positive. r~ an. index Ic can be. found such. that..
(a.7a) bk~÷l ~
There. are. a~ infinite. ziumbe.r~ of ways. to do this. One. way to
do. thisisto..taka b~1÷q~. Tharix0 isatraris
cendental number. ro prove. this we. must show. that however
~ > Q~ is chosen ani h.awever~ large. ii is. chosen, positive.
integers p ani cj cart he found. such. that.. e.~uation (3.~3) fails
to hold. Suppose. tha-t satis1~Lea sri irreducible. algabraic
equati.ozi of degree a with rational coefficients. Then by
e.cjuatirni (3.22) we have.
p(a.74) Ic —x 1
0 ~
44
Vroai equation. (3.5) we get ~ b~q~ ÷ ~ gives
us the. inequality q < b~ q, Using this inequalityk~i-l i~
in equation (3.74) we. get
(3.75) ____ — x -
k÷1
Using (3.73). in (3.75) we. get
(3.76) ____ x 10
for infinitely many ~raiuas of k. But no matter how. small, we.
choose c ~ 0, 1. will he. lass than e for all sufficient—p
:Ly large values of k. Benca by the. lemma,. x~ must be. a trans—
candantal number • The. numbers given by this coristruc.tion are.
caile.d~ Liouville. numbers.. A remarkable property of these.
numbers, is that if x is a. Liouv~i1le. ~ ~ ax.~ + box0
is a. Liouville. number for’ arbitrary integral values of a, b,.
e., and d with ad — be. ~ 0. It follows that Liouville
numbers’ are. everywhere dense in the. set of real numbers;.
Applications to Diophantine. Equations
When an equation, or syatem. of equai.ions is. indeterm.—
mate ,.. we may often obtain a solution or so.lutions of the..
kind desired by imposing outside. conditions on the equation
or systeia of equations.. It is. sometimes possible to eliminate
or lessen the. degree of the indeterminateness. Then it some.—
times; happens. that there is rio solution or set of solutions
of the,. kind desired. Th~is if the. kinds of solutions desired
are rational, integral,. or positive integral,. the. equations
are called Diopharatmne equations. We. shall confine, our
selves mainly to~ positive. integral solutions. Li; 474~/.
First we. will consider scjuations of the first. degree
in two. variables • Since we. desire only the. positive integral
solutions, we need only consIder eq~uations of the form.
(3.77) aX~bY,c
where a, b, and c are positive. integers. W~ may eonf ins our—
se.l~s to the eases where a is. prime. to b.. If a and b have a
factor in e.ornmoxi, it rxinst also he a factor of e and may be
divided, out. W~ may always find a; particular integral solution
of
(3.78) &~—‘~!~c
If we convert a/b. into a, continued, fraction,, ~ is the. athq
p aconvergent and a a so. ~ a and q~ = b. E4uation
b. n—i
(3.12) gives us pq~1 — p1q~ (—1) but since. = a
and. q~ b we. get
(a.79) aq~1 — bP1 ~ c.a._l = ± I
~ltiplying (3.79) by ~ a we get
(a.ao). a(± ecj) b(± CPa...i) C
Hence. x’ ~ eq1 and y’ ± CPa1 are particular solutions
at equation (3.78~. Let. X~ and r he. any integral solution of
e.quatiQr~ (3.78) anci subtraetiari aq~uatio~. (a.8G) from. acluatiori
(a.7a1(a..ai) - Ci cq~j - b/El - (± cp = 0
Dividing aq~uation (a.afl by~ a[~ — C± cp~~)J we get
X-C~c~ )~(.a.aa) ______________ _____
— a
Since a. and h are. prima, the numerators ani the denominators
of the left anci right members of equation (3.821,. with the
possible exception of a proportionality factor, are equal.
Therefore 2(— (+cq )i —bE, anci~— (±cp ) —atwhe.re— -n--i—
t is an integer. .aver~j possible. integral solution of equation
Ca.7s) is. Included in
(3.83) X = ± eq1 i- ht and
(3.84) Y=±c:Prili~aE
Wham~ ‘~-~ ~ ~-~--~ ~ t ~ ~ cofor positive
values of r and I. When ~~ CPrii ,~ for positive
—
valesofXandYwehave- ÷ ~ ~ t~4 i-co. Thareareb
obvioualy an infinite number of sueb solutions.
Now we wish to find all the integral solutions of the
equation
(3.85) aXi-bi
and. to separate the positive integral solutions. We. may always
47
find an zitegral solution for~ e~quationi (a.8~)L if p~, q~, ~~11
a~-~ ~ have the same meaning that they had in the last
paragraph. We. get.
Ca.a6) (± a ~t- (ePai)b ~
where x’ ± = F ~ is a partieular~ solution
of e~1uation (3.85~. B~j subtracting (3.86) f~om Ca.85) we. get
(3.aZ) - (± cq~1~j’ - (~ cp~1~j = a
Dividing both sides of equation (3.87) by a~ (~ cp1).7
we get
(3.882: ______________
a
Since the numerators and deriomiziators of the left. and right
members of equation (a.8aI are. equal with the possible ax
c~aption of a praportio.naIity~ factor we have
±1)-btandi-(~cp~)at. Thea
(a..891 x ±
(3.9Q) ~ = ~ cp~~ + at
A1EI of the integral solutions are given b~j equation (3.89)
and (3.9Q1. ~be~ra is also an infinite number of integral
solutions of equation (3.85).. ~I~o get. the positive. integral
solutions of equation (a.a51 we use the following limits.
4aWlieii c~q~ ~ t _______
_ ___ - a b
eq epWhan~ u—i ~1 ± ep ~ ±
b. a then ri-i - -
a
In~ both eases. the number of positive. selutions is cle.ariy
baited.
~I& may also use. the determinaxital. prope.rty of a&~
Jaeent~ conivergenits to. solve, quadratic equations ia two
variabs for integral or positive. integral vaiue.s of the.
‘~ariahiss. We. may ~aae. this same. property to find integral
so.lu.tio.ris of the variabias whare. we. have, a systeni. of equations.
in. which we. have caere. variabie.a than we. have. e.quations.
CKAPTER IV
GEEERAL CONTINUED FR~.C~fIONS
The theory of the: general continued fractions as. de~
fined in equation Ca.i), is g~nara1iy considered lass mi
portent. than the: theory~ of simple continued fractions. it is~
alec mu~ch less corizplete. ~ Haweve.r,, there: are certain particular
neral continued fractions that have received considerable
attention.
Iii dealing with the general continued fraction,, all
of the numerators are not necessarily positive units nor are
the denominators necessarily positive,, it must be remembered
that the: chain of operations indicated in. the definition of
the: general continued fraction ma~ fail to have an~j definite
e~riing even when the number of operations i1s finite. Thus
in~ forming the third convergent of?
(4.1) F 1 .~- 1 1 :i 11-1-1-1--
we: are led to g~t
___ ~ _____
1—i
In forming’ the fourtK convergent of the right member of (4.1)
we get.
P 1÷ 1(4.a) ___ 1.-i(a4 1-i.
The fourth convergent of the right: member of (4.1), as given
4a
5G
by~ equation C4.3i is clearly meaningless. Thareiore. we. cmi—
aider the right member of equation (a.4) merely as represent
ing the. asaemb1ag~. of 0~e.Z1t5 P0/Q0~ P~fCi~ ~a1”a’
-—-—,, p/Q, —--—--- whose. component parts se. found by the
u~se of: equations (a.iO) and. (all): whenever any difficulty
arises regarding the. meaning or c~arwergency of the continued
&action taken iii its primary sense. Ll~ 491j. It must
als:o be. remeothered that no piece of reasoning that involves
the. use. of the. val of a nm~tarminating continued fraction
is legitimate until we have. shown that the value in question
is finite and definite
Properties of Corwergents
We have, seen froni equations (a.lQ) and (2.11) that
P. = b~ ~ ± a~aa and ± r~%-a’~ ~roa equation
(a.i9) we see that Pn5L~ p~_1q1•• ~ a
F~rom. eçuation (a.IO) we get
(4.4) P~ - p =~ ± -
Thea
(~ ~ ~n-.i ~n — UPa~ ±
If we divide. both members of eq~uatian (a.19) by q1q, we get
(4.6) ___ ~n—l ~F 1)na~aa a11
Equation (2.19) also indicates the fact that we. have no
51
assurance. that~ an~~r coavei~ge~zit as compu~tad. ~y aquatians Ca.io)and. (a.lli is in its lowest te.r~ris.. We may write as
p p /p1 p 7f~- p 7____ ____ ~ /___ — ____ / ÷ ____ — 1 1 ÷
q0 L ~ L q~
p 7+/rL — n~—i ,
p pWe see. that every bracket c~ontains~ fl zil for
particular values of a. Substiteting the: value. of pfq~ and.
the. value, for each brackat, witb~ the. aid of equation (4.6).,
we. may~ reduce. eQuation (4.7) to
~ _~ ~ ± - a1 aiaa (~ 1)tla1a2__a°
Since. equation (4.81 is valid, for every positive. integral
finite value of n, by letting n be. (n~—2) in (4.8) we get
~4.9) = b~ ± a1 a1a2 ± -- ~, l)~~aiaa~-a~
Sbbtracting equation (4.9) f±~oni equation (4.8) we. get
(4.IQ) p — p2 —~ 1)~1a1a~---a1
n—(~ 1) a1a2a3------a~1
which. may be. written, as
5a
____ _____ — —Cv~ ± (~ 1)E~a1a2__a~11q
Then.
(4.11) p ~ (ç 1}~Ca~aaaa___a ~)(-. a~q~_2 ±c~)
in—a
But froa equation (2.11) we see that. —aq2 ± = ~
S~ibstitating this in equation (4.11) we~ get
p p~ (~ 1)~(a1a~a3-—-a ~ (± b~q~_1)
q11-~2q~_1q~
Simplifying the above equation we have.
(4.13) ___ - _______ = C;1)~’1aaaa~b
If: we multiply bath members of equation (4.13) by q~ q~
we have
(4.14) ~ P~~_2Q11 z — (~ 1~’~aja2a~ a1b
When we divide, equation (4.61 by equation (4.121 we get
____ — ~a-1, .4- _________ a1~q~2
- rn-a
Snbstituting for b~q~1 the equal given by equation (2.11): in.
53
(4.15) we. get
pn— _____
Convergenca of an Infinite; Continaed Fraction
The value of an infinite. continued fraction is the
limit, if any suc:h limit exi~ta, that the nth. convergent,
~ approaches as a approaches infinity. fi; 5c1&j. A
general continued fraction may coznrerge, diverge or~ oscillate.
If the Lini p /q is finite, and definite., the continu.eda—oo~ a
fractioa converges • If the Lii ~ ± ~, the fraction
is divergent, If the. Lie ~ fIi~ictuatas between a certaina
finite number of valuas according to the. integral character
of n, the fraction oscillates.
All simple continued fractions converge.. The. fraction
1 — .~ 1 1, oscilate& for itsl—l--1-1-
value is i, Ct, or — 00 depending on whether- a = 3a ÷ 1.,
3m~ -~- a or ~ 3. The; fractiazi 1 — 1 1 1p1/a i- 1/2 /5) 1 i- I
diverges to CD, for 1 1 1 1 —-
1- l+i+i+li
can. be shown to. converge to —1/2 i- 1/2. /~ The ilitistration
of the divergent continued fraction implies an important fact..
The; divergence of a continued fraction is. different from. the
(4. .16)± a~q~2
— ~ aq~
divergeace of an. infinite. series and in general will disappear
if a coniponent fraction is omitted. Since. this is: true., it is
not safe to say that a continued fraction. does act diverge
because. the eoatiriuad~ fraction, formed by taking all of Its
Component fractions after a. certain order, converges.
A. continued, fraction. of the. first class with positive
cQmporie~iit fractions. cannot diverge. it will be convergent, or
oscillating if any one. of its. complete quotients converges or
oscillates;. Froni. equation (a.15). we get.a—i
P P (—a.) a~ a~-~a~a(4.17) xi ~—1. — .L.~ ~. ~ 4.
for a eontinued~ fraction. of the. first class • Since. the. coni—
porient fractions are. positive., a. and h~ (I ~ 1, 2~, 3, ---, n)
a~aaa3a~~~ -~are. positive. and therefore _____________________ ~> c~.
When xi is. o&i, the right member of equation (4.17) is positive..
When n is even., the right member of equation (4.17) is neg
ative.. This shows that every even convergent is lass than.
every odd convergent. ~quation. (4.17) also shows that the
even corwerge.n.ts: forni an increasing sequence and the. odd con.—
vergents: fern a decreasing sequence. Thus it follaws that
Lini p~. pa—--a ~ -A. and Lini 2ji1 = B, where A. and. B are.
xl— 00
two finite quantities arid A. T B. If A~ ~ B, the fraction
a-onverge.a. If A. ~> B, the. continued fraction oscillates.
55
L ~aatiriue~d fraction of the first class witki positive
elamarita is earivar~ent if the series _________ isL
divergetit. Since; all the elements are positive we. have
+
÷ an i% a ‘ q11_1 ~
q ~ ~i—a~n--’i ‘ %~—a ~
~ ~
= baQ~ ÷ ~,
~ (bb~1 ÷
(bii~~ibn~~a +
~, (b4b3 ÷ , ~ (b3b2 i- a3)ci~ Therefore;
~ + ani
q~q~ Qq1(a~ ÷ biba)~ ÷ taba) (a~ -‘-
Sieq~=iamiq~1—b1wegat
h~ f i~, 7T hh 7(4..ls) - U—iLL ‘~ //1÷ 23
a~a2a3~a~ a1L a~fL a~j
f~ b. b 7~/i+ fl~—11~ /
L a~J
b b /b. b 7h~t since — ri-1 ri is divergexit, + n~—1 ~. /
I a~ L a~j
ci 1q~1diverges to t- co, axil Lim. _________ : ÷ CD • Thus
a—co a a. -~-—aia a
p211_-t 7 a.a~—a(4~j9) Lini /____ ~ / Lini 1 ~ L~m I
ri—CD L ci~ q21J ri—co q~q zi—co
and
pc~1 7(4.2Q) Liixi /_~ —. _________ / = 1 0
n~—CD L ci~ ÷ ~
Therefore the continued fraction is convergent. if
b. bLim _____ ~ Q,aontinuedfractjonafti~fjj~sta—co a.
class witft positive elements is convergent. We readily se.e.
that the right. membe.r~ of aq~atiQn (4.la) becomes~ b1/a1 times
ari infinite product. in which sack factor of the prothict is
greater thazi one. Similarly we see that.. the right member of
squatiozi (4.18) is divergent 52: ii approaches. infinity if the
bLija - ~, U and h is. divergent. Thus if Liiu ____
fl—CD a I ri—co aU. a
is.: greater than aero. an~ is diverg nt, the continued
5-7
f~action. is c~Qnvergent. W~ alsa se~ that the caritnue(i
f±~acticn is convergent if Urn. i~i1~ 1.
if we convert a continued fraction of the first class
witkL positive. elements into sri equtvalent continued. fraction
of the forn
(~.ai) &~÷i I I _id1i- &a# c13+ +
then it. is. convergent, if at least one. of the. series.
(4.22) Sd ~÷d7i- ------
or~
(4.23)
is. divergent, oscillating if botft these. series are. convergent.
wa kriaw,that ~bu~t~~j ~ Q~~SO
~ (i~~j)~ + and’
(4.a4) ~ ~ q~1(d~ I)
By’ replacing n by (ri—Il, (xi—2), (n—Si, - —-, 1 and
substituting successively izi e.q~aation. (4.24.) we get
(4.a&i q~ < q1(i ÷ d~i(i~ + da)Ci i d.~) (1 +
hut. (L~ 1 hence
—(i÷ç
~I readily see that
5a
C4..a7~ ~
an~
(4.aai
Since; all the numerators are 1, ecLuatioxl (4.6a becomes
(4.2k) ~2ri—1 — _____ ____________
(12fl~_l
If we suppose ~ U, neither liar- can. ~rania1i.
Therefore. if both Lini q~ and Urn. ~ are finite, the.li—CO
fraction. will oscillate and if one. of theaL is infinite, the
coatinueci fraction will riverge b~y e.quatiaa C4.29).
Thare baa been. rio. eonverg~aca thearent offered for the
rn~st. ae~nsral kind of continued fractiQn.. There are several
general converge flee theorems for very particular kinds~ of
general continued fractions but with the exceptiQa of the.
general coatinued fraction of the. first class with positive
coniponezit fractiona, the criterion. for can.ve~rgen.ce is inconi-~
plate. The proofs for niost of these theorems are; both too
length~ and to.o. advanced for this treatise.
C~aaversion. of Seriea into Coatinuad F~raction~
W~ wish to. co-avert the series
(4.ao.i Sn.~U=+U~a~u;+
into an. equivalent. continued fraction of the forni
a, a a,. a(4.31) ___ a
b~ ba~ b~
A continued fraction is said to. be. equivalent to a series
when the uth convergent of the cortinuedt fraction is equal
to the. sum.. of a terms of the. series for all ~ralue.s of a.
Since. only the cmive.rgeats are. given, we. may leave the. de—
aotninatQrs-, q. U.. ~ i, a, a, — -, ii), arbitrary altkougk
we must still take q~ — 1. From~ eQuation (4.30) we get the0
SUflL of a terms. prom. equation. (4.31) we get
p11 ~ri..-i a1a2~a3~a4-- -a
a a-i
W~ kncrn~ frOQI equation (a.il) that q~ b q - a q~ • Wena—i a1
also know that:
(4.33) p11 u~~-u. +na+_2
by the. dofini.tioa of equivalence. From~ equations (.32). and
(4.33) we get
&a a~a a(4.34) ~ = I2~L4
a
F~om.. equation (4.34). we. get~
___ aala2 aiaaaa aala2a3a41 a qq~ q~q4
-— __ S a
Solving for a. ft i, a, a, , xi) by using successive
equations we get
___ u~q1q~ U2~a — , a~ — a1 = _____
a - ______ ______ - ______a — alaa -~ ______ — _______
LtQ(4.a5) a _____
xiri-1 xi-a
W~ a1s~ knaw~ that
(4.361 q~ = - aq~ (xi = 1, a, a, —--s n)
Coinbiriir~ ~iaUons (4.351 and (4.3€) wa get~
3 Cia
s_—a—a
Cpflaaaaanaaaaa..
(4.a~z1 xi fl—i
U~sing equations (4.311, (4.351 an≤i (4.371 we get
(4.38) a I~i - __________
ci1—~÷tQ_ ~a’a~?q1u1
61
—~51ri--a(u~1i-uj
traing th~ property af ejui~ta1aat ~ontiziuad fractiozis w~ get
(4.321 = 1 (u~ ~
UI
U.3
—(u~÷a3) —
(4.40i S 1 a11 ÷ — +
if we. give u~ (a 1, 2, 3, ———, n) various valuesa
and: modify the contiruied fractioa ui equation (4.4(1), ~ can
derive a. variety of forma of coatiaued: fractious. if we let
ii. v~ we get.a a
(4.42.1 S~ ____ ______ ______
—(ii ÷~ )a—i xi.
E~cLuatiaa
praparty
(4.391 may be simplified even raore by again using the
of equivalent caat~riued fractions arid: we get
Lia—an
— ~%—i + Li)
w~iereSt —~rx÷vx~.~-vrx3÷-— —
ni a a a
If we let u~ ___ wea
a
a a aV~. .~x(4.4ai S” ___ _______ a rL—L.a (v~x + v~) — (VaX + — •4VaiX+ 1t~)
a + ___
a V1 ~ta ~a
a.aaiaa ,wageta
ax hex~4.4aY Si” — ____ 1a a~.
a ~
L
— ÷ a~xIa
a x. a,a~~—--a x‘where. s’ ~ — 1. ÷ .L. ~. + .L.
a — b. hba
it shatili be z~em ioered~. that~ an~ series that. we c~oa-~
vert tc~ a. c~ontinued. fraetiorz will d~tarm.ine the can rgence
G~; di~raen~e of the e~ui~,a2entt eontinued fractiaa. If the
series ~oa~re~rge~a to a limit, the oaritinuad. fraction eanverges
to the same limit. If the se~iea divarg~s, the continued.
f~acticiri diverges. This. fallows from. the definition, of
equivalance • The series u•al in the ahave corwersiaxis ma
he series of constants or’ functional series.
63
CQnVe.rSLOU of. a Contimied Product, into a Continued. F~a.ction
The. section on the conversion of series into. continued..
fractions enables u.s to c o.nstru.c.t continued, fractions, of the..
fcirrn given by equation C4.. 31), whese. first a convergents
shall be any given quantities f~ f2, -,
respectively. All. we. need to do is replace. by f1, u~ by
fa—fi,~Tharequire.d.
fraction is
f f f f. Cf f)i. a i ~ a1. f~ — (f~—flI —
~n—3~ ~n. —
(f—f)n—2.
From. equation (4.441 ~we. may express. any continued produ.ct
of the f orm~
d d d -d(~ 45) P = ~2 3 fl
xi a e a1 2 3 fi.
d d. iBy- replacing f1~ by 1 ~a by 1 2. ,
a1 a1 a2
d..df by -L 2. a an~ again using the property
a a —a1 2 a.
of equivalent continued fractions we. ge.t~
(4.4~6J = ____ ea) Ya~a(d2c13
~(3~ ea~
—
a—1~n-.a ~n—2~ ~SL
BrBLIR~y~
ClJ G. W. Cr~sta1, ~ext. Rook of: AIg&nr~’, London, &. andC. Biac1~, 1900.
LaJ Eall and. K~iigrtt, igher Algebra, London, I~t[acmillaaand. C~cimparty-, iaai..
EaJ L.. M~ Milne. — TIiOUiSQZi, The. Caic1ilnp of. Finite. D.iff’-.erences, Laiiciozi, Ma cmiilaii and.C.ampan~y, 1933.
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ACKNOWDGME~
I wish~ ta expre~ss. mj gratitud~. anã~ sincere thanks to
Dr. Jose~pK. T Dannis for~ his kelp, guidance, suggestions,
arxd c.onstru~tive critihis~.. ~itkout these a~d narly ether
carisidaratLous, this treatise w~oulc1 riot. have. been possible.
Lawrence Edgar iazicey