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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