Tim th eo reti c al part o f this little book i s an elementary
o f b e xposition of the nature the num er concept, of the posi
o f a o f b tive integer, and the four artifici l fo rms num er w v th e “ b hich, with the positi e integer, constitute num er ” th f b v i z . e system o alge ra, the negative , fraction, the irra
h e o f i nal . T t o , and the imaginary discussion the artificial
a numbers follows, in gener l, the same lines as my pam i phlet : On the Fo rms of N umber ari si ng i n Co mmo n
Al ebra but x g , it is much more e haustive and thorough
n Th e v o ne b goi g . point of iew is the first suggested y P b eacock and Gregory, and accepted ymathematicians gen erally since the discovery of quaterni ons and th e Ausdeh nun sleh re G a b a fi g of rassmann, that lge r is completely de ned formally by the laws o f combinati o n to which its funda b mental Operations are su j ect ; that, speaking generally,
' these laws alo ne define th e Operations and the operations
b t bo the various artificial num ers , as heir formal or sym lic
. T v for results his doctrine was fully de eloped the negative,
b H Co m lexe the fraction, and the imaginary y ankel, in his p
’ a hlens stemen 1 867 and e m b C Z y , in , mad co plete y antor s b o f 1 871 but eautiful theory the irrational in , it has not as yet received adequate tre atment in English . Any large degree o f originality in work o f this kind is f I naturally out o the question . have borrowed from a i v PR E FA CE .
Pe G n g reat many sources , especially from acock, rassman , C and T Theo ri e d er' Hankel, Weierstrass , antor, homae (
einer co m lemen era nd erli hen I ana lyti schen Functi o nen p V c ) . o r may mention, however, as more less distinctive features um b i 1 Of my discussion , the treatment of n er, count ng
a 4 th e and the equ tion , and the prominence given n f b t laws o f the d etermi nate e ss o su trac ion and division . Much care and labor have been expended o n the hi s
T o ut to ri cal chapters o f th e book . hese were meant at the set to contain only a brief account o f the origin and history
a But I l d no t b o f the artifici l numbers . cou ring myself to ignore primitive counting and the dev elopment of numeral I notation, and soon found that a clear and connected accoun t o f the origin o f the negative and im aginary is possible only when embodied in a sketch of the early his I tory o f the equation . have thus been led to write a r’ é sumé of th e history of th e most important parts of ele mentary arithmetic and algebra ’ Moritz Cantor s Vo rlesung en uber di e Geschi chte derMa the
mati k . b fo r , Vol I, has een my principal authority the h n 1 2 A D . Fo r t e v i . 0 e t e 0 . tire period which it co ers, to
I sa o n th e 1 200 1 600 I e little have to y period to , hav
fl b m b o n depended chie y, though y no eans a solutely, H ankel : Z ur Geschichte der Mathematik i n Altertum und Th f m M ittela lter . e remainder o y sketch is for the most
o n th e part based original sources .
HE NRY B FINE . . A 1 89 1 . PRINCETON , pril ,
In n f this second editio 3. number o important corrections n hav e b een made . But there h as bee no attempt at a f complete revision o the book .
Y FINE . HENR B .
I S 1 902 . PR NCETON , eptember, C ON T E N TS .
THEOR E TIC A L I . .
I I V IN TH E S . I . PO T E TEGER
Th e number co ncept
Additi o n and its laws
A ND NE GATIv E I II . SUBTRACTION THE NTEGER .
o Numerical subtracti n . Determinateness o f numerical subtractio n
Limitatio ns o f numerical subtractio n
o f a o i a o Principle perm nence . S ymb l c subtr cti n Zero
Recapitulatio n o f th e arg ument o f the chapter
“ III . DIVI SION AND THE FRACTION .
Determinateness o f numerical di visio n Fo a o f d vi rm l rules i si o n . Limitatio ns o f numerical di vi sio n
Symbo li c di visio n . Th e fractio n Negative fractio ns General test o f th e equality o r inequality o f fractio ns Indeterminateness o f di visio n by zero vi CON TE N TS .
Determi nateness o f symbo li c di visio n Th e vanishing o f a pro duct The system o f ratio nal numbers
V TH E I I . RRATIONAL .
Inadequateness o f the system o f ratio nal numbers ” ul a i a o a Numbers defined by reg r sequences . The rr ti n l d d n o o f z o o a Generaliz e efi iti ns er , p sitive , neg tive Of the fo ur fund amental Operatio ns Of equality and g reater and lesser in equality Th e number defined by a reg ular sequence its limiting value Divisio n by z ero
The number- system defined by reg ular sequences o f ratio nals a c lo sed and c o ntinuo us system
V . TH E IM MP X N M . AGINARY . CO LE U BERS
Co mplex numbers
The fundamental Operatio ns o n complex numbers . Numerical compariso n o f co mplex numbers Adequateness o f the system o f c omplex numbers F f undamental ch aracteristics o f the algebra o number .
V I P I R P I N M . THE V . I . GRA H CAL E RESENTAT ON OF U BER S AR ABLE
Co rrespo ndence between the real number - system and the po ints o f a line Th e co ntinuo us vari able
C o rrespo ndence between the co mplex number- System and the p o ints o f a plane Defin itio ns o f mo dulus and argument o f a c omplex number and o f o and a a r o f an a 45 48 Sine , c sine , circul r me su e ngle ,
D o a o a a £6 c o s 0 i 0 ei 9 45 48 em nstr ti n th t p ( sin ) p , o o o f th e o n th e sum d ff C nstructi n p i ts which represent , i erence , o d c and o o f two o 46 47 pr u t, qu tient c mplex numbers ,
VII . TH E M OF A FUNDAMENTAL THEORE LGEBRA .
Definitio ns o f the algebraic equati o n and its ro o ts Demo nstratio n that an algebraic equati o n o f th e nth degree h as CON TEN TS . vii
I N M D I D I N . VII . U BERS EF NE BY NFI ITE SERIES PAGE
R S I S . I . EAL ER E D n o o f sum o and di efi iti ns , c nvergence, vergence . General test o f
A so and o d o a o b lute c n iti n l c nvergence . S a s o f o . peci l te ts c nvergence 0 0 0 0 0 0 0 0 C O O - 1 0 0 0 0 0 0 0 0 0 Limits o f co nvergence Th e fu da a o a o o n i n ment l per ti ns infin te series . Q O O O Q O O O O O O O O O
II . M P X I S CO LE SER E . test Of o n r a c ve enc e o o 0 - 0 Gener l g o o c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 A o and o d o a o bs lute c n iti n l c nvergence . Th e regio n o f co nvergence A theo rem respecting c omplex series The fun d amental o peratio ns o n complex series
IX THE ExP ONE NTI A L A N D L M I F . OGARITH C UNCTIONS.
M D I I S . IN V I A ND V TH E UNDETER INE COEFF C ENT OLUT ON E OLUTION . I M I M B NO AL THEORE .
f n o Definitio n o f u cti n . Functio nal equatio n o f th e expo nential functio n d d o ffi Un etermine c e cients . Th a f o e exp o nenti l uncti n . Th e functio ns sine and
Perio dicity o f these functio ns . Th e lo garithmic functio n Indeterm inateness o f lo garithms Permanence o f th e laws o f expo nents Permanence o f th e l aws o f l o garithms
Invo lutio n and evo lutio n . r The bino mi al theo rem fo co mplex expo nents .
H I T R C A L II . S O I .
I I MI IV N M S . . PR T E U ERAL
Spo ken symb o ls Written symb o ls I I S S MS O F N . II . H STOR C Y TE OTATION
E gyptian and Phoeni cian
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
II I. TH E F I RACT ON . Primitive fractio ns
R o man fractio ns . E gyptian (the Bo o k o f Ahmes) Babyloni an o r sexagesimal
IV . O I I OF THE I I R G N RRAT ONAL .
a o a . Discovery o f irratio nal lines . Pyth g r s C o nsequences o f this disco very in Greek mathematics Greek appro ximate values o f i rratio nals
I O F N V A O I ND IM I . V . R GIN THE EGAT E THE AG NARY
H T E E QUAT ION . Th e equatio n in Egyptian mathematics In th e earlier Greek mathematics Hero o f Alexandria Di o phantus o f Alexandria A h tt Th e I di a a a . r ab a a B a a a B a a a n n m them tics y , r hm gupt , h sk r Its algebraic symb o li sm Its inventio n o f th e negative t u o f z o I s se er . Its use o f irrati o nal numbers Its treatment o f determinate and indeterminate equatio ns Th e A a a Alkh wari zmi Alk ar h i Al h m a a . c c a i r bi n m them tics , , yya A a a a a I d r bi n lgebra Greek r ther than n ian . Mathematics in Eur o pe befo re th e twelfth century G erb ert
E a c o f th e A a a a a L o a do ntr n e r bi n m them tics . e n r Mathematics durin g th e ag e o f Sch o lasticism Th e R a a S o o o f th e and ad a a o en iss nce . luti n cubic biqu r tic equ ti ns Th e a a a o f d F a a a o f neg tive in the lgebr this perio . irst ppe r nce th e imaginary o Algebraic symb o lism . Vieta and Harri t nd d Th e fund amental theo rem o f algebra . Harrio t a Girar C ON TE N TS . i X
O F N V I I . A P I V CCE TANCE THE EGAT E , THE GENERAL RRATIONAL , A N D T H E IMA G INARY A S N UM BERS . o
’ ’ Descartes Gé o métri e and th e negative ’ Descartes geo metric algebra
a a N o . E Th e co ntinuous v ri ble . ewt n uler
Th e general irratio nal . Th e a a a o z d a a a r im gin ry , rec gni e n lytic l inst ument ’ Argand s geo metric representatio n o f the imaginary
Gauss . Th e co mplex number
R OF S M I OF VII . ECOGNITION THE PURELY Y BOL C CHARACTER
ALGEBRA .
I S . TH E SD N S QUATERN ON AU EHNU G LEHRE .
f ma a o The principle o per nence . Pe c ck ”
h f d a a o f a a S o a a a . T e un ament l l ws lgebr . ymb lic l lgebr s Grego ry ’ Hamilto n s quaternio ns ’ Grassmann s Ausdehnungslehre d d o f ia fo m f m Th e fully develo pe o ctrine the artific l r s o nu ber .
H . W a a G . a o . nkel eierstr ss . C nt r Recent literature
F OTN OTES PR IN C IPA L O .
Instances o f quinary and Vigesimal systems o f no tatio n Instances o f digit numerals Summary o f th e histo ry o f Greek mathematics Old Greek demo nstratio n that th e Side and di ago nal o f a square are incommensurable . o d o f a Greek meth s ppro ximatio n . Dio phantine equatio ns ’ Alch ayyami s metho d o f so lving cubics by the intersectio ns o f c Oni c s
Jo rd anus Nemo rarius . Th e S umma o f Luca Paci o li Regio mo ntanus A lgebraic symbo lism . Th e a o a o f and L d a irr ti n lity e 1 r. in em nn
. THE OS T E NT I P I IV I EGER ,
‘ AND T HE LAWS WHIC H REGULAT E T HE ADDIT IO N AND A O N F POS IT I E INT ER M ULT IPLIC I O EG S . _ T V A35 7 8 . 0 di i i N mbe . sa ai 1 . u r We y of cert n st nct th ngs that they * form a gro up when we make them collectiv ely a single obj ect o f o ur attention . Th e number of thi ng s i n a group i s that property o f the group which rem ains un changed during every change in ' the group which does not destro y th e separateness o f th e things from o ne another o r their common separateness from all other things . Such changes may be c h ang es in the characteristics of the A a . a things or in their rrangement within the group g in , changes o f arrangement may be changes either in the order o f the things o r in the manner in which they are associated with one another In smaller groups . We may therefore s ay The number of thi ng s i n a ny g r o up of di sti nct thi ng s i s i nde end ent o the cha r acters o th se thi n s o the o rd er i n p f f e g , f whi ch the ma be a rr a n ed i n the ro u and o the ma nner y y g g p, f i n whi ch they ma y be asso ci a ted wi th o ne ano ther i n sma ller g r o ups .
N me E Th 2 . u rical quality . e number o f thi ngs in any o f d a a two groups istinct things is the s me , when for e ch o o ne d an d thing in the first gr up there is in the secon , r a a d r recip oc lly , for e ch thing in the secon g oup , one in the
first . , T th e mb ro A B C hus , nu er of letters in the two g ups , , , ; D E F . d a , , , is the Same In the secon group there is letter
at B o w e m a ni te o a o ne a o b e y g r up e n fi gr up , th t is , which c nn t
o o o n e - t - ne br ught int o o c o rrespo ndence 2) with any part o f itself. M R —S S M F A 4 N U BE Y TE O LGEBR A . which may b e assigned to each of the letters in the first ' to A E B F C an d a a as D , to , to ; reciproc lly, letter in the fir st which may be assigned to each in the second : as A to
B E C F. D, to , to — Two groups thus related are said to be in o ne- to - o ne (1 1 ) co rrespo nd ence. U nderlying the statement just made is the assumption that if the two groups correspond in the manner described
a for one order of the things in e ch, they will correspond if be an i n the things taken in y other order also ; thus , the x n E ad o f D b e a A e ample give , that if inste ssigned to ,
e ll n be D E F vi z . D o r ther wi agai a letter in the group , , , F B and C and re c i ro , for each of the remaining letters , p
T i 1 - an a . cally . h s is immedi te consequence of , foot note Th e number of things in the first group is g rea ter than in d b that the secon , or the num er of things in the second less than o ne that in the first, when there is thing in the
n but no t first group for each thing in the seco d, reciprocally o ne in the second for each in the first .
i mer mbo s . As d b 3 . Nu al Sy l regar s the num er of th ngs a be d which it contains , therefore , group may represente f b e . o y any other group, g the fingers or of simple marks , ’ d a o f I s, which stan s to it in the rel tion correspondence Thi t o f re re described in § 2 . s is the primi ive method p a and senting the number of things in group , like the b a al modern method, makes it possi le to comp re numeric ly groups which are separated in time o r space . f Th e modern method o f representin g the number o things ubstitu in a group differs from the primitive only in the s m a d as o ne 1 2 3 . o r tion o f symbols , as , , , etc , nu er l wor s, ,
fo r o f a I ll two three . , , , , , etc , the various groups m rks III T b o f etc . hese sym ols are the positive integers arith metic . h n umber o thin s i n a A pos iti ve i nteg er i s a symbo l fo r t e f g g rqup of di sti nct thi ng s . 5 THE P OS I TI VE IN TEGER .
Fo r convenience we shall c all the positive integer which represents the number o f thi ngs in any group its numeral .
o r f b symbol, when not likely to cause con usion, its num er — b use d simply, this eing, in fact, the primary of the wor ” number i n arithmetic . th e a o f v In the following discussion, for s ke gi ing our ll statements a general form, we sha represent these numeral
b a b 0 . symbols y letters, , , , etc
n b o f The E o . The 4. quati n umeral sym ols two groups being a and b ; whe n the number of things in the groups is i x b e uati o n the same , th s relation is e pressed y the q
a = b ;
r n b when the fi st group is greater tha the second, y the i nequa li ty a b
fir b th e i ne when the st group is less than the second, y qua lity
a b.
A numeri ca l equa ti o n i s thus a d ecla rati o n i n terms of the numer a l symbo ls of two g ro up s a nd the symbo l tha t these g ro ups a re i n o ne- to - o ne co r respo ndence
h Co i . T e da l f m 5 . unt ng fun menta operation o arith etic is counting .
To count a group is to set up a one - to - one correspondence between the individuals o f this groupand the individuals o f some representative group . Counting leads to an e xpression for the number o f things in any group in terms of the representati Ve group : if the
a be t o a o represent tive group the fingers , gr up of fingers ;
a to a o f th e a o d o r if m rks , group marks ; if numer l w r s b f o o d o r bo . sym ols in common use , to one these w r s sym ls There is a difference between counting with numeral
d a nd a d due a wor s the e rlier metho s of counting, to the f ct M R - s YS TE M OF A R A 6 N U BE L GEB .
a d a ai z that the numer l wor s h ve a cert n recogni ed order .
As in fing er- counting one finger is att ached to e ach thi ng o u d so d but a d c nte , here one wor ; th t wor represents numer i c all i a a d but e y not the thing to wh ch it is tt che , the entir Th m group of which this is the last . e sa e sort of coun ting may be done on the . fin gers when there i s an agreement as to the order in which the fin gers ar e to be used ; thus if it were understood that the fin gers were always to be taken d b n fi in normal or er from thum to little fi ger, the little nger
d be mb fo r 5 as woul as good a sy ol the entire hand .
A i o . o r be b 6 . ddit n If two more groups of things rought so a b together as to form a single group, the numer l sym ol Of this group i s c alled the sum ”o f t h e numbers of the sepa rate groups .
If sum he s an d b o f a the , the num ers the sep rate groups m a b c . a b s , , , etc , respectively, the rel tion etween them is y bo li c ally e xpressed by the equation
where the sum - group is Supposed to b e formed by j oining — b b — the second g roup to which elongs to the first, the — hi 0 b — n third group to w ch elongs to the resulti g group, and so on .
3 a b . Th e Operation of finding when , , c, etc , are known, is
a ddi ti o n .
Addition is abbreviated counting .
Addition is subj ect to the two following laws , called the
i ti ve v V iz co mmutati ve and a sso c a laws respecti ely,
T a i s to a b. I . o add b to the same as add to
o sum b c to a as II . T add the of and is the same to add n 0 to the sum o f a a d b.
8 M R- S S M OF A R A N U BE Y TE LGEB .
e b m a i ve b ba . T d b ba qual to , or herefore , y the com ut t v fo r n and associati e laws additio .
i The Asso ci t e La w. IV. a v
a bc c sums such - as (a a to b terms) a + a + a + to bc terms (by th e associativ e law fo r addition) a (bc)
: The Di stri uti e w V b v La .
a + a + a + m to (b + c ) terms (a a + to b terms) (a a to 0 terms) b ( y the associative law for addition) ,
a b a c .
Th e t b commutative, associative, and dis ri utive laws for sums o f any number Of terms and products of any number — T d o f factors follow immedi ately from I V. hus the pro uct o f a a b c d an the f ctors , , , , taken in y two orders, is the same, since the o ne order can be transformed into th e other by u successive interchanges of consec tive letters .
T TI N A D THE E AT E T II SUB RAG O N N G IV IN EGER .
m r b r c o . C 8 . Nu e ical Su t a ti n orresponding to every math emati c al n n operation there is another, commo ly called its i verse x d a , which e actly un oes wh t the Operation itself does .
b a a o ad and Su tr ction stands in this rel ti n to dition, division to multiplication . To subtra ct b from a is to find a number to which if b be
sum be a . Th e W a b b added, the will result is ritten , y n a definitio , it identic lly satisfies the equation
( a S B R A U T CTI ON. 9
a i s sa a b u b b th t to y, is the n m er elonging to the group
b- a - which with the group m kes up the a group . Obviously subtraction is always possible when b is less
an a but . U add I n a a a th , then only nlike ition, e ch pplic tion o f this operation regard must be h ad to the relative siz e of the two numbers concerned .
D m s of N meri a . b r Subtra 9 eter inatenes u c l Su t actio n . c
b d etermi na te T IS tion, when possi le , is a operation . here but o ne b a: b : a num er which will satisfy the equation ,
' o ne b sum o f and i but num er the which b s a . In other
- d a b o ne a d . wor s, is v lue ‘ Fo r c d b a a: b a if and oth satisfy the equ tion , since th en c b = a d b = a th at i s a o ne - + and + , , to o ne correspondence may be set up between the individuals o f the (c + b) and (d b) groups Th e s ame sort o f b an b d d a correspondence , however, exists etween y in ivi u ls o f the first group and any b individuals o f th e second ; it b n c o f must, therefore, exist etween the remaini g the first
and d o f d o r c d . the remaining the secon , This characteristic of subtraction is of the same order o f
a and a at and importance as the commut tive ssoci ive laws , we shall add to the group of laws I— V and definition VI as b i e i n the eing, like them, a fundamental pr ncipl follow ing discussion the theorem
If a + c = b + c = b a , which may also be stated in the form : If o ne term of a sum
h h e a a . c anges while t other rem ins const nt, the sum changes Th e a same reasoning proves, also, th t
An o f sub 1 orm s of b c i o . 0. F al Rule Su tra t n the rules traction are purely fo rma l consequences o f the fundamental —S M F A 1 0 N UMBER YS TE O LGEBRA .
I
— w I d . T la s V, VII, and efinition VI hey must follow what
a o f b a b c a ever the me ning the sym ols , , , a f ct a a which h as an import nt be ring on the following discussion . It will be sufficient to consider the e quations which fo l
Fo r b o f low . , properly com ined, they determine the result any series of subtractions o r of any complex operation made f b and o . up additions , su tractions , multiplications
— — = — — 1 . a b c d c b.
— — z — 2 . a b e a b c ( ) + .
3 . a b b a .
— = — 4. a + (b c d c + b. — = — a b c a b . 5 . ( ) ac
a b Fo r 1 . c is the form to which if first c and then b be am b added ; or, what is the s e thing ( y I) , b and o r n first then c ; , what is agai the sam e
b b at - sum thing ( y II) , c once , the pro ‘ a b duc e d is a (by VI) . c is therefore the a b same as c , which is as it stands the b c be sum form to which if , then , added the is
a as a b c ; also the same ( + ) , which is the b e th e um i form to which if b c added s s a .
a — b — c a — b — c — c c 2 . ( ) ( ) + , a — (b — — a b + c .
3 . a + b But — = a + b b a . — a + b c + c) — — a + (b c )
— — a b ac = a (b c + c ) — a b — c a c — a c ( ) + , — — a (b c ) . S UB TR A C TI ON . 1
Equation 3 is part icularly interesting in that it defines as E 1 addi tion the inverse of subtraction . quation declares
b ma a are that two consecutive su tractions y change pl ces,
. E 1 2 4 law commutative quations , , together supplement
a as ad II, constituting with it complete sociative law of di tion and subtraction ; and e quation 5 in like manner supple ments law V.
L mi i f o o m r cal b a . 1 1 . i tat ns Nu e i Su tr ctio n Judged by a 1 — 5 b a o i s xa u a o f the equ tions , su tr cti n the e ct co nterp rt nf a ddition . It co orms to the s ame general laws as that and o d a be ad operation, the two c ul with f irness m e to interchange their roles of di rect and inverse operation But this equality proves to be onl y apparent when we Th attempt to interpret thes e equations . e requirement that subtrahend be le ss than minuend then bec o mes a ak the serious re striction . It m es range Of subtraction much
d a narrower than that o f addition . It ren ers the e qu tions — f 1 5 a b fo r a o a o f a b c o . av ila le speci l classes v lues , , nly be d o n so n as If it must insiste , even simple an i ference
a a a b 2 b a b c a n be d th t ( ) is equ l to n ot rawn , and the use o f subtraction in any reckoning with symbols who se relative values are not at all times known must be pro no un c ed unw arranted . One is thus naturally led to ask whether to be valid an a b ni be ab e a and lge raic recko ng must interpret l numeric lly , if . no t to subtra ti o n th e c , to seek free p and rules of re koning with the results o f subtr action from a restriction which we have found to be so serious .
mb ri of erma c . S m o E a i o s . c 1 2 . Sy lic qu t n P n iple P nen e y
' b li c a e o f th i s o ne u o Subtraction . In pur su nc inquiry t rns
r a b b a as a fi st to the equation ( ) , which serves definition o f subtraction when b is less than a . This is an equation in th e primary sense 4) onl y when a b But th e b t b . is a num er in roader sense , tha — M 1 2 N UMB ER S YS TE OF A LGEB RA .
An equati o n i s a ny d ecla r a ti o n of the equi va lence of d efi ni te co mbi na ti o ns of symbo ls — equi va lence i n the sense tha t o ne
ma substi tuted o r the o ther y be f ,
a b b a ma ' be an v a ( ) y equation, whate er the v lues
o f a and b.
And d a n h as b a a b if no ifferent me ni g een att ched to , and it is declared that a b is the symbol which associated
b b a — b b v a with in the com ination ( ) + is equi alent to , o r e ua ti o n this declaration , the q
(Cl
* de ni ti o n is a fi of this symbol . By the assumption o f the permanence of fo rm of th e num erical equation in which the definition o f subtraction
o ne d o f resulted, is thus put imme iately in possession a
s mbo li f n y c definition o subtractio which is general . Th e numerical definition is subordinate to the symbolic b n o f definition, ei g the interpretation which it admits when b is less than a .
But o f b fin from the standpoint the sym olic de ition, inter pretabili ty — the question whether a — b i s a number o r — not is irrele vant ; only such proper ti es may be attached a b b flo w to , y itself considered, as immediately from the generaliz ed equation (a
o f I— V In like manner each the fundamental laws , VII o n the assumption o f the perma nence of i ts fo rm after it d be b l b a has cease to interpreta le numerica ly, ecomes declaration of the equivalence o f certain definite combi . n o f b a o f natio s sym ols, and the form l consequences these laws — th e equations 1 — 5 o f 1 0— become definitions o f a di b i n and a d tion, su traction, multipl catio , their mutu l
A d i addi o Th o s o f o no t a . e efiniti n in term symb l c , numeric l ti n c an o f o n d a m a addi o o o Sign , c urse , i ic te nu eric l ti n nly when b th th e symb o ls which it co nnects are numbers . S UB TRA C TION . 1 3
— defini ti o n s are b ma relations which purely sym olic, it y r d e but a e a a . b , which unrestricte in their pplic tion These d efinitions ar e legitimate from a logical point of w Fo r are a I— and V ie . they merely the l ws VII, we may assume that these l aws are mu tually c o nsi stent since we have
d a d d fo r o . H prove th t they hol goo p sitive integers ence , if u sed c o rrectl o o y, there is no m re p ssibility of their lea ding to false results than there is of the more tangible numerical d d to efinitions le a ing false results . Th e l a ws of c o rrect
nk b b a thi ing are as applica le to mere sym ols s to numbers .
a a b d a Wh t the v lue of these sym olic efinitions is, to wh t e xtent they add to the power to dra w inferences concerning
umb a b a ab da n ers , the element ry alge r un ntly illustrates . One o f their immediate c o nse quences is the intr o duction
a b a o f b zero and ne ative into lge r two new sym ols , the g , b a l m which contri ute gre t y to increase the simplicity, co pre h ensi v eness and i ts a , power of Oper tions .
r W b set a o . a 1 3 . Ze hen is equ l to in the general equation a b b a ( ) , it takes o ne of the forms
a a a a ( ) ,
(b b) b b. It may proved that
a — a = b — b
(a Law
a b ,
(a a ) a a .
b) (a + b) _ (b Laws — b + d (b b) b T herefore a a b b. — 1 4 N UMBER S YS TEM OF ALGEB R A .
a — a is therefore altogether independent o f a and may h properly be represented by a symbol unrelated to a . T e
b 0 l zero . symbol whi ch has een chosen for it is , cal ed
Additi o n is defined for this symbol by the equations
n f 0 O a a O . 1 . , defi ition L w a . a 0 a . I
ubtra ti n b th e n S c o (partially) , y e quatio
2 . Ct O a .
Def . . Fo r (a 0) O a . VI
Multi li ati o n b th e e p c (partially) , y quations
0 O a O. 3 . a x x
Fo r a 0 a b b o f O. x ( ) , definition
a b a b 1 0 5 . , ,
o f . 0. definition O
T N b n he v . a e 1 4. eg ati e When is greater tha , qual say a d so a b — a = d to + , th t , then
a — b = a
— — a a d,
— — 0 d . ni defi tion of 0. For 0 d the briefer symbol d has been substituted
a V o f a with propriety, cert inly, in iew the lack Of signific nce f Th o 0in relation to addition and subtraction . e equation O — d — d m o f sub , moreover, supplies the issing rule
0. C a 1 3 traction for ( omp re , Th e b d i s ne a tive and i n sym ol called the g , opposition
b d o s iti ve. to it, the num er is called p Though i n its origin a Sign o f operation (subtraction from the Sign is here to be regarded merely as part f o the symbol d .
d i s e b b a b a b as s rvicea le a su stitute for when , i s as a single numeral symbol when a b.
1 6 N M R — S S M OF A E U BE Y TE LG BRA .
= O b o f ( ) , definition O. — a b — a b 1 0 5 ) ) , § , . O . 1 4 6 5 o r C . , , and ,
— a) ( b) = a b
— a b — b ) ( ) ,
— a — b ) ( ) , — a b + < — a i < — b>
— a 1 4 1 ) ( § , ;
m o f t By this ethod one is led, also, to definitions equali y u i t o f T and greater o r lesser i neq al y negatives . hus
— o r — 9 . a b,
* according as b o r a .
b a For as > , < ,
' a — b Law o r , VII VII .
In like manner a 0 b.
h R c a i o . Th e t e 1 5 . e pitulat n nature of argument which has been de veloped i n th e present chapter should be care fully Observed . F ni o f th e v rom the defi tions positi e integer, addition, b v a v and and su traction , the associati e and commut ti e laws Th as m the determinateness of subtraction followed . e su p o f a b as d b tion the permanence of the result , define y
a b b a o f a b n ( ) , for all values and , led to defi itions
On th e o a d a a d to numeric a ll a a a ther h n , is s i be y g re ter th n , equ l t o o r a b acco d n as a is s f a a a to o r , less th n , r i g it el gre ter th n, equ l , a b less th n . S UB TRA CTI ON 1 7
b 0 d z an d a and o f the two sym ols , , ero the neg tive ; from
I - V VI the assumption o f the perm anence of the laws , I dd b a andI were derived definitions of the a ition, su tr ction , b — b multiplication of these sym ols , the assumptions just sufficient to determine the me anings o f these operatic
unambiguously . a I—V and In the case of numbers, the l ws , VII, definition VI were deduced from the characteristics o f numbers and the definitions o f their Operations ; in the case o f the sym
O d o n d a o f bols , , the other han , the ch racteristics these symbols and the definitions o f their operations were deduced
from the laws . With the acceptance of the negative the character o f * a ad arithmetic un dergoes a radical change . It was lre y in x d and i n e uali a sense symbolic, e presse itself in equations q
and a l . But ties , investig ted the resu ts of certain Operations b a n and a o all b its sym ols, equ tio s, Oper ti ns were interpreta le f a i b o f in terms o the re l ty which gave rise to it , the num er
f o things in actually e xisting groups o things . Its c nnec tion with this reality was as immediate as that o f the ele x mentary geometry with actually e isting space relati o ns . h v But the negative severs this connection . T e negati e is a symbol for the result o f an Operation which cannot be d a o f effecte with actu lly existing groups things , which is , An f nd m n b . d u a e therefore, purely sym olic not only do the tal Operations and the symbols o n which they are performed a a a d all lose reality ; the equ tion , the fund ment l j u gment in
a a a . F b mathem tical re soning, suffers the s me loss rom eing
a declaration that two groups o f things are in o ne - to - o ne d b a d a d correspon ence , it ecomes mere ecl ration regar ing
” b a b a i o ne two com in tions of sym ols , th t in any reckon ng
may be substituted fo r the other .
In this co nnecti o n see 25 . 1 8 M R—S S OF A N U BE Y TEM L GEBRA .
. D O AND T III IVISI N HE FRACTION . m D 1 6 . v s o Th e Nu erical i i i n . inverse operation to multi
plication is division . To v b b di ide a y is to find a number which multiplied by b a Th e . th e a b b produces result is called quotient of y , and is written 3 By definition
. b VIII a .
L bt d a be ff ike su raction, ivision c nnot always e ected .
Only in e xceptional cases c an the a- group be subdivided i b nto equal groups .
D 1 . rmi D 7 ete nateness o f Numerical ivisio n . When divis be ff but ion can e ected at all, it can lead to a single result ; i s it determi na te. Fo r there can be but o ne number the product of which b b a e y is ; in oth r words ,
Cb d b If ,
* 6 d .
Fo r b groups each containing 0 individuals cannot be e qual to b groups each containing d individuals unless c d
T be his is a theorem of fundamental importance . It may d a c alled the law o f determinateness o f division . It ecl res a o ne o f be d th t if a product and its factors etermined, the remaining factor is definitely determined also ; or that if o ne o f the factors o f a product changes while the other
a . a rem ins unchanged, the product changes It alone m kes v Th e di ision in the arithmetical sense possible . fact that
Th e a b O d d 0no t a th e c se is exclu e , being number in sense in a o d d which th t w r is here use . DI VI SI ON AN D THE ERA C TI ON . 1 9
no t d fo r b 0 but it does hol the sym ol , that rather a product remains unchanged (being always 0) when one of i ts factors
0 . o a be a d is , however the ther f ctor ch nge , makes divi sion b O b d n a y impossi le, ren ering u justifi ble the conclusions c an be a o f d which dr wn in the case other ivisors . Th e reasoning which proved law IX proves also that
AS cb o r db > < ,
c > o r < d .
orm R s of D i v s o . Th 1 8 . F al ule i i n e fundamental laws o f the multiplication o f numbers are
Oi l) ba ,
. a be a bc IV ( ) ,
V .
Of these , the definition
unless
ad and b and the corresponding laws Of dition su traction ,
Of d ar e o rmal d d the rules ivision purely f consequences , e u cible precisely as the rules of subtraction 1 — 5 o f 1 0in the d T d prece ing chapter . hey follow without regar to the 6 ni o f b a b c a b — T mea ng the sym ols , , , , 5 hus
d bd
bd = 9 b d L ’ aws IV, III . v R
c D f . a e . , VIII
D f . e . VIII — 20 N UMBER S YS TEM OF ALGEBRA .
b IX Th e theorem follows y law .
f De . VIII .
0 a d c 1 L w 1 8 a I . , , , v bc d b col S , b d L w Ix do 1 Def. a cd x VIII, cg
Th e theorem follows by law IX .
a a d j: be i c Z h bd
- - — a: bd b d i d b Laws III V § 1 0 5 . % g ; ,
a d bc Def. . ziz , VIII
Def. . bd a d j: be. VIII
Th e theorem follows by law IX .
By the same method it may be inferred that c 4. < ’ d
'
D f. L . bc . e VIII, aws III, IV, IX, IX
L m . o o f me D i o bol c D v s o . 1 9 . i itati ns Nu rical ivis n Sym i i i i n
The Fracti o n . General as is Of the preceding
a a ab equ tions , they are c p le interpretation only a c are b a a i a when 5 ’ 5 num ers , case of compar t vely r re occur
Th r rence . e narrow limits s et the quotient in the nume ical definition render division an unimport ant Operation as I I O AN D THE FR A I ON DI V S N CT .
add o a o r a z compared with iti n, multiplic tion, the gener li ed subtraction discussed in the precedi ng chapter . But the way which l ed to an unrestricted subtraction lies Open also to the removal of thi s restriction ; and the reasons v fo r following it there are e en more cogent here . o a di d d b b b We accept as the qu tient of vi e y any num er ,
no t 0 b 3 b n which is , the sym ol 2defined y the equatio
regarding this equation merely as a declaration o f th e
a o f b b a o f sub equiv lence the sym ols and , the right to
fo i stitute one r the other in any reckon ng .
Whether be a number o r no t i s to this definition i rrele
a, 0 a a b i s a d a ra cti o n and In vant . When mere sym ol, c lle f , i Oppo sition to this a number s called an i nteg er . We then put Ourselves in immediate possessi o n o f defi ni i n o f add o b a o n a and t o s the iti n, su tr cti , multiplic tion,
o f b as as o f a o f a division this sym ol, well the rel tions e qu l ity and greater and lesser inequality — definiti o ns which are consistent with the corresponding numerical d efinitions and with o ne another — by assuming the perm anence o f
o f a o 1 2 3 and 4 o f 1 8 as form the equ ti ns , , of the test b a a to be b as sym olic st tements , when they ce se interpreta le numerical st atements Th e purely symbolic character of g and its Operations d a a and ab etr cts nothing from their legitim cy, they est lish division on a footing o f at le a st formal equality with the f * other three fundamental Operations o arithmetic .
Th e do ctrine o f symbo lic divi sio n admits o f being presented i n the very same fo rm as that o f symb o lic subtractio n . Th e equatio ns o f Chapter II immediately pa ss o ver into theo rems 22 UMB R —S S M OF AL R A N E Y TE GEB .
N 2 ve r c o s. 0. eg ati F a ti n Inasmuch as negatives conform
a and I— a 1 2 3 an to the l ws definitions IX, the equ tions , , d
4 o f 1 8 are a an o f mb a b c the test v lid when y the nu ers , , , d
d b a v . o th e are replace y neg ti es In particular , it f llows from
o f a definition quotient and its determin teness, that
a a a a a a . . ’ ’ b b b b b b
a b e a s It ought, perh ps, to said th t the determinatenes f o f o di vision negatives has n o t been formally demonstrated .
Th e :t a :t c ziz b j: c theorem, however, that if ( ) ) ) ) , d: a j: b o f zt , follows for every selection the signs from
o ne b 1 4 6 8. the selection y , ,
2 1 G er Tes of E i or I l of r o s . . en al t the qual ty nequa ity F acti n 6 G v :t — i i en any two fractions 5, g a c :t o r :t b 01 according as :t a d or j: be . ’
4 1 4 . L . C a 9 aws IX IX omp re , , respecting divisio n when th e sig ns o f multiplicatio n and divisio n are d fo r o o f add o and a o n s o fo r a substitute th se iti n subtr cti , inst nce,
a a b C z a — c — b
a C
bc 6 9 In a l a to a a a a o s o d a a . a p rticu r, ( ) c rre p n s Thus purely a It a th e a rOle symb o lic defin itio n m ay b e given 1 . pl ys s me in multipli A ai h as th e sa o a a a ddi o . catio n as 0in a ti n g n , it me excepti n l ch r cter mul in invo lutio n an o perat io n related to multiplicatio n quite as tipli m — fo r 1 a catio n to addi tio n as O in multipli catio n ; wh tever the values o f m and n . 0 o r 0 a a z 0 o Si ilarly to th e a i o a a , , c rr m , equ t n ) ( ) e
1 a 1 a as a d o o f i f a o spo nd s , which nswers efiniti n the un t r cti n (a ) 1 and in m o f a o and all o f a o 7 ter s these unit fr cti ns integers ther r cti ns a may b e expressed .
24 N M R - S YS TE M OF A RA U BE LGEB .
o f which, therefore , the property numerical determinateness Th h as no possible connection . e s ame i s true Of the prod
uet o r di fler en c e and , sum of two fractions , of the quotient n b of o e fraction y another .
As b m a for sym olic deter in teness , it needs no justification d when assume , as in the case of the fraction and the d 1 —4 b d emonstrations , of sym ols whose efinitions d o not Th e preclude it . inference, for instance, that because
which depends o n this principle o f symbolic determinate n o f ess, is precisely the same character as the inference that c l‘ bot z b 9 d b d ,
which depends o n the associative and commutative laws . Both are pure a ssumptions made of the und efined symbol 9? C fo r the sake o f securing it a definition identical i n form b d f * with that o the product o f two numerical quotients .
T a b he h of ro . 2 4. Vanis ing a P duct It has alre dy een i 1 3 3 1 4 7 1 8 1 th at th e d shown , , , , , ) sufficient con ition for the vanishing Of a product is the vanishing o f one o f its
a F o f d f ctors . rom the determinateness ivision it follows
a i s d at th t this also the necessary con ition, th is to say
i h I a ro duct a ni sh o n o i ts a cto rs must va n s . f p v , e f f
Let m 0 a ma b o f y , where , y y represent num ers or any d the symbols we have been consi ering .
a muta ti s muta ndis a a o to sub These rem rks , , pply with equ l f rce ac o tr ti n . DI V1 SI ON AN D THE R A I O 25 F CT N .
m : 0 y ,
m azz wz y , o r
a be 0 whence, if not , o r
Th s m f o mb rs. T 25 . e o b Sy te Rati nal Nu e hree sym ols , 6 0 — d b d c an be d , , 5 , have thus een foun which reckone
b a as b and o f with y the s me rules num ers, in terms which I s b o f a d sub it possi le to express the result every d ition,
a n a d o n tr ctio , multiplic tion or division, whether performe
- b b a num ers or on these sym ols themselves ; therefore , lso, the result o f any complex Operation which c an be resolved b o f a into a finite com ination these four oper tions . In asmuch as these symbols play the s ame role as numbers da a o f a in relation to the fun mental Oper tions rithmetic, it Th ” a a b . e d b is n tural to cl ss them with num ers wor num er,
a a ab to o h as origin lly pplic le the p sitive integer only, come
z v and to apply to ero, the negati e integer, the positive nega
a a o f b b a tive fr ction lso, this entire group sym ols eing c lled * T ra ti o na l numbers . the system of , his involves, of course ,
ad a o f b o f a r ic l change the num er concept, in cons equence which numbers become merely part o f the symbolic equip
a a d m o f ment of cert in oper tions, a mitting, for the ost part , only such definitions as these Operations lend them .
It a d d b e a d a f a o z o and a h r ly nee s i th t the r cti n , er , the neg tive actually made their w ay into th e number - system fo r quite a different ” reaso n fro m this because they admitted o f certain “ real interpre t i n h f a o a o f th e a in d at o s t e r c i i s r s li s , g i i , t n n me u ement ne , ne t ve eb t where th e co rrespo nding p o sitive meant credi t o r in a leng t h measured to th e left where th e co rrespo ndi ng po sitive meant a lengt h measured to th e S a o o r o o d to right . uch interpret ti ns , c rresp n ences existing li e o d o f a are o d in th e things which entirely utsi e pure rithmetic , ign re present discussio n as being irrelevant to a pure arithmetical d o ctrine o f th e a a o o f rtifici l f rms number. 6 - M A 2 N UMB ER S YS TE OF ALGEB R .
a b as b a In ccepting these sym ols its num ers, rithmetic ce ases to be occupied exclusively o r even pri nc ipally w ith f b the properties o numbers in the strict sense . It ecomes an a lg ebra whose immediate concern is with certain Opera
d dd b a tions efined, as a ition y the equ tions a b c z a b c a ( ) , form lly only, without reference to * o f o n the meaning the symbols Operated .
THE T IV. IRRA IONAL .
T m f N Th e he o o mbe s d a . 2 6 . Syste Rati nal u r Ina equ te
o f b fi fo r system rational num ers, while it suf ces the four fun dame nt al Operations o f arithmetic and finite combinations of
d no t d o f a b a these Operations , oes fully meet the nee s lge r . Th e a a b o f a b a gre t centr l pro lem lge r is the equation, and that only is an ade quate number - system for algebra which supplies the means o f expressing the roots o f all possible
Th e a a b equations . system of r tion l num ers, however, is equal to the requirements o f e quations o f the first degree only ; it cont ains symbols no t even for the roots o f such 2 2 f x 2 x a o 1 . elementary equ tions higher degrees as , But how is the system of rational numbers to be enl arged into an algebraic system which shall b e adequate and at the same time sufficiently Simple Th e roots Of the equation
‘ 1 x" fv o x 0 + r i M n — i
Th e o d a a d th e a th e w r lgebr is here use in gener l sense , sense i n Whi ua ternio ns and th e Aus d ehun slehr e see 1 27 1 2 8 are a ch q g ( , ) lge I m a a as a a l o d a . a as br s n s uch element ry rithmetic , ctu l y c nstitute , a th e f a o no a d ff and ccepts r cti n , there is essenti l i erence between it elementary algebra with respect to th e kinds o f number wi th which it deals algebra merely go es further in th e use o f artifici al numbers . Th e elementary algebra differs fro m arithmetic in empl o ying literal o fo r but fl a th e a o an o f symb ls numbers , chie y in m king equ ti n Object investigatio n . TH I 2 E R R A TI ON AL . 7
re no t a a as a the results of single element ry oper tions, are the negative Of subtractio n and the fraction of di vision ; fo r ” ad a are “ though the roots of the qu r tic results of evolution, and the s ame Operation often enough repe ated yields th e
o f b and b ad a a roots the cu ic iqu r tic lso, it fails to yield the roots o f higher equations . A system built up as the w as b b a i d a rational system uilt, y ccepting n iscrimin tely
b d fo r every new sym ol which coul Show cause recognition , f wn o o . would, therefore , fall in pieces its weight Th e most general characteristics o f the roots must be discovered and defined and e mbodied in symbols — by a method which does no t depend on processes fo r solving a T b o f a equ tions . hese sym ols, course, however char cter i zed m a a otherwise , ust st nd in consistent rel tions with the
f n o system o rational numbers a d their Operati ns . An investigation show s that the forms o f number meces sary to complete the algebraic system m ay be reduced to
: b 1 a d i ma i na r an nd a two the sym ol V , c lle the g y ( i ic ted 2 ro o t o f the e quation x 1 z : and the cla ss o f symbols 2 a i rra ti o nal o f a T — 2 =O c lled , to which the roots the equ tion belong.
Th Irr 2 N mb D ed b r e e es. e 7 . u ers efin y Reg ula S qu nc a
n a o 2 d d f r x a ti onal . O pplying t the or inary metho o e tr cting the square root o f a number,there is Obtained the follow o f b th e ing sequence num ers, the results of carrying reck
o ut O 1 2 3 4 a o f d viz oning to , , , , , pl ces ecimals ,
1 ,
These numbers are rational ; the first o f them difi ers from
a a b an 1 d b a e ch th t follows it y less th , the secon y less th n 1 b a nth b , the third y less th n ’ the y less than 1 0 1 00 n — l
And i s o be an n — l a fracti n which may made less than y
ab b a b n assign le num er wh tsoever y taking great enough. 8 N UMB E R —S YS TE M OF A L B R A GE .
This sequence m ay be regarded as a definition of the
square root of 2 . It is such in the sense that a term may be a Of as o f found in it the s qu re which, well as each fol m 2 b lowing ter , differs from y less than any assignable
number .
Any sequence of ra ti o na l numbers
a G a a “ ], 27 3 , “ , a wl ;
i n whi ch a s i n the a bo ve se uence the term a ma b ta lc , q , “ y, y i n n u h ma d e to d i er n umer i ca ll . rea t e o be ro m ea ch g p g g , jf y f
term that o llo ws i t b less tha n a n a ssi na ble number so f y y g ,
tha t o r a ll va lues o v the d erence a a i s numer i ca ll , f f , izf , w , “ , y
less tha n 8 ho wever small 8 be ta ken i s ca lled a r e ula r , , g
sequence .
Th e entire class o f Operations which lead to regular
be re ular se u en e - ui ldi n E sequences may called g q c b g . volution
n f o is only o e o many perations belonging to this class . “ ” An r e ular se uence i s s ai d to d e ne a number y g q fi , this “ ” b b b d a o f num er eing merely the sym olic, i e l, result the
b e Operation which led to the sequence . It will sometimes convenient to represent numbers thus defined by the single
a b 0 . letters , , , etc , which have heretofore represented posi
tive integers only .
A (1 fter some particular term all terms of the sequence 1 , Th e b d b th e a ma b e sa a . 2, y the same, y num er define y A d fo r sequence is then a itself . place is thus provi ed rational numbers in the general scheme o f numbers which the d efini tion contemplates . h n t a mb b W en o a ration l, the nu er defined y a regular
sequence is called i rra ti o na l .
. 33 m limi ti n va lu v i z Th e . 3 . regular sequence , , , has a g e, ,
; which is to say that a term c an be found in this sequence 3 as e a d which itself, well as ch term which follows it, iffers
m b t n an n b b . t fro g y less ha y assig a le num er In o her words, L 29 TH E I R R A TI ON A .
th e di fference between an d the nth term of the sequence may be made less th an any assignable number whatsoever by taking u gre at enough . It will be shown presently
a b d d b an a a a th t the num er efine y y regul r sequence, ], 2, n a a a d a . st s in this s me rel tion to its term “
In ' n N a e . a 2 . e o o s v v a e e 8 Z r , P iti e , eg ti y regul r se qu nc a (1 a a m a a a be o n d hi as l, 2, term “ y lw ys f u w ch itself,
as a well e ch term which follows it, is either
1 a th a an a ab b ( ) numeric lly less n y ssign le num er, 2 a a d a b or ( ) gre ter th n some efinite positive ration l num er, o r 3 a o d ni a a b ( ) less th n s me efi te negative r tion l num er .
r b a hi de In the fi st case the num er , w ch the sequence i s a d to be zero i n d o si ti ve d fines , s i , the secon p , in the thir t neg a i ve .
The Fo r F m O e s O the numbe s 2 9 . u unda ental p ration . f r d efined by the two sequences :
“ 0- a “ 1 7 37 , “ IL 3 8 8 3 8 1 1; 1 27 1 3, 1 m 1 74 44 , Bn + y ,
(1 ) T he s um i s th e number d efi ned by the sequ en c e
8 a a “ “ 1 1 : s + 32, Bu; n + 1 i Bn + 1 , + Bn + u,
(2 ) T he d ifi erenc e i s th e number d efin ed by the sequence
— — a — 8 a Bb 2 B2’ 1 y + y
(3 ) The p ro d uct i s the n umber d efi ned by the sequen c e
“ B a 8 a - B - - IL/ IL) n + 1 1 u + 1 2 IL r V I u i vi
(4) T he quo ti ent i s the number d efi ned by the sequ enc e
3 “ 2 , , BI 6 2 Be 3
Fo r these definitions ar e c o nsistent with the correspond ing definitions fo r r ati o nal numbers ; they reduce to these
a d a element ry efinitions , in f ct, whenever the se quences 30 MB R — S YS M OF A L B A N U E TE GE R .
a to a a 1, B,, 32, either reduce the forms , , 8 8 o r a a a v al , , , , h ve r tion l limiting ues .
T f da — hey con orm to the fun mental laws I IX . This is
d a b t o o a a imme i tely o vious with respect the c mmut tive, sso c i ati ve and di b v a d f , stri uti e l ws, the correspon ing terms o
(l 3 a fo r a the two sequences l / 1 , Bl h BM , inst nce , b d a a b a law I eing i entic lly equ l, y the commut tive for a ti n l o a s .
But a a d n Fo r g in division as just efi ed is determinate . division c an be indeterminate only when a product may
i n 24 a v anish without either factor va nish g (cf . ) where s c an d 0 nth a efine , or its terms after the f ll b an ab b a v elow y assign le num er wh tsoe er, only when the a o f o ne o f (1 (1 s me is true the sequences 1 , 2, 32, a a It only rem ins to prove , therefore , th t the sequences (4) are qualified to define numbers
1 and 2 a a ( ) ( ) Since the sequences ], 2, B], 32, are , b as d n b di y hypothesis , such efine um ers , correspon ng terms
a be a in the two, “ , B“ , may found, such th t
a a. a 8 “, “ is numeric lly , nd a 8 a . BM B“ is numeric lly , and a ziz d r!: 2 8 , therefore , ( w L v ) ( ,, B“ ) , m fo r all a o f v an d a a 8 a b e . v lues , th t however sm ll y
T a o f a 8 herefore e ch the sequences 2 + , 2,
a i s a . B1 , 2 regul r
3 Let on and b e n b . ( ) ,L B“ chose as efore
T a “ - l hen 444 + v BAH v ll - Bi )
Since it i s identically e qual to
“ a 8 8 a ' O n + v (1 # + v 1 0+ ( M t V HI O,
It is wo rth no ticing that th e determin ateness o f di vi sio n is here no t an d d a o but a o o f d o o f in epen ent ssumpti n , c nsequence the efiniti n l a o and th e d a o f th e d o o f a o a mu tiplic ti n etermin teness ivisi n r ti n ls . Th e a i u o f th e o u d a a a I—V V II s me th ng is tr e ther f n ment l l ws , .
— 3 2 N UMB ER S YS TE M OF ALGEB R A .
Fo r it is only a rest atement of the definition of a regular
(1 sa sequence 1 , to y that the se quence
_ 0 a a c o o a 0 0 0 1 z M, , fl + y ,
n i a a 29 o ne which defi es the d fference “, , is whose terms after the nth can be made less than any assignable b b a num er y choosing M gre t enough, and which, therefore, b as a ecomes , p. is indefinitely increased, sequence which defines O — d li mi t a a . In other words , the of ,, as p is in efinitely
: H O o r a a . increased is , limit ( n) ence
The number d efined by a reg ular sequence i s the li mit to
h h the th ter o thi s se uen e a ro a hes a s i s i nd e w ic m c u. n f q c pp , fi ni tely i ncrea sed ? ie
Th e n 4 o f 2 9 defi itions ( ) may, therefore, be stated in the form
u :t a ziz limit ( fl ) limit (BM) limit ( n
a = li mit a limit ( n) limit (BM) ( y fifl ) ,
limit limit (BM)
What the abo ve demo nstratio n pro ves is that a stands in th e same
a o to at r a o a as a o a Th e o f a rel ti n M when i r ti n l when r ti n l . principle perm
nenc e cf. fo o ne in a d n a as th e d a ( there re, justifies reg r i g i e l limit in th e fo a th e a a im a a rmer c se since it is ctu l l it in the l tter , when
a o a i i a th e a a th e o irr ti n l , is l m t ( n) in precisely s me sense th t is qu tient It i no t o ai n d It fo o f o o f c d c a o . by , when is p sitive nteger c nt ni g ll ws r m th e d o a o a if a a o o di to a as em nstr ti n th t there be re lity c rresp n ng , in geo metry we assume there is th at reality will b e th e a ctual h i h limit o f t e real ty o f t e s ame kind co rresp o nding to a n. Th e no tio n o f irratio nal li miting values w as no t immedi ately avail a a o to 28 29 30 th e a o f di and a ble bec use , pri r , , , me ning fference g re ter and lesser inequali ty h ad no t been determined fo r numbers defined by sequences . TH E IRRA TI ON AL .
1 i Fo r b a a limit the more complete sym ol 5 3 ; ( M) is lso “ d d a a a a a h use , rea the limit which “ ppro ches as u ppro c es ” “ ” a a o a a infinity , the phr se ppr ches infinity me ning only, ” “ a an a ab becomes greater th n y ssign le number .
h 1 2 D v o b o . 1 T e 3 2 . i isi n y Zer ( ) sequence , , cannot 21 22 d b b d b Is 0 efine a num er when the num er efined y B1 , 32, ,
b n b a (1 . be a o 0. unless the num er defi ed y ], 2, ls In this a “ ma ma as case it y ; y approach a definite limit u increases , BM
an d b o a . But however small “ ec me this number is not to be regarded as the mere quo tient 3 Its value is n o t at all ' d b a b d b a etermined y the f ct that the num ers define y ],
ar e 0 an b o f di f BI, 32 , ; for there is indefinite num er f e re nt d 0 and b sequences which efine , y properly choosing a a o f ], B1 , 62, from mong them , the terms the
a may be m ade to take any value what El [72
1 2 2 .Th e I s a ( ) se quence , , not regul r when Bl , B2, 21 22
fi 0and a d b d a O. de nes b 2, efines a num er ifferent from N 0 term ? c an be found which differs from the terms I p. following it by less than any assign able number ; but
a a b a a “ c an be a r ther, y t king ja gre t enough made gre ter fin than any assignable number whatsoever .
Though n o t regular and though they d o not define b d l num ers, such se quences are foun usefu in the higher i T i n in t . mathematics . They may be s aid to define f y heir due d a usefulness is to their etermin te form, which makes it possible to bring them into combinatio n with other sequences o f like character or even with regular se quences .
T an a hus the quotient of y regul r sequence y,, y, 34 UMB R —S S M OF A B RA N E Y TE LGE .
by is a regular sequence and defines 0; and 2 0L 7—1 1 2 o f . . b a the quotient , y similar sequence , , BI BJ 8 8 I I 1 2 — m a a be a and if a 8 i : 1 2 y lso regul r serve ,, B,, y,, , ( , , — to d n an b a be properly chosen efi e y num er wh tsoever .
a “ “ i . e a Th e term approaches infinity ( . incre ses with [i s o ut d d o r limit) as M is indefinitely increase , in a efinite
“ 1 “ 2 a a so determin te m nner ; that the infinity which , 3 3 . 1 [ 2 “ d b o f 22 defines is not in eterminate like the mere sym ol O . But here again it is to be said that this determinateness
no t due a a 3 d O is to the mere f ct th t BI, [ 2 efines , which
F r is all that the unqualified symbol e xpresses . o there is an indefinite number o f different sequences which like
0 and b o f a BI, 62, define , g is a sym ol for the quotient by any o ne of them .
The s em d fi ed b e e es of Ra o a s 3 3 . Sy t e n y R g ular S quenc ti n l , A e ular se uence o i rrati o na ls Closed and Continuous. r g q f
al (11 c o 0 a a/ o o 0 l , 2, m) m+1 )
in whi ch the d erences a a ma be ma de numeri c a ll ( if m. “ m y y less tha n a ny assig na ble number by ta ki ng m g rea t eno ug h) d e nes a number but never a num er whi h fi , b c may no t a lso be d e ned b a se uence o ra ti o na l n umbers fi y q f .
‘ Fo r 8 b an o f rati o nals whi ch B1 , , 2, eing y sequence
0 a a a (1 defines , construct sequence Of r tionals l , 2, such — a d ot a 3 and a e th t l 1 is numerically less th n 1 1 in the s m — T = a a a . a sense 2 3 3 < B3, etc hen limit ( m ) O
28 a a . , or limit ( m) limit ( m) This theorem j ustifies the use o f regular sequences o f a d b an d so irr tionals for efining num ers, makes possible a simple e xpression o f the results o f some very complex I A OM X N MB R S . TH E IMA G N R Y. C PLE U E 35
T m a . a a a b Oper tions hus where is irr tion l, is num er ;
b a a a l a a z d the num er, n mely, which the se quence , , efines , f a a o a o a d m. when ], 2, is any se quence r ti n ls efining But the importance of the theorem in the present discus sion lies in its declaration that the number - system defined by regular sequences of rationals cont ains all numbers which result from the Operations o f regular sequence - building in a clo sed general . It is system with respect to the four
a a a o and a fund ment l oper ti ns this new Oper tion, exactly as the rational numbers constitute a closed system with respect a a a o to the four fund ment l oper ti ns only (cf. Th e system o f numbers defined by regul ar sequences of a — rea l b as are a d — ration ls num ers , they c lle therefore po ssesses the following two properties : (1 ) between every two unequal re al numbers there are other re al numbers ; (2) a v ariable which runs through any regula r sequence of b a a a a a real num ers , r tion l or irration l , will approach re l d l number as limit . We in icate a l this by saying that th e b i system Of real num ers s c o nti nuo us.
THE MA NA . OM LE M V. I GI RY C P X NU BERS .
T m r Th bo 4 he e . e 3 . Pur I ag ina y other sym l which is
d d b - o f a b a e nee e to complete the num er system lge r , unlik
i a a but a and a o ad the rr tion l like the neg tive the fr cti n, mits
d b a n a v i z . of efinition y si gle equation of very simple form, , N + 1 = 0
th e b a i s 1 b 1 It is sym ol whose squ re , the sym ol ,
“ ’ now commonly written i fi It is called the u n i t of i mag
i na ri es .
In contradistinction to i all th e forms o f number hitherto “ ” “ d r a a and a ~ d a e l. T consi ere called r ea hese n mes , re l im gi
f V l Gauss intro duced th e use o i to represent . 3 6 M R —S S M OF A B N U BE Y TE LGE R A .
a an o nary, are unfortun te, for they suggest Opp sition which
d oes not exist . Judged by the only st andards which are admissible in a pure doctrine of numbers i is im aginary in
a as a and the s me sense the negative, the fr ction , the irra
ti o nal but all ar , in no other sense ; e alike mere symbols d evised for the s ake o f representing the results of opera tions even when these resul ts are not numbers (positive i integers) . got the name imagin ary from the di fficulty
once found in di scovering some e xtra - arithmetical re ality
to correspon d to it . As the only property att ached to i by definition i s that — 1 a d w a o f i ts b its square is , nothing st n s in the y eing ” “ b a b a i d multiplied y any re l num er ; the product, , is
called a pure i mag i na ry.
An b a c o entire new system Of num ers is thus cre ted, ex a b but tensive with the system of re l num ers , distinct from E n . 0 b o e it xcept , there is no num er in the which is at * N b the s ame tim e cont ained in the other . um ers in either system m ay be compared with each other by the definitions o f e qu ality and greater and lesser inequality i a being
Z— i b a Z b but b o ne called , as ; a num er in system cannot
be a d to be a o r a s i either gre ter than, e qual to less than
number in the other system .
Th e sum a i b co m Com ex N mb s . 3 5 . pl u er + is called a
b to o f lex n umber . p Its terms elong two distinct systems,
which the fundament al units are 1 and i . Th e g enera l complex number a i b is defined b y a co m plex sequence
i a i “ i + Bl ) 2 + 62, F fi
a a . where l , 2, 32, are regular sequences
Thro ughout this discussi o n 00 is n o t regarded as bel o nging to th e
- b ut as a o f th e o a number system, limit system, lying with ut it , sym b o l fo r s o mething g reater than any number o f th e system . AR Y C OM X N UMB R S 3 TH E I MA GIN . PLE E . 7
r nd i O i b all a i O 36 3 Co . a b Since a , , ) , real
mb a and a i b are a d nu ers , , pure imagin ries, , cont ine in the system of complex numbers a i b
a i b c an a b a O and b O. 4, v nish only when oth
The Fo r d m O er i ons on Com lex Num 3 6 . u Fun a ental p at p Th e a n a n o f fundamen bers . ssumptio of the perm ne ce the tal laws le ads immediately to the following defin itions o f
add b u t and di v o f the ition, su traction, m ltiplica ion, ision
complex numbers .
1 .
' ' ' ' i b a i b a i b Law Fo r a i b a . ( ) ( ) , II
Law I . L a . ws II, V
a i b 2 . ( + )
finiti On b 36 1 . By de Of su traction (VI) and ,
The necessa r a s well a s the su ci ent co ndi ti o n o r C OR . y ffi f ' ’ i o two co m lex numbers a i b a i b i s th at the equa l ty f p , ' ' z = a a and b b .
' ' ' ' Fo r a i b a i b a a i b b O if ( + ) ( + ) + ( ) ,
' ' o r d = a = a b , b b .
’ ' " ' ' a i b a i b d a bb i a b 3 . ( ) ( ) (
’ ' ’ ' i i a i a a i b i b Law a b a b b . For ( ) ( + ) ( ) + ( ) , V
' ’ ' ’ i d a i b i b i b Law a a b . + + + , V ' ’ L I— (d a bb ) aws V.
” ’ I ei ther ac to r o a ro duct va ni sh the r duct COR . f f f p , p o ni va shes .
Fo r i O= i b — b = i b — i b 1 o = 0 x ( ) (§ , H ence
La 28 29 3 . ws V, IV, § , § , M R —S YS TEM OF AL B RA N U BE GE .
' ' ' ' a i b d a + bb ba a b
t l2 l2 l2 l2 a a + b a b
’ ' Fo r f o a i b a i b b e .v i let the quotient b y y.
B o f y the definition division (VIII) ,
’ ' (9: i y) (a + i b ) a ib.
' cva § 36 3
' ' ’ '
sca b a cc a 2 o r . b b. y , y 36 , , C
H a; and b ence, solving for y etween these two equations ,
’ ' ’ ' d a bb ba a b
re 12 vz a 6 b
T i n a o f a b d v i s herefore, as the c se re l num ers , i ision a d a x d 0 i s eterminate Oper tion, e cept when the ivisor is ; it
d Fo r a: and ar e d b then in eterminate . y eterminate ( y IX) '2 '2 ' ’ ’ ’ l a b 0 a b = O o r a i b O un ess , that is , unless , + ; ' ' ” '2 a b b n a a an d b b v o ne for and ei g re l, are oth positi e, and * H n d . b a 24 can ot estroy the other ence, y the re soning in ,
I a ro duct o two o m lex nu ers ani sh o ne o C OR . mb v f p f c p , f i h the fa cto rs must va n s .
m m f . Tw 3 7 . Nu eri cal Co pari so n o Co mplex Numbers o ' ’ x b a i b a i b do a comple num ers, , , not, gener lly speak adm o f a as do ing, it direct comparison with e ch other, two re al numbers o r two pure imaginaries ; fo r a may b e gre ater ’ '
a a b a b . th n , while is less th n
T are a d numeri ca ll o b a o f hey comp re y, h wever, y me ns
What is here proven is that in th e system Of co mplex numbers ‘ fo m d o th e fu d a a ni 1 and i o ne and b ut o ne r e fr m n ment l u ts there is , , ' ’ number which i s th e quo tient o f a ib by a i h thi s being a co use quenc e o f th e determi nateness o f the di vi sio n o f real numbers and the peculiar relatio n (i 2 1 ) h o ldi ng between th e fun d amental ni Fo r th e a o f a o f IX w e a th e a o u ts . s ke the perm nence m ke ssumpti n, o i a a th e o a o f th e o therw se irrelev nt, th t this is nly v lue qu tient whether w n o r o th e e fo d o th e ni 1 and i ithi with ut syst m rme fr m u ts .
MB R - S S M OF AL B 40 N U E Y TE GE RA .
a d b a most general line r, associative , istri utive, commut tive al b a mb are x b o f o ge r , whose nu ers comple num ers the f rm zv e a e ac e b il n da a i e e l 1 g , n , ” u t from fun ment l un ts 1 , 2, is reducible to the algebra o f the complex numb er
Fo r n a w a i b. Weierstrass has Show th t any t o complex
b a and b o f a e e. x e num ers the form l , 2 + n n, whose sum i d and are b o f , d fference, pro uct, quotient num ers this a and a and d I— d s me form, for which the l ws efinitions IX hol d ma b ab a a be goo , y y suit le tr nsform tions resolved into com o nents a a a b b b p l, 2, , l , 2, , , such that
= m a b a 1 b1 + a 2b2 +
a a E a m . b r
Th e a b c d components ,, , are constru te either from one tal dam a n dame n k . unit g , or from two fun ent l u its g , , , T Fo r components of the first ki nd the multipli cation fo r mul a i s
a 8 Q “ ( g , ) (1 9 1 ) ( B) g t
Z ur Theo rie d er aus n Haupteinheiten gebildeten c o mplex en G ttin er N a r Nr 1 0 1 884 o . O . . Gr ssen g ch ichten , Weierstrass finds that these general c o mplex numbers differ i n o nly o n o a o th e o m i If th e num e imp rt nt respect fr m c mplex nu ber a b. b er o f f n da a ni b e a a 2 a a i num u ment l u ts g re ter th n , there lw ys ex st diff f o 0 th e o d o f i a o um bers , erent r m , pr uct wh ch by cert in ther n bers d i f 0 Th e o 0 W a a l o o . f o is . eierstr ss c l s them iv s rs number excepti ns di v o fini n d t o th e determinateness o f isi n is in te i stea o f o ne .
' ni are a s ealnn no t e e e but l n a T These u ts , gener lly p g , , , , , n, i e r o a o Of as 6 e e K e x c c mbin ti ns them, 7 1 1 w , yn n , l l 2 , An set o f n i d d a o a o o f th e n c y n epen ent line r c mbin ti ns u its l , ma be a d d as o a s et o f u d a a all y reg r e c nstituting f n ment l units , since
m o f th e fo axe a e ci e ma b e d a nu bers rm l 1 , 2 n ” y expresse line rly o f in terms them. P M R S 41 GRAPHI CAL R E RES ENTA TI ON OF N U BE .
For components o f the second kind th e multiplication formula is
' ' a k a 75 a a 3 03 k ( g , B i ) ( g , 5 5 ) ( 1 ) r
And these formulas are evidently identical with the mul tipli c ati o n formulas
' ' ' (a 1 Bi ) (a a M301 (G B o f common algebra .
A CA E E E TAT O OP N BE . VI . GR PHI L R PR S N I N UM RS
THE VARIABLE.
Corr s o d b w the N mb r - em 40. e p n ence et een Real u e Syst
nd the o s o f a. . Let a be o a P int Line right line ch sen, and o n x b e d - a a x d it a fi ed point , to calle the null point lso fi e unit fo r the me asurement o f lengths . Lengths may be measured o n this line either from left to
o r a a d right from right to left, and equ l lengths me sure in ad Opposite directions , when ded, annul each other ; Opposite
b a b e d . alge r ic signs may, therefore, properly attache to them Let the sign + t be attached to lengths me asured to the right, the Sign to lengths measured to the left . The enti re system of r ea l n umbers may be represented by the o i nts o the li ne b d a b p f , y taking to correspon to e ch num er that point whose distance from the null - point is represented
b b . Fo r as d a y the num er , we procee to demonstr te, the d o f — istance every point of the line from the null point, d o f d a b measure in terms the fixe unit, is a re l num er ; and we may assume that for each re al number there is such a point .
1 . The di stance of any po i nt o n the li ne fro m the null- po i nt i s a real number .
Let be a any point on the line t ken, and suppose the seg ment of the line lying between this point and the null - point MB R - S S M OF A RA 42 N U E Y TE LGEB .
0. a al to contain the unit line times, with a rem inder l , this a a o f rem ainder to cont in the tenth p rt the unit line B times ,
a a d d d with rem in er 2, 2 to contain the hundredth part of the
1 d . unit line 7 times, with a remainder 3, etc
The o f a b v i z . sequence ration l num ers thus constructed, , a a a i s a , B, By, (adopting the decimal notation) regul r ; for the difference betwee n its nth term and each succeedi ng 1 term is less than a fraction which may be made less 1 0“ 1
a an ab b b and th n y assign le num er y taking p. great enough ; , b b o f th e y construction, this num er represents the distance
d - point un er consideration from the null point . By the convention m ade respecting th e algebraic signs o f lengths this number will be positive when the point lies to
o f - the right the null point, negative when it lies to the left .
2 o rres o ndi n to ever rea l number ' there i s a o i nt . C p g y p o n th li ne who se di sta nce a nd di recti o n ro m the null- o i nt a e , f p re th um i ndi ca ted by e n ber.
a b a c an ( ) If the num er is r tional , we construct the point . Fo r every rational number can be reduced to the form o f
‘ An 2 v . d b a simple fraction if denote the gi en num er , when B x find e thus e pressed , to the corresponding point we hav only to lay off the Bth part Of the u nit segment a times
- along the line , from the null point to the right, if g is
- 9 a v . positive, from the null point to the left, if is neg ti e B b b ( ) If the num er is irrational, we usually cannot con
o r x . struct the point, even prove that it e ists
a d b d m a But let enote the num er, and l , , n, any
o f so a regular sequence rationals which defines it , that n will approach a as limit when n is indefinitely increased .
T b a o f e hen , y ( ) , there is a sequence points on the lin a corresponding to this sequence o f rationals . C ll this A A m A sequence Of points , It has the property I A R R S N A I N O GRAPH C L EP E E T T O F N UMBERS . 43 that the length of the segment will approach 0 as n nd i a limit when is i efinitely ncre sed .
a ad When n is m e to run through the sequence of values d m n A l , , the correspo ding point , will run through the
o f A A And a ssum sequence positions 2, we e that just
i s a d b a as there in the re l system a efinite num er a which ,
a o n is appro ching as a limit, so also is there the line a A A as . definite point which , approaches limit It is this d point A which we make correspon to a . Of course there are infinitely many regular se quences o f
a 06 n and as o f ration ls 1 , defi ing a, many sequences cor A A a responding points 1 , 2, We ssume that the limit point A is the same for all these sequences .
Th a f o - - The Co o s V r b e . e o ne 0ne 41 . ntinu u a ia l rel tion to correspondence between the system of real numbers and the points of a line i s o f great importance both to geometry d o n a d to an b . b to alge ra It ena les us , the one h n , express a a o n to geometric l relations numeric lly, the other, picture
r i u complicated num erical relations geometrically . In pa t c a b i b d to lar, lge ra s inde te it for the very useful notion of the continuous variable . One of o ur most familiar intuitions is that o f continuous motion .
A P
Suppose the point P to be movi ng continuously from A B a OAB b and x d e th e to long the line and let a, , enot o f A B and OP 0 lengths the segments O , O , respectively,
- being the null point . It will then follow from o ur assumption that the segment
' AB ai f r v r b and b cont ns a point o e e y num er between a , a A B x ma be th t as P moves continuously from to , y regarded as increasing from the value a to the value b r ll To e th ough a intermediate values . indicat this we call x co nti n u u i a o s va r a ble . - 44 N UMBER S YS TEM OF AL GEB RA .
Corr s o d ce b w Com x mb r- s m 42 . e p n en et een the ple Nu e Sy te 0 f Th and the Po ints o a Plane . e entire system o f complex b be d b o f num ers may represente y the points a plane, as follows ' ' In the plane let two right lines X OX and Y OY be a dr wn intersecting at right angles at the point 0. ’ ” Ma OX “ x b ke X the a is of real num ers, using its points n b b to represent real um ers, after the manner descri ed in ' 40 and e Y Y th e x , mak O a is o f pure imaginaries , represent ing i b by th e point o f OY whose distance from O is b n b v and b whe is positi e, y the corresponding point of OY '
when b is negative . Th e point taken to represent x b a i b P the comple num er + is , c o nstructed by drawi ng thro ug h B A and , the points which rep a i b resent and , parallels to
FI G 1 . . ’ ’ O a OX v . Y Y nd X , respecti ely Th e correspondence between the complex numbers and
h f - - To t e points o the plane is a o ne to one correspondence . every point o f the plane there is a comple x number corre s o ndi n an d but o ne b p g , , while to each num er there corre * spo nds a single point of the plane .
A reality h as thus been fo und to co rrespo nd t o the hitherto unin ter reted But a i h as no o o p symb o l a i b. this re l ty c nnecti n with the a hi a t o a th e Of a o re lity w ch g ve rise rithmetic, number things in gr up o f di and d o not at all th e o a stinct things , es lessen purely symb lic ch r a o f a i b a d d o th e a d o o f a a cter when reg r e fr m st n p int th t re lity , the standpo int which must be taken in a purely arithmetical study o f th e n nd f o o rigi a natur e o the number c ncept . The co nn ectio n between the numbers a + i b and th e po ints o f a a Th e a o a o f pl ne is purely artificial . t ngible ge metric l pictures the relatio ns amo ng complex numbers to which it leads are nevertheless a a id th d o f a o v luable a in e stu y these rel ti ns . 4 GRAPHI CAL R EP RES ENTA TI ON OF N UMB ERS . 5
If the point P be m ade to move along any curve in its
a d b a: be d d a s pl ne, the correspon ing num er may regar e a x a ch nging through a continuous system of comple v lues ,
nd d a co nti nuo us co a is calle mplex vari able. (Compare
h e f M s . T o o P Fi . i . e . 43 . dulu length the line O ( g it b i s l mo dulus o f a i L b. e t be re re , cal ed the it p sented by p.
Ar m . Th e X OP b P 44. g u ent ang le made y O with the positive half o f the axis of real numbers is called the ang le L o f a i b o r ar ume t. et be + , its g n its numerical measure represented by 0.
Th e angle is always to be measured counter - clockwise from the positive half o f the axis o f real numbers to the l i modu us l ne .
Th f th e di P 4 . e o PA 5 . Sine ratio , perpen cular from 1 3 i d x o f l b P i . e. s to the a is rea num ers , to O , , calle the p in o f in s e 0 S 9 . , written Sin 9 is by this defin ition positive when P lies above the
x o f b a P b hi . a is real num ers , neg tive when lies elow t s line
Th d l a P Co i . e o f PB 46 . s ne ratio , the perpen icu r from
x f t Q a l co si ne o P e . to the a is imaginaries, to O , . , is c l ed the f P o 6 t 0. , wri ten cos Gos d is positive o r negative accordi ng as P lies to the right o r the left o f the axis o f imaginaries .
ms o i ts T or m. The ex ressi o n o a i b i n ter 47 . he e p f f modulus a nd a ng le i s p(oo s 9 i si n 9 ) a F b 46 _ 9 or y § cos , p
b 45 9 _ e and y s , sin , p
Therefore a i b 5 3- p(o os 9 i sin N UMB E R —S YS TE M OF A R A LGEB .
Th e factor cos 9 + i sin 0h as the same sort o f geometrical meaning as the algebraic signs and whi ch are indeed but particular cases of it : it indicates the di recti o n o f the
b n - point which represents the num er from the ull point . a e It is the other f ctor, the modulus p, the distanc from
ul - o f b the n l point the point which corresponds to the num er, ” i “ b o f b which ndicates the a solute value the num er , and may represent it whe n compared numerically wi th other numbers 3 7) — that o ne of two numbers being numeri cally the greater whose corresponding point is the more
h - distant from t e null point .
’ rob m I . Gi ven the o ints P and P re resenti n 48 . P le p , p g ’ ' ' a i h a nd a i h respecti vely requi red the po i nt represent i ng ” Th e d P o f point require is , the intersection the parallel P P ’ P ’ to O through with the parallel to O through P.
Fo r b completing the construction indicated y the figure, ’ ” ” ’ OD PE DD and OD : OD D we have , therefore O ; ’’ " ’ ' and similarly P D PD P D . T COR . I . o get the point corresponding to a ’ ’ produce OP to P making ”' ’ OP OP and , complete the
u OP P parallelogram ,
Th u u th COR . II . e mo d l s of e sum o r d if erence of two co mplex numbers i s less than (a t g rea test
equa l to ) the sum of thei r mo duli .
” Fo r OP is less than OP + " PP and OP , therefore, than P’ “ O P P ’ O , unless , , are in e . 2 .
the same straight line , when ” ’ ’ : PP OP OP + OP . Similarly, , which is equal to the modulus o f the difi erenc e of the numbers represented by ’ ’ P P OP OP . and , is less than, at greatest e qual to, +
48 N UMBE R—S YS TE OF A RA M LGEB .
F OR . f C . II rom the definition o division an d the preceding demonstration it follows that
a i b 9 i Si n 9 ’ ’ 7 [cos ( ( a i b £ the construction for the point representing is there a i o b . f re, o vious
. C r r M Let a 5 0 i cula easure of Ang le . circle of unit radius b e d f constructe with the vertex o f any angle o r centre . Th e length o f the arc of this circle which is in tercepted between the legs Of the angle i s called the ci rcular mea sure f o the angle .
T . e ex ressed i n 5 1 . heo rem Any c o mplex number may b p i ” the fo rm pe ; where p i s i ts mo dulus an d 9 the c i rcula r meas ure o i ts a n le f g . It has alre ady been proven that a comple x number may be written in the form p(o os 9 i sin where p and 9 have n Th e b e the meani gs j ust given them . theorem will demon
“ strate d a b , therefore, when it sh ll have een shown that i " e cos 9 i sin 9 .
n be v b 36 th e If any positi e integer, we have, y and b inomial theorem,
(i 9) 3 +
Let n be indefinitely increased ; the limit o f the right side of this e quation will be the s ame as that o f the left . I A RE P R S N A I F N MB R 49 GR APH C L E E T T ON O U E S .
But the limit o f the right side is
l i i " at + 9 + i . e . e
n “ T e i s as n a a 00 herefore the limit of ppro ches .
fi To construct the point representing 1 +
On the axis o f re al numbers lay Off OA 1 . Draw AP equal to 9 and par a to OB and d d n llel , ivi e it into Let AA be o ne equal parts . 1 T f . A O these parts hen I is the
1 0 point 1 — n
T A d a A A a hrough I r w I 2 t A an d right angles to O 1 con A struct the triangle OA1 2 simi
lar A . to OA l
FIG . 3 . A2 i s then the point
F01 ‘ A A 2 O Q A 01 4] ; and OA : OA OA : OA and OA 1 since , l , , , the length OA the s quare of length OA1
a o In like m nner c nstruct A3 to represent 1 A n for
Let n b e d Th e b AA A in efinitely increased . roken line I 2 A ” will approach as limit an ar c of leng th 9 o f the circle ad OA and x A a a Of r ius , therefore , its e tremity, n , will ppro ch as limit the point representing cos 9 i sin 9
f h m i 9 d 3 This use o t e sy b o l e will b e fully ju stifie in 7 . MB R - S S M OF A B R A 5 0 N U E Y TE LGE .
1 ! Therefore the limit o f 1 + as n is indefinitely i n
o s i si n cre ased is c 9 9 . “ But thi s s ame limit h as already been proved to be e ” H ence ei c o s 9 i sin
T T P . VII . HE FUNDAMENTAL HEOREM O ALGEBRA
The Ge er Th e r m 2 o . 5 . n al e If
w = a z" a z a z o + 1 + 2z + a n — l
n a and d d a b where is positive integer, o, l , n any num ers, d d z a a real or complex, in epen ent of , to e ch v lue of z corre n f spo ds a single value o w. We proceed to demonstrate that conv ersely to each v alue
f r o w set n v a z i . e. a a e corresponds a of lues of , th t there n b b d a num ers which, su stitute for z in the polynomi l
" ” * 1 d z a z a a an v a o l n, will give this polynomi l y lue, w b e a O, which may ssigned . It will be sufficient to prove that there are n values o f z
” ‘ 1 a z a a a a O a as which render o l n + ,, equ l to , in smuch from this it would immediately follow that the polynomial f a an a o r n a z v i z . t kes y other v lue , w) , v lues of ; , for the a a a d v lues which render the polynomi l of the s me egree ,
" ‘ 1 a z a a a w a . O. o l n ( n o) , e qu l to
f E A f o o o an o . a o fo r 5 3 . R t quati n v lue z which
a i s O a d a ro o t + ,, is c lle of this poly o r n o f a l ebrai c e uati o n nomial, more commo ly a root the g q
" — 1 a r a 0. W i .
Ma a l F Am a Jo a o f Vo . VII r F. a D . r nklin , eric n urn l them tics , ,
A o MObius o d Wo V o l . IV . 726 . 76 . p . 3 ls , C llecte rks, , p FUNDAMEN TAL THEOR EM OF AL GEBRA . 5 1
l ebra i e u ti n ha s a ro o t. T eo m. E ver a a o 5 4. h re y g c q
" n * 1 v w a z a e a . Gi en o l n
d w e Let | w | denote the mo ulus of . We shall assum be d a a o f w though this can prove , th t mong the values [ ]cor b least responding to all possi le values of z there is a value ,
o a z and that this least value c rresponds to finite value of . L w d a et | o | enote this le st o f w and z a value | | , o the v lue o f z to which it corres ponds . T w O. hen I o | Fo r if w be re re not, O will p sented in the pl ane of com plex numbers by some point P distinct from th e null point 0. Through P draw a circl e havi ng its centre in the null h o O. T n b p int he , y the y FI G . 4. o th esi s p made , no value can be given z which will bring th e correspondin g w- point within this circle .
- But the w point ca n be bro ug ht wi thi n thi s c i rcle. Fo r z and w b o f an d w , o O eing the values z which corre s o nd to P a b z 8 p , ch nge z y adding to o a small increment , and A n w T A i s let represent the consequent cha ge in . hen defined by the equation
- l w A a z a z ( e ) o( o 1 < o
n _ 2 a z 8 a e 3 a . 2( o ) _ l ( o ) n
On applyi ng the bin omin al theorem an d arranging the o f 8 b o f terms with reference to powers , the right mem er this equation becomes
n—1 “ z (l z “ “ a a O on i o i n — l n — 1 —2 W’LZ n n 1 (I [ O O ( ) n 2 3
8 8 . terms involving , , etc 5 2 UMB R - S YS TE M OF A R A N E LGEB .
— 1 w d z a z a 2 a o o en r on rt - 1 0 n
‘ 1 " “ 2 A n a z n 1 a z a 8 [ o 0 ( ) 1 0 n 1 ] 2 3
v 8 8 . + terms in olving , , etc
’ ’ ' Let p (c o s 9 + i si n be the comple x number
n—1 ma n 1 a s o ( ) w a s .
x a e pressed in terms of its modulus and ngle , and
p (c o s 9 i si n 9)
th e d x correspon ing e pression for 8. Then = ’ ’ A p (c o s 9 + i sin x p (cos 9 i sin 9) 2 3 terms involving p , p , etc .
’ ’ pp [c o s (9 i sin (9 2 3 terms involving p , p , etc .
’ Th e point which represents pp [c o s (9 i sin (9 for any particular value of p can be m ade to describe a ’ circle o f radius pp about the null - point by causing 9 to e O 4 increas continuously from to right angles . In the same circumstances the point representing w ’ i O pp [cos (9 sin (9
b a ab P will descri e an equ l circle out the point and, there hi P fore, come wit n the circle O . b a A ma be ad ff But y taking p sm ll enough, y m e to di er as ’ as c o s 9 i 9 little we please from pp [ ( sin ( and , ' v b P w A therefore, the cur e traced out y (which represents o + , 9 o f v as as as runs through its cycle alues) , to differ little ’ ad we ple ase from the circle o f centre P and r ius pp . T b and 9 w herefore y assigning proper values to p , the ’ P point (P ) may be brought within the circle O .
' 2 8 In A B (7 . a o o f all fol the series p + p p etc , the r ti the terms
o th e to r i . e . l wing first the fi st,
which by taking p small eno ugh may evidently be made as small as we please . A H E E M F A B R A 5 3 FUN DAMENT L T OR O LGE .
- Th e w- point nearest th e null point must therefore be the ‘ " null - point itselffi
T eo em. I a be a ro o t o 5 5 . h r f f thi s po lyno mi al i s di vi si ble by z a .
” “ 1 Fo r d d a z t x a b a i ivi e o a n y z , continu ng the u i ai d and a division ntil z d sappears from the rem n er, c ll this
a d R fo r rem in er , the quotient Q, and, convenience , the poly nomial f (z) . Then we have immedi a tely
fl e = e — OQ+ R
f z holding for all values o .
Let a a z a as z t ke the value ; then f ( ) v nishes, also the product (z a ) Q.
T 7: a R O and b d d o f 2 herefore when , , eing in epen ent it is hence always 0.
he nu ber o the r o o ts Th e d me a T eorem. T m 5 6 . Fun a nt l h f
” "—1 o the o l no mi a l d z a t! a i s n . f p y o l n
Fo r b 5 4 at a o n e o s a , y , it has le st r ot ; call thi ; then , b 5 5 i b b a y , it is d visi le y z , the degree Of the quotient being n 1 . Therefore we have
n : — a z a z a Z a b — ° 0 + l + n ( ) + n l )
n ‘ l " ‘ 2 A a b 5 4 a a b z b g in , y , the polynomi l (x l “ 1 h as a h and di as b a root ; call t is B, divi ng efore, we h ve
— z a z e f C ( ) ( l + u
In the ab o ve demo nstratio n it is as sumed that the co effi cient o f 6 n 0 r o t . If 0 let A5 d o e o f A n is it be , en te the first t rm which is o t 0. ” ” If A p (c o s 9 i sin w e then h ave
" A f c o s r 9 i r9 in 6h ” p p [ ( sin ( terms , f o hi a o o o o as a o r m w ch the s me c nclusi n f ll ws b ve . 4 MB R - S YS M OF A B RA N U E TE LGE .
Since the degree o f the quotient is lowered by 1 by each n 1 repetition Of this process, repetitions reduce it to the
firSt d o r egree, we have — — — a z a z " - z V a( ) ( r) ( ) ,
d o f n a o f d . pro uct factors , each the first egree N o w a product vanishes when o ne o f its factors vanishes — = 3 6 3 a a a 7. a , , and the f ctor z v nishes when ,
” ’“ 1 h n T a a x m v w e v. z z z when , z z herefore o + 1
‘ f a va fo r n a a v o z. n nishes the v lues, , B, y, , F a o ne Of urthermore, a product c nnot vanish unless its a v 3 6 4 and o ne o f a f ctors anishes , , not the f ctors
— a v a o ne o f z , z z v, anishes unless z equ ls the
b a 8 v. num ers , , , Th e n polynomial has therefore and but n root s . Th e theorem that the number Of roots o f an algebraic equation i s the same as i ts degree is called th e fundamental o f b theorem alge ra.
N TE E E VIII . I FINI S RI S .
An o f ad D fi i o . 5 7 . e n ti n y Operation which is the limit diti o ns indefinitely repeated produces an infinite series . We ar e to determine the conditions which an infinite series must fulfil to represent a number . m b If the ter s Of a series are real num ers, it is called a
real seri es x co m lex s eri es . ; if comple , a p
I. REAL S ERIES .
An n Sum. C r c D . e o v . v 5 8 . n e g en e i erg ence i finit series
a a a d l + 2 + 3 + + a +
b as e represents a num er or not, according the sequenc
8 8 S 0 0 0 g 0 0 0 0 0 0 1) 2, 3, m, ,
W (l 8 : a a (1 (1 here l , 2 l + 2, 1 + 2 + 0
re ular o r is g not .
5 6 M R—S S OF A A N U BE Y TEM LGEBR .
convergent series which become divergent on this change of
o f a are signs . Series the first cl ss s aid to b e a bso lutely con
v o f o co ndi ti o na ll ergent ; those the second class, nly y con
v ergent . Abso lutely c o nverg ent seri es have the cha ra cter of o rdi na ry
th r s ums ; i . e . e o d er of the terms may be chang ed witho ut
a lter i n the sum o the s e i g f r es .
Fo r o d th e a a a c nsi er series 1 2 3
o d to b e a o o and to a th e sum S supp se bs lutely c nvergent h ve , when th e are th a o d o f th n d terms in e no rm l r er e i ices . It is immediately Obvio us that no change c an b e made in th e sum o f th e seri es by interchanging terms with fin ite indices fo r n may be
' taken greater than th e index o f any o f th e i nterch ang ed terms . Then S h as n o t a f d th e a a fini sumand n been f ecte by ch nge , since it is te i t a a a o d th e o f a ni sum are add d . is imm teri l in wh t r er terms fi te e ; and as fo r th e o f th e s no a h as ad i n th e rest serie , ch nge been m e
o rder o f its terms .
But a a (1 ma a a d o a o f i i 1 2 3 y be sep r te int number nfin te series
as fo r s a o th e a a a and a a a , in t nce , int series 1 3 s 2 4 s and s m d a a Let b e a a d o l these eries sum e sep r tely . it sep r te int such th e o f i all b e a o o as series , sums wh ch they must bs lutely c nvergent, a s o f an a so o — are S O) S O” being p rt b lutely c nvergent series , , respectively it is to b e proven that
1 2 3 : ( ) ( ) ( ) . H) S S + S S 4 S ,
) ) Let S S b e th e o f th e r m o f th e 23, 53, sums fi st terms series 0) S . , respectively th e o a th e e a a a o u Then , by hyp thesis th t seri s 1 2 is bs l tely o m ma b e a s o a a th e sum c nvergent , y t ken l rge th t
1 . 2 z
sh all differ fro m S by less than any assignable number 8 fo r all values o f n fo th e o f sum there re limit this is S . m u a it ma b e s o a a a l diff f o “ ) B t ag i , y k h S sh l r r S y n t en t t m+n e m b ) Q) a f o S a and o th e sum less th n , fr m by less th n , theref re i Si Ht 5 “ ) Q) “ ) S f o S a I S S r S S y l ss h , 21” ig n ig n m b e t n “ ) ”) . . H um i e by less than 6 . ence th e limit o f this s is S S Therefo re S and S “ ) S ”) S “ ) are limits o f th e same fin ite
sum and a We o h o f fo r h e a hence equ l . ( mit t e pro t c se I infinite . ) 5 IN FINI TE S ER IES . 7
h Co o Co ve e . On t e 6 1 . nditi nal n rg enc the other hand, terms of a co nd i ti o na lly co nverg ent seri es ca n be so a rra ng ed tha t the s um of the ser i es may ta ke a ny rea l va lue wha tso ever .
In a co ndi tio nally co nvergent series th e p o sitive and th e negative terms each c o nstitute a di vergent series havi ng 0 fo r th e limit o f its a m l st ter . If fo b e an o u and S b e o d , there re, C y p sitive n mber , , c nstructe by first addi n g po sitive terms (beginning with th e fir st) until their sum is a a to a u i sum i s a a a gre ter th n C, these neg tive terms ntil the r g in less th n o ll th e sum a ai a a and so o n C, then p sitive terms ti is g n gre ter th n C, ’ d th e im o f S as n d ni a d O. in efinitely l it n , is in efi tely incre se , is
1 . I e h h t e T of C v . ac o t e erms 6 2 . Sp cial ests on erg ence f f
seri es a a be numeri c a ll less tha n a t rea test of a 1 + 2 + y ( g equa l to ) the co rresp o ndi ng term of a n a bso lu tely co nverg ent seri es o r i the r a ti o o ea ch term o a a to the co r re , f f f l 2 sp o nd i ng ter m of a n a bso lu tely co nverg ent ser i es n ever exceed so me n i te number the ser i es a a m i s a bso lutel fi 0, 1 + 2 + y co nverg ent.
I o n the o th er hand ea ch term o a a be numeri ], , f 1 2 ca lly g r ea ter tha n (a t the lo west equ a l to ) the c o rrespo nd i ng term o a di ver ent ser i es o r i the r a tio o eac h term o f g , f f f
a to th o rr es n di n i r en s ri es be a l 2 e c p o g term of a d ve g t e ’ never numeri ca ll less tha n s o me ni te number d erent y fi C , ifi
h s i s ro m O t e er e a a is di ver ent. f , l 2 g
2 T he seri es a a a a the terms o whi ch a re . m 1 2 3 4 , f a lterna el o si ti ve a nd ne ati ve i s co nver ent i a ter so me t y p g , g , f f term a ea c h ter m be numer ica ll less o r a t lea st n o t r eater , y , , g than the term wh ich i mmedi a tel reced es i t a nd the li mi t o y p , f
i s a a s n i s i nde ni tel ncrea ed be 0. n, fi y ,
Fo r here
m n — l sm+n 3 m 1 ) [a m+ 1 d m+ 2 l ) a m+n ]
Th e expressio n with in brackets m ay be written in either o f th e fo rms
a d a a ( m- H m+z) ( m+3 m+4)
a m+l (a m a m+3> B R - s YS TE M OF A R A 5 8 N UM E LGEB .
It o o and a a and is theref re p sitive, less th n m“ , hence by a m a o ma b e ad a a an a t king l rge en ugh , y m e numeric lly less th n y ssign able number whatso ever . 1 1 1 Th e 1 b series 5 5 1 is, y this theorem, con v ergent . 1 1 1 Th ri i 1 i s d ver ent. 3 . e se es 5 g 4 g
Fo r th e first 2 A terms after the first may be written 1 1 1 1 1 1 1 + 2
o o ea o f o hi a a where , bvi usly , ch the expressi ns wit n p rentheses is g re ter 1 than 2 Th e sum o f th e first 2 A terms after th e fir st is therefo re greater than
and ma b e ad to d an a a o tak 3, y m e excee y finite qu ntity wh ts ever by ing A great eno ugh i o o a d th e a m o Th s series is c mm nly c lle h r nic series . By a similar method o f proof it m ay be shown that the 1 1 i f 1 > . series ; is convergent p 21 i
I I 2 1 1 1 l 4 2 2 z — p > and th e sum Of th e o s a a o f th e d a series is , theref re , le s th n th t ecre sing 2 2 2 — geo metri c seri es 1 - 2 p (27 ) l 1 Th e 1 — — if 1 th e e series + + is divergent p , t rms 2p g p being then greater than the correspondi ng terms o f 1 1 + 2 3
h ser i s a a a i s abso lutel co nver ent i 4. T e e 1 + 2 + 3 + y g f a ter so me ter m o nite i nd ex a the ra ti o o ea ch term to f f fi , , , f tha t wh i ch i mmed ia tely precedes i t be numeri ca lly less tha n 1 a nd a s the i nd ex o the term i s i nd e ni tel i ncrea s d a ro a ch , f fi y e , pp 5 9 IN FI N I TE S ERIES .
th n 1 ut di er ent i thi s rati o and a limi t whi ch i s less a ; b v g , f i ts li mi t be g reater than 1 .
— Fo r to co nsider the first hypo thesis suppo se that after the
a o a a a a 0. d o a a term a, this r ti is lw ys less th n , where en tes cert in m a 1 po sitive nu ber less th n .
o d h ” a Then, t “ a ,
a a —— — “ l +k S i f (k l )
Th e given series is therefo re
2 3 " S a t t [a + a a a
And thi s is an abso lutely co nvergent series .
2 " d + a + - a + m
i l mit a. ' n z oo — l a.
a since a is a fractio n . l — a
Th e o a o o 62 1 given series is theref re bs lutely c nvergent, , . Th e same co urse o f reaso nin g wo uld pro ve that the seri es is diver a o m a th e a o o f a to a gent when fter s me ter , r ti e ch term th t which d a o a hi f a prece es it is never less th n s me qu ntity , a , w ch is itsel gre ter a 1 th n .
When the limit o f the ratio o f each term o f the series to
d a di i s 1 the term imme i tely prece ng it , the series is some h i d . T e times convergent, somet mes ivergent series con
ider i n 2 l s ed 6 3 a . , are il ustr tions of this statement
An L m s of Co v r e . 6 3 . i it n e g enc important application Of the theorem just demonstrated is i n determining what are called the limits of convergence of infinite series Of the form )t “ 3 ‘ “ 0 05 1 3? ‘ i “ 293 “ 3517 R - S YS M F AL A 60 N UMBE TE O GEBR .
e x va ab but o a a . wher is supposed ri le , the c efficients o, 1 , etc , a constants as in the precedi ng discussi o n . Such series will be a o f x ffi convergent for very sm ll values , if the coe cients be all b e and a d finite, as will supposed, gener lly ivergent fo r very gre a t values Of x ; and by the limits of conver gence o f the series are meant the v alues o f x for which it ceases to be convergent an d becomes divergent . By the preceding theorem the series will b e c o nverg ent if the limit o f the ratio o f any term to that which precedes it
e 1 i . b numerically less than ; e . if
fl 71 2 Cl )
11m “ a i x be numer i ca ll and di ver ent i th t is, f y g , f n g
li i x be numeri ca lly mt n I oo
T 1 . hus the infinite series
hi a b b o f w ch is the exp nsion, y the inomial theorem,
a a a m for other th n positive integr l v lues Of , is
o o f x a a a di c nvergent for values numeric lly less th n , ver a gent fo r values o f x numeric lly greater than a .
Fo r in thi s case
i a l mit n n 00 a n“ E R IE 61 IN FIN I T S E S .
9 ” f 1 x c o n A a o o e i . e . m 2 . gain the exp nsi n , + , is , 5] ll fi a vergent fo r a nite v lues of x. 1 limit (n ) I Fo r h ere n i oo 1 (n l )
Th e s ame is true for the series which is the expansion ” o f a .
1 Th O r o s on n e e s . . e sum o two 6 4 . pe ati n I finit S rie f c o nver ent seri es a a a nd b b m i s the ser i es g , 1 , 1 2 ,
a b and thei r d erence i s the seri es (a 1 bl ) ( 2 2 ) ifi
‘ ‘ 05 b “ bl ) i ( 2 e) i
Th e sum o f th e ri a a th e m d n d 3 S se es l 2 is nu ber efi e by 1 , 2 , and th e sum o f th e b b th e m d d t t series 1 2 is nu ber efine by ,, , ,
: s : a a a and t b b b . Th e sum o f th e two where , 1 , , , 1 2 ,
fo th e u d n d 3 t 3 t 2 9 series is there re n mber efi e by 1 , , 2 , , ,
B ut i f a b a 6 a we a S : s t S i ( 1 l ) ( 2 2 ) ( , h ve t , + i
‘ di fo r a fo r all a o f i . a o v o o f i v lues This is imme tely b i us finite v lues , and c an b e no d f S and s t as i a o a 00 there i ference between , , , ppr ches , f r since it wo uld b e a difi eren c e having 0 o its li mit . fo th e m d n d 3 t s t th e sum o f There re nu ber efi e by 1 , , , , , is the s a b a b eries ( 1 1 ) + ( 2 , )
h wo utel 2 . T e pro duct of t a bso l y co nverg ent s eri es
a. + a 2 + a nd i s the seri es a b a a b u b a Z) 06 5 l 1 ( b s 2 l ) ( a 3 Q , 3 1 )
“ a b a’ b a a b i ( l u 2 n — 1 n — l n l )
E ach set o f terms within parentheses is to be regarded as c o nstitut ing a sin gle term o f th e pro duct ; and it will be no ticed that th e first o f them co nsists o f th e o ne partial pro duct in which th e sum o f th e indices 2 th o ll in h um o f th e d 3 e d o f a t e s etc . is , sec n which in ices is , h l B 29 th e o d a a b b i s y , pr c o f y u t 1 2 b 1 2 n E- i
s and t th e m o f th e n o f a a where n n represent su s first terms 1 2 b b l 2 respectively . S o a th e r o f a a and b b are upp se first th t te ms 1 2 1 2 all o i f b e th e sum Of th e n m p sitive . Then S n first ter s o f — 6 2 N UMBER S YS TEM OF ALGEB RA .
n — l a b a b a b ) + and m r pr when n is even and l 1 ( l 2 2 1 e esent 3 n Odd when is ,
d s t S s t . evi ently n n n m m
(sutu ) (smtm)
o S s t Theref re ( n) ( n n)
If o f a a b b no t all o f a the terms l 2 l 2 be the s me sign , call th e sums o f the first n terms o f th e series g o t by mak ing all the ’ ’ ' s and t a o S th e o f r 71 signs plus , n n respectively ; ls n , sum the fi st terms o f th e series whi ch is their pro duct . Then by th e demo nstratio n just given li i m t 8 t ( n n ) , n <1 ) n (D
a di o s t a at r a as as but S n alw ys ffers fr m n n by less th n ( g e test by much ) ’ ’ ’ S f o s t o as o n r m n n theref re, bef re h m . t S n sun n ; ( ) ( )
Th e uo ti ent a a x b 3 . q of the series o l y the series b b x m b 0 a m 0 + 1 + ( 0 not ) is series of a similar for ,
c c x m a a x m as o + l + , which converges when 0 + l + is and b x n absolutely convergent 1 + is umerically less than bo.
C O MPL ll . EX S ERIES .
Th e sum co nver ent di ver ent e terms , g , g , have the sam mean ings in connection with comple x as in connection with real series .
G r T s f Co r . 6 5 . o v c A co m lex s r i es ene al e t n e g en e p e , a a m i s co nver ent when th e u 1 + , , g mo dul s of sm may be mad e less tha n any a ssig nable number 8 by ta ki ng m g rea t eno u h a nd tha t o r a ll va lues o n di ver ent when thi s g , f f ; g , c ndi ti o n i s no t i sat ed . 4 o r 8 . 5 . o sfi See , C II ; 9
Of Abs 6 6 . o Co r Let lute nve g ence .
a a bea x 1 2 comple series ,
A A m o f I 2 , the series of the moduli its terms.
64 — NUMBER S YS TEM OF ALGEBRA .
Fo r o o s a n NOTE . ther p int th n Z o the ci rc umference Of this circle th e i s no t a o through Z series necess rily c nvergent . z 2 1 o h s h s ri z 1 . But T u t e e es g § c nverges when Z
o n th e o th e o 1 o 1 a o nd circle thr ugh p int , the p int ls lies ; a the d 1 series iverges when z .
Th T or m. e n 6 8 . he e following is a theorem o which many b Of the properties Of functions defined y series depend .
I the seri es a a z a z" f o l n
ha ve a ci rcle o co nver ence rea ter than the null- o i nt i tsel f g g p f , a nd run thro u h a re ula r se uence o values x Z d e z g g q f l, 2, ni n O the su o n h t i m a ll terms o llo wi t e rs v z . fi g , f f g fi , ,
2 a z a z a z" l z n
will run thro ug h a sequence of va lues li kewi se reg ular a nd
d e ni n O o r the enti re ser i es ma e ma d to d e er a littl fi g , y b e fi s e
a s o ne cho o ses ro m i ts r f fi st term d o.
Th e z z are o f o all o d to lie h th e numbers , , , , , c urse , supp se wit in It e o n o f o and fo r o to b e a . b c circle c nvergence , c nvenience , re l will
i ent z z z i ven a so to o etc . . e . a a a a l supp se l > 2 > a, ; th t e ch is g re ter th n o ne f the o llo wi ng it .
S a a z a z2 ince o l 2
o a o fo r z 2 so a o d o s c nverges bs lutely 1 , ls e
2 "’ a z a z a z l 2 n and o a a z a zu , theref re , l 2 n
’ A A z A z n Hence , 2 1 n l
A o d a o and a M can o d (where , m ulus .) is c nvergent, number be f un a r a it gre te th n s sum .
And fo r z z z i di d a Of since , , 3 , the n vi u l terms
A A z A z " I z n are a o o d o f A A z A s less th n the c rresp n ing terms l z , n ,n and o o d a a z ai a a this series , theref re , m ulus ( l 2 rem n lw ys less
a M as z th e Of a z th n runs thro ugh sequence v lues 3 , 'z th e a o f o d a z a z o o d to Hence v lues m ulus ( l z which c rresp n z z z z o a a d 0 a , , , c nstitute regul r sequence efining , e ch term being numerically less th an th e co rrespo nding term Of th e regul ar sequence
z M d O. l , Z , which efines I I I S RI S 65 NF N TE E E .
Th e a COR . same argument proves th t if
be o f m 1 th o n the sum all terms of the series from the ( ) , a a z be ad d f the series m m+1 can m e to i fer as little as o ne may please from its first term a m .
h o f O r o on Com x e . T e 6 9 . pe ati ns ple Seri s definitions
’ sum d i erence ro d uct x , fi , and p of two convergent comple series a a viz are the s me as those already given for re l series ,
The sum o two o nver ent s eri es a a nd 1 . a f c g , 1 2 b b m i s the seri es a b thei r 1 2 , ( l z) d er enc e the seri es a b a b ilf , ( 1 , ) ( 2 ,)
Fo r if a a a and i z b b 3 , l + 2 + , , , + , +
o d s t S t m ulus [( m+n 4 w i n) ( m 4 m) ]
o d l S S o d f t m u us ( m m) m ulus ( m+n m) , and ma o be ad a an a a tak y , theref re , m e less th n y ssig n ble number by m Th o ing great enough . e theo rem therefo re fo ll ws by th e reas o nin g o f 64, 1 .
T he r duct o two a bso lutel o nver ent seri es 2 . o c p f y g ,
a1 + a 2 + af 3 + and b1 + b2 + b3 +
th a b Ci/ b (1 1 6 CLO (1 6 a b e l l ( l z 2 1 ) + ( 1 3 2 2 s l )
Fo r S A A A and T 8 1 3 B , letting . I 2 , , 1 + 2 “ A B are o d o f a b and n where ” the m uli , , , , respectively , representi g th e o f th e 72 r o f by a n sum first te ms the series
a b a b l l z i ) and o f th e n o f th e by 2 7. the sum first terms series A 3 4 3 A 1 1 ( 1 2 2 3 1 ) we a o d l s t h ve m u us ( n n o n ) S n Tn 5 )
But the limit o f the right member o f this inequality (o r equatio n) is O 64 o , theref re limit n i 00 66 R - S S M OF A R A N UMBE Y TE LGEB .
IX. THE E ONE T A AND OGA T C CT XP N I L L RI HMI FUN IONS.
R D L D UNDET E M INE C O EFFIC lENT S . INVO UT IO N AN
E O LUT IO N. T HE BIN M AL HEO RE V O I T M.
A o . v ab e w be a un 70. Functi n ari l is said to f cti o n of a
second vari able z for th e area A Of th e z - plane when to the z belonging to every point o f A there corresponds a
e o r v determinat value set Of alues of w.
if w : 2 z w fu o f z F r z w z a o . o 1 2 Thus , is ncti n when , ; when z z 2 w z 4 and ik a a d a a o f w r , ; there is in l e m nner etermin te v lue fo
- a o f z . In a A o wi z a every v lue this c se is c extensive th the entire pl ne .
S a w a fu o o f z if imil rly is ncti n ,
2 / n w (1 ‘ " (1 2 61 2 G z “ ” 0 i 1 2 ri i
so o as n i s o i . e . fo r o o Of l ng this infi ite series c nvergent, the p rti n the z - a o d d a a i th e - o fo r and fo r pl ne b un e by circle h v ng null p int centre, radius th e mo dulus o f the smallest value o f z fo r which th e series d i verges .
It i s customary to use fo r w when a function of z th e 1 ‘ ” mb z n z. sy ol f( ), read functio
1 u c i o al E a i o of the Ex o c i o . 7 . F n t n qu t n p nential Fun t n z ‘ z “ - Th r v t a a d . e Fo positive integral alues of z and ,
o f question naturally suggests itself, is there a function z n x b n which will satisfy the conditio e pressed y this equatio , ” o r = z t a ll the functional equation f ( ) , for values of z and t ? We proceed to the investigation Of this question and
nl b defi another which it suggests , not o y ecause they lead to ” ni ti o ns a lo z x Of the important functions and g a for comple v o f a z v o f alues and , and so gi e the Operations involution, v o f e olution, and the taking logarithms the perfectly gen eral character already secured to the four fundamental — but b ff x Operations , ecause they a ord simple e amples Of a " large class o f m athematical i nvestig ati o nsfi
An applicatio n o f th e principle o f permanence 1 2) is invo lved Th e a o in the use o f functio nal equatio ns to define functio ns . equ ti n UN DE TE R MIN ED COEFFI CIENTS . 67
ffi n f e e m Co s . a o 72 . Und t r ined e cient In i vestig tions this d o o y d o ne o r a sort , the metho c mm nl use in form nother is i that Of und etermi ned co efii c ents . This method consists in assuming for the function s o ught an e xpression involving a
o f n but a a — c o effic i ents series unk own const nt qu ntities , in substituting this ex pression in the equation o r equations which embody the conditions which the function must and so n k n satisfy , in determini g these un own constants
a a be i denti ca ll a th t these equations sh ll y s tisfied, that is to d fo r f sa a o a b a ab . y, s tisfie all values the v ria le or v ri les Th e b o n a method is ased the following theorem, c lled “ ” o f n d i vi z the theorem u eterm ned coefficients ,
2 ’ ' If the seri es A + Bz + 0z + m be equal to the seri es A E x ’ z O e o r a ll va lues o whi ch ma ke bo th o nver ent + f f z c g , a nd the co e i ci ents be i nde endent o z the co e i ci ents o like fi p f , fi f
w s po er of z in the two are equal.
Fo r , since
2 2 A + Bz + 0z + 06 +
’ 2 A C ) x + throughout the circle o f convergence common to the 67 69 given series , ,
An d b hi eing convergent within t s circle , the series
’ ' ’ 2 A A + (B E ) x (O C ) x
‘ ‘ t a a : a “ fo r a o l o a n o a a o , inst nce , n y bec mes fu cti n l equ ti n when its per ma nenc e i s assumed fo r o ther values o f z and t than tho se fo r Whi ch it h as been actually demo nstrated . In thi s respect th e metho d s o f defini tio n o f th e negative an d th e f a o o n th e o ne a d and fu o a Z l z o n th e o are r cti n h n , the ncti ns , og , , ther,
d a but i a o a b b a f d as defi i entic l , wh le the equ ti n ( ) itsel serve niti o n o f a — b i no o Of , there be ng simpler symb ls in terms which a 6 o u d b e d f o th e a o a Z a t a “ t a c l expresse , r m equ ti n series 73 ma b e d d d i d fi a z o f o f , y e uce wh ch e nes in terms numbers th e a i h system . — B A 68 N UMB ER S YS TEM OF ALGE R .
t c an be made to differ as little as we please from its firs A term, — ’ = ’ A = O 30 A A . A , or Therefore
’ 2 (B O ) z + m 0
o f e throughout th e common ci rcle convergence, and henc
fo r v ff O (at least, alues of z di erent from )
B 0.
Therefore by the reasoning which proved that
' A B o r B = B
’ " ’ C C D D . In like manner it may be proved that , etc
2 2 A E x t —Dz E xt Ft C OR . If C +
’ ' ’ ’ ’ z A + B z C t E zt E t
o r a ll va lues o a nd t whi ch make bo th ser i es co nver ent and f f z g , be i nd e endent o t a nd th e co e i ci ents i nd e endent o bo th z p f , fi p f z a nd t the co e ici ents o like wers o and t i n the two seri es , fi f po f z a re equal.
Fo r e b e o f , arrang oth series with referenc to the powers Th e f either variable . coefficients o like powers o f this re b di T v ariable a then e qual, y the prece ng theorem . hese va b b coefficients are series in the other ria le, and y applying the theorem to each equation betwee n them the corollary is demonstrated .
T The Ex o F c i o . O hi 73 . p nential un t n apply t s method to the case in hand, assume
2 z = f ( ) A0 + 21 1 Z + A2z + Anzn +
' o f and determine whether values the coefficients A, c an be “ ” a ab o f f e found c p le satis ying the functional quation,
' z t = z t f ( ) f ( ) f ( ) . for all values of z and t. E X P O IA F I 69 THE NENT L UNCT ON .
v fo r v e On substituting in this equation, we ha e, all alu s nd t v Of z a for which the series con erge,
o r x a d e , e p n ing and arranging the terms with referenc to the and t powers of z ,
2 z AOAO 441 1 402 AOAl t A2A0z A1 A1 zt Ao Azt
- 1 t A - k + . r +
? ‘ ' Ac i Alz + Al t Ag e + 2 A2zt A26
” ’ H n ' k " “ A x A nz t A n z t u n n k Ant where k I
E quating the coefficients o f like powers of z and t in the b o f a two mem ers this equ tion, we get
a a to A n . equ l lways n k A A A A A 1 . In particular O 0 O, therefore O lso A A 2 A A A 3 A I 1 2, 2 1 3, A A 4 ”A 3 I A Art— 1 1 n ; o r mu n b b membe , ltiplyi g these equation s together mem er y r,
1 " A A n o r A 1 I n n $
A a a ef part Of the equ tions mong the co ficients are, there suffic i ent f fore , to determine the values o all of them in m o ne . ter s Of the coefficient A1 But these values will satisfy the remaining equations ; for substituting them in the gen eral equation
A — A A n n k k n ie)
n — k k A1 A we g et X __L (n k) ! k i
i s b which o viously an identical equation . N MB R - S YS TE M OF AL U E GEBRA .
Th e AI o r A coefficient , more simply written, , remains 0 undetermined .
h as b d d to It een emonstrate , therefore, that satisfy equa tion it is only necessary that f (z) be the sum o f an infin ite series o f the form
A h as sum t. where is undetermined ; a series which a , e. is v v f , o and A . 63 2 con ergent for all finite alues z , , B y A z m a be properly determining , f ( ) y identified with ’ a fo r an a f , y particul r value o a .
‘ If a is to be identically equal to the s eries A must hav e such a value that 2 3 — A Ct 1 A + 2
2 z — 4 2 ! $ 1 — — l Where e + + 2 ! 3 !
2 S A A A
“ Therefore a e o r b , calling any num er which satisfies the equation
” e a
lo ari thm a b e n lo a the g of to the ase and writi g it g , ,
A lo a . g c
i m e th e a of th e Na a o f o a m Th s nu ber , b se peri n system l g rith s , is ” a “ a d a rra o a a d a th e a tr nscen ent l i ti n l , tr nscen ent l in sense th t there is no algebraic equatio n wi th integral co efficients o f which it c an b e a o o see r o R d LXX II h as th e a a r t ( He mite, C mptes en us , V ) . 1 r s me ch r a as L d a o d 1 882 d d at th e a th e cter , in em nn pr ve in , e ucing s me time first actual demo nstrati o n o f th e imp o ssibility o f th e famo us o ld pro b lem o f squaring th e circle by aid o f the straight edge and co mpasses o see Ma a A a XX nly ( them tische nn len, ) .
M R - S YS TE M OF A RA 72 N U BE LGEB .
o c . 9 75 . Peri di ity When is real, evidently neither its sine nor its cosine will be changed if it b e increased or
b 2 1 r o r diminished y any multiple of four right angles, or ; , i f n be any positive integer,
9 c o s 9 9 j: 2 n1 r si n 9 cos ( , sin ( ) ,
e ei mi z n”) i f’ and henc e .
i " Th e e 9 si n 9 o n d functions , cos , , are this account calle
eri o di n mo dulus o eri o di i t Z r . p c fu ctions, with the f p c y
z nd t : e The o r hm c i o . e a 76 . L g a it ic Fun t n If z
Z T z zt e e e “ 73 ,
l o z 7 t z t . g e loge log , ( )
Th e question again is whether a function exists capable “ o f o r l th e satisfying this equation, , more general y, func ” ti o nal equation,
f (zt) =f (z) f fo r comple x values o z and t.
W : O 7 b hen e , ( ) ecomes
lo o 0 t g e log , log, , an e quation which cannot hold for any value o f t fo r whi ch loge t is no t z ero unless log o is numerically greater than
T lo O . b v . any finite num er whate er herefore g e is infinite On 1 7 b m the other hand, when z , ( eco es
1 0 l lo t t loge g , ,
so that loge l is z ero . m m Instead, therefore, of assu ing a series with undeter ined
z coefficients for f ( ) itself, we assume one for setting
1 A zn f ( + n +
n f A o f and i quire whether the coe ficients , admit values which s atisfy th e functional equation (8) for comple x v f alues O z and t. THE O ARI MI N I 3 L G TH C FU CT ON. 7
N o w 1 t 1 z 1 z ( ) identically .
+ 1 +
? A z t A z t A z t " 1 ( + ) + z ( ) + n ( )
2 n A 27 A 2 0 ” A z 1 + 2 + + n
Equating th e coefficients Of the first power o f t 72) i n th e m b two em ers of this equation,
z A 2 A z 3 A z n 1 A zn 1 + 2 + 3 + ( ) n+1
— 2 3 " n A 1 - - — 1 ( z i z z 1 ) z +
z whence , equating the coefficients of like powers Of ,
A = A Z A — A ”A = 1 D 2 D n (
As o f x o f in the case the e ponential function, a part the equations among the coefficients are sufficie nt to determine n But as a them all in terms of the o e coefficient A1 . in th t case (by assuming the truth o f th e bi nomial theorem for n egative integral values of the e xponent) it c an b e re a dily shown that these v alues will s atisfy th e remaining e qua tions also . 2 3 n ’ 2 Z _ i _ n— 1 Th e seri es z 1 2 3 n
l conv erges fo r all values o f z whos e modu i are less 62 3 (s . ) Fo r v e th e such alues, therefor , function
l ) n - 1 % satisfies th e functional equation
f [(1 z) (1 + z) +f (1 + 0 - M OF A R A 74 N UMBER S YS TE LGEB .
l — l — z ant I — l — t And si nc e z s ( ) ( ) , th e function 2 satisfies this equation when written in the simpler form = f(zt) f(z) for values of 1 — z and 1 - t whose modul i are both less than 1 .
TO 1 2 1 . Lo b. g , identify the general function f ( ) with the particular function loge (1 + z) it is only necessary to giv e the undetermined coefficient A the v alue 1 .
Fo r 1 z b fu n since log , ( + ) elongs to the class Of nctio s which satisfy the equation
l0 1 Z A g 6 ( ) Therefore A lo 1 +2 e ge( ) 3
1 +A
1° z g e a+ l But e 1 z. H ence 2 2 z 1 z
1 o r th e z A . , equating coefficients of the first power of , Th e coefficients of the higher powers Of z in th e right number are then identically 0. b lo b b It has thus een demonstrated that g e is a num er
. o r x b w 1 z (real comple ) , if when is ritten in the form + , To v i the absolute value of z is less than 1 . pro e that it s a b o f b b ew num er for other than such values , let p b m ul v b . where p, as eing the od us Of , is positi e T i 9 hen log, loge p ,
b . and it only remains to prove that log, p is a num er Let be e" en e e" the p written in the form ( p) , wher is o f e first integral power greater than p. O ARI H MI I O 5 THE L G T C FUNCT N . 7
" Then since (e p)
” e i i p and lo g s a number S nce i s less than 1 .
b 2 . Lo b. v g “ It ha ing now een fully demonstrated that
Z Z T Z ‘ H ” a i s a number satisfyi ng the e quation a a = a fo r all z T n a T l t a = z a = t and th e fi ite values Of , Z, ; e , , call Z lo ari thm o to the base a o r lo z i n e r g f z , g a , and lik manne
T t. , log ,
z T T e zt a a hen, sinc
lo zt lo z lo t g a ( ) g a g a , o r lo z b z to g a elongs, like loge , the class of functions which satisfy the functional equation P r d a o f lo b u suing the method followe in the c se g , , it will 2 z be u 1 z A fo nd that log a ( ) is equal to the series 1 when A This number is called the mo d ulus Of 1 oge a the system o f logarithm s Of which a is bas e .
m . n ex nu 77 Indetermi nateness of log a . Since a y compl ber a ma be h n ew y t row into the form p ,
a lo 1 0 loge g e p ( )
T wev l o ne an n e o f b e his , ho er, is on y of i finite s ries possi l v f i e i wi zn’f ) o a . Fo r c e alues loge , since i H a lo e fl lo i 9 zt 2 n7r loge g e p g e p ( )
Lo a e e n be an . wher may y positive integer ge is, th refore , to a certain ex tent indetermin ate ; a fact which must be care “ ful ly regarded i n using and studying this func ti o nfi Th e
Fo r instance lo ge (z t) is no t equal t o lo ge z lo ge t fo r arbitraril y
o a Of o a but to o z 10 mi 2 n1 r W n ch sen v lues these l g rithms , l ge t , here is o s me po sitive integer . R —S YS M OF A B RA N UMBE TE LGE .
v v n = O i s a d a alue gi en it in for which , c lle its princip l
Wh n a i s a b 0 so a e positive real num er, 9 , th t the prin c i al v o f a o n h a d a p alue loge is real ; the ot er h n , when is a
e b 9 7r o r o f a negativ real num er, , the principal value loge i s m o f n b a the logarith the positive um er corresponding to , plus i rr.
'
rm c o f m ws of Ex o s . 78 . Pe anen e the Re aining La p nent
z t z t law a a a + i ts th e Besides the which led to definition, function a ’ is subj ect to the laws
z ‘ ” a a 1 . ( ) .
2 a b z . ( )
z t
1 . a M 2 2 Fo r 1 lo a z 0 73 4 ( g , ) , ( )
“ “ e °g e 73 3 , ( )
m “ z ge l a e o a . , and g e z loge
From thes e results it follows that
‘ z c lo wz ) (a ) e g e
t lo a z e g e
tz lo a e g e
“ a .
b ‘ c o m d d a o da a a , whi h is s meti es inclu e m ng the fun ment l a t Z t z t z is c fo o m d a fro a a z a + laws to whi ch a subje t, ll ws i me i tely m by
th e definitio n o f divi sio n . I N OL UTI ON A D O I V N EV L UT ON . 77
m f o the m Law o f o r hms . 79 . Per anence Re aining L g a it a n o lo b n th e In like m n er , the functi n w is su j ect not o ly to law xt t log , ( ) log, z log, , but also to th e law
‘ l lo o z t z. g , g a
lo z a g a z ,
t lo z a g a § 78 1
Evo i o . C e x n be i z 80. lut n onsider thre comple um rs f, , Z, Z c onnected by the e quation { z .
T a i i b a his equ tion g ves r se to three pro lems, e ch of which
Fo r n m is the inverse of the other two . Z a d 2; ay be given and and be and z sought ; or Z z may given Z sought ; or, ma n a be a d . fin lly, z and Z y given Z sought Th e e xponential function is the general solution o f th e
b i nvo luti o n and n o f first pro lem ( ) , the logarithmic fu ction the second .
r h d o luti o n f h a d Fo t e thir (ev ) the symbol i ; s been evised . This symbol does not represent a new function ; for it is b x/ i z n defined y the equation ( Y , a equation which is x ! satisfied by the e ponential function z . L m / z ike the logarith ic function, f is indeterminate , though x no t always to the same e tent . When Z is a positive 2 i : b n and b 5 6 h as nteger, £ z is an alge raic equatio , y Z Z / fo r o ne § i s b d n a b . roots any Of which \ , y efinitio , sym ol
th e a z : t a n be i n From mere fact th t , therefore, it c n ot Z Z / / t but o ne o f a o f ferred that \ x , only that the v lues 7 Th e a a 7 o ne o f o f 5 . { 3is equal to the values < s me rem rk, 1 1 ! i o f a b z t . course, pplies to the equivalent sym ols ,
1 erm e of the B omi T eo em. B ai d 8 . P an nce in al h r y of the b d ma d l b e d d a results just O taine , it y rea i y emonstrate th t the binomial theorem is valid for general complex as well x as fo r rational values Of the e ponent . 8 UM R - s YS TE M OF A B R A 7 N BE LGE .
Fo r b b n x b v and th e ei g any comple num er whatsoe er, v e 2:b 1 absolute alu Of eing supposed less than ,
— 1 bz terms n v h o f z + + i olving igher powers . Therefore let
b 2 1 z = 1 bz ( + ) + + A2z +
” a Since, then, if 2be substituted for z in and the equation be multi b b a plied throughout y ,
” ” b ‘ l ”‘ 2 2 ” ” " a z a ba e A a z A a z 1 2 ( ) z n ( ) Starting with the identity
” ” developin g (1 + z t) by (1 1 ) and (1 z t) by e quating th e coefficients o f th e first power o f t in thes e l i th e e e a n b 1 z developments, mu tiply ng r sultant qu tio y + , and equating th e co effi cients o f like powers o f z in this
d b va be pro uct, e quations are O tained from which lues may d A e erived for the coefficients , identical in form with thos ” occurring i n the d evelopment fo r (1 z) when b is a posi v ti e integer . It may also be shown that these values o f the coefficients s atisfy the equations whi ch result from equating the c o effi c i ent o f t s Of higher powers .
T E I . PRIMI IV NUMERALS .
G s r mbo s. T d b v 82 . e tu e Sy l here is little ou t that primiti e n w as d o n n a counti g one the fingers, that the earliest umer l symbols were groups o f the fingers formed by associating a single finger with each indi vidual thing in the group o f w a d things whose number it s desire to represent . Of course the most immediate method of representing the number o f things in a group and doubtless the method first used — is by the presentation of the things themselves o r the recital of their names . But to present the things themsel ves or to recite their names is not in a proper sense to count them ; for either the things or their names represent all the properties o f the group and not simply th e
f w as d number o things in it . Counting first one when a group was used to represent the number of things in som e other group ; o f that group it would represent the number an d be b only , therefore, a true numeral sym ol, which it is the sole Obj ect of counting to re ach . Counting ignores all the properties of a group except the distinctness or separateness of the things in it and pre supposes whatever intelligence is required consciously o r unconsciously to abstract this from its remaining properties . On a b this ccount, that group serves est to represent num b d a d f o f b ers , in which the indivi u l if erences the mem ers
a Th a fin e r - b . e are le st o trusive n turalness of g counting,
no t a b therefore, lies only in the ccessi ility of the fingers, b but in their eing always present to the counter, in this that the fingers are so similar in form an d function that it is almost easier to ignore than to take account of their n differe ces . But there is other evidence than its intrinsic probability 81 2 B R —S S M OF A B RA 8 N UM E Y TE L GE .
th e o f fin er - v N for priority g counting o er any other . early e very system o f numeral notation o f which we have any V o r mi x knowledge is either quinary, decimal, igesimal, a 3“ o f a sa x b ture these ; th t is to y, e presses num ers which are 5 o f 5 b greater than in terms and lesser num ers, or makes 1 0 r T a similar use of o 20. hes e systems point to primi o ne d tive methods of reckoning with the fingers of han , the o f b a d all fin and fingers oth h n s , the gers toes , respectively .
F - inger counting, furthermore, is universal among uncivil i z e d b v far o tri es of the present day, e en those not en ugh developed to hav e numeral words beyond 2 o r 3 represent ing higher numbers by holding up the appropriate number o f fing ers j
— o mbo s. N b 83 . Sp ken Sy l umeral words spoken sym ols — d a a a woul n turally rise much l ter than gesture symbols . v o f be t a d Where er the origin such a word can r ce , it is found to be either descriptive o f the corresponding finger — symbol o r when there is nothi ng characteristic enough b fin b a a out the ger sym ol to suggest a word, as is p rticularly the case with the smaller numbers — the name o f some T a familiar group of things . hus in the l nguages of numer o us 5 i s d 1 0 tribes the numeral simply the word for han ,
at a and Vi a s are a if d d o Pure quin ry gesim l sy tems r re , in ee they ccur ll As an a o f fo o P 1 at a . ex mple the rmer , Tyl r ( rimitive Culture , , 26 1 a a o a 1 2 3 4 p . ) inst nces P lynesi n number series which runs , , , ,
5 5 2 and as an a o f - a a o s i , , ex mple the l tter, C nt r (Ge ch chte w o th e o a o o f th e Ma a d r M a k . o o e athem ti , p f ll ing P tt, cites n t ti n y s a s a o d fo r 20 400 8000 o f Yucatan wh o h ve peci l w r s , , , The
n a o i th e I do - Ar a a fo d s an a o f a Hebrew o t ti n , l ke n bic , f r ex mple pure
M d are o o . s th e R o a decimal no tatio n . ixe systems c mm n Thu m n is
and ui a Az d Vi a and u a . mixed decimal q n ry , the tec mixe gesim l q in ry a th e a and V a are o i re Speaking gener lly , quin ry igesim l systems m re
h e o a d a a o th e . P quent amo ng t l wer r ces, the ecim l m ng higher ( rimi
C I . tive ulture , , p o fo r i a th e a o o f V o a and th e Bo o o o f 1 S , nst nce , b rig ines ict ri r r s
B az I . r il (Primitive Culture , , p A S 83 PR IMI TI VE N UMER L .
“ ” b nd s 20 an a and for both a , for entire man (h nds feet) ; * d a o r . while 2 is the wor for the eyes , the e rs , wings As its original meaning is a distinct encumbrance to such
as a no t a th e a word i n its use numeral, it is surprising th t numeral words of the highly developed languages have been so modified that it is for the most part impossible to trace their origin . Th e practice o f counting with numeral words probably T i s an arose much later than the words themselves . here artificial element in this sort o f counting which does not appert ain to primitive counting T (see One fact is worth reiterating with reference to both th e n primitive gesture symbols and word symbols fo r umbers . There is nothing in either symbol to represent the indi vidual characteristics of the things counted or their arrange
Th e us o f b a ment . e such sym ols , therefore, presupposes conviction that the number o f things in a group does not
In th e language o f th e Tamanacs o n the Orino co th e wo rd fo r 5 ” a “ a o a d th e o d fo r 6 “ o ne o f o a d me ns wh le h n , w r , the ther h n , ” and so o n u to 9 th e o d fo r 1 0 a “ o a d 1 1 “ o n e p ; w r me ns b th h n s , , ” ” to th e fo o and s o o n u to 1 4 1 5 “ a o fo o 1 6 “ o ne to t , p ; is wh le t, , ” ” th e o fo o and so o n u to 1 9 20 “ One I d a 40 “ tw o ther t, p ; is n i n, , ” r d i h riri I d a etc . O a a m a are t e C a n i ns , ther l ng u ges ich in ig t nu er ls y , Abi o ne and a i o f S o Am r a E o A z and Tupi , p , C r b uth e ic the skim , tec , 1 u P C . Z lu ( rimitive ulture , , p Tw o a o d m a i a in Chinese is w r e n ng e rs , in Thibet wing, in ”
Ho o a d . o w H M G S o o r o f a a . ttent t h n ( , h rt ist y Greek them tics , p — S ee a o r I . 2 5 2 2 5 9 . ls Primitive Cultu e, , pp 1 Were there any reaso n fo r suppo sing that primi tive co unting was d o a o d o d b e o a a th e o d a no t ne with numer l w r s , it w ul pr b ble th t r in ls , F r h f th e a di a th e a i a . o t e o ma o d o c r n ls , were e rl est numer ls n r l r er th e cardinals must have been fully reco gniz ed befo re they c o uld be d use in co unting . In o o K o U d en a ff Jo ur this c nnecti n, see r necker , eber Z hlbegri n l f d M V l 1 . a r a 0 33 7. ii di e i und a a a o . 1 re ne ngew n te them tik , , p Kro necker go es s o far as t o declare that h e finds in th e o rdin al num bers th e natural po int o f departur e fo r th e develo pment o f the number co ncept . 4 R - s YS TEM OF A R A 8 N UMBE L GEB . depend o n the character o f the things themselves o r on a o but o n a their colloc ti n, solely their maintaining their sep rateness and integrity .
m h Wr t bo s . T e wri tten b 84. it en Sy l earliest sym ols for n b d be e o f um er woul naturally mer groups strokes I, II, b va etc . ve b v III, Such sym ols ha a dou le ad ntage o er
b : be e are gesture sym ols they can mad permanent, and b o f x — b capa le indefinite e tension there eing, of course, no the n m limit to u bers of strokes which may be drawn .
T Y TE OF OTAT O . II . HIS ORIC S S MS N I N
E oe i . T b 85 . g yptian and Ph n cian his written sym olism did no t assume the complicated character it might have h ad th , had counting with written strokes and not with e
P a fingers been the primitive method . erh ps the written strokes were employed in connection with counting num bers higher than 1 0o n the fingers to indicate how often all the fingers had been used ; o r if each stroke corresponded to v a th e an indi idu l in group counted, they were arranged as
o f 1 0 b was they were drawn in groups , so that the num er represented by the number of these complete groups and the strokes in a remaining group of less than 1 0. At d d x all events, the decimal i ea very early foun e pres
a b fo r 1 0 1 00 and be sion in speci l sym ols , , if need , of higher f powers o 1 0. Such signs are already at hand in the earliest known writin gs Of the E gyptian s and Phoenician s in which numbers are represented by unit strokes and the signs for 1 0 1 00 1 000 v , , , and e en each repeated up to 9 times .
Gr In o f n n o f 86 . eek . two the best know otations a n t — ntiquity, the Old Greek ota ion called sometimes the R I S YS MS N A I N HI S TO C TE OF OT T O . 5
o A a H erodianic, s metimes the ttic and the Roman, primi tive system o f co unting o n the fingers o f a single hand h as
b fo r left its impress in special sym ols 5 . In the H ero dianic notation the only symbols — apart from certain abbreviations fo r pr o ducts of 5 by the powers — ' ’ l O are r WEI/ r e A 36 Ka H éKa r' Ov Of ( , ( , ( , ' M v tOt all o f x i (p p , them, e cept , it be a o f T will noticed, initi l letters numeral words . his is nl be d d A the o y notation, it may a de , found in any ttic n o f a da b h n C . T e a a d inscriptio te efore hrist l ter , for the
o o f a o a purp ses rithmetic, much inferi r not tion , in which the 24 letters of the Greek alph abet with three inserted n d b 1 2 1 0 stra ge letters represent in or er the num ers , , ,
20 1 00 200 9 00 was a a o d , , , , pp rently first empl ye in
A x B d a a a a 3d . C . an b b le ndri e rly in the century , pro ly originated in that city .
Th e a o i b b o f Etr Rom . s us 87 . an R om n notati n pro a ly It h as o n e d e a can origin . very istinctiv peculi rity the subtractive meaning o f a symbol o f lesser value when it pre
o ne o f a as in W : 4 an d a i n cedes gre ter value, in e rly sc ripti o ns IIX 8. In nearly every other known system o f notation the principle is recogniz ed that the symbo l o f lesser a o f a and be add d value shall follow th t gre ter value e to it . In this connection it i s worth noticing that two o f the four fun dam ental Operations of arithmetic — addition and multiplication — are involved in the very use Of speci al
fo r 1 0 and 1 00 fo r but a b fo r symbols , the one is sym ol the sum 1 0 b fo r 1 0 o f 1 0 of units , the other a sym ol sums units d add o ma o r fo r ro duct 1 0x 1 0. d each, the p In ee , iti n is pri
o a n abb a d rily only abbreviate d c unting ; multiplic tio , revi te
d Th e a a b Of n a dition . represent tion of num er in terms te s and o x o o f th e units , moreover, inv lves the e pressi n result o f a division (by 1 0) in the number of its tens and the result of a subtraction in the number of its units . It does o o f a o o f a h ad not foll w, course, th t the invent rs the not tion 86 N UMB E R —S YS TE BI OF A B R A LGE . any such n o tion of its me aning or that these inverse Opera
n ar e add and as o ld as tio s , like ition multiplication, the
E - o symbolis m itself . Yet the trusco R om an notati n testifies
b b a . to the very respecta le antiquity of one of them, su tr ction
I o - Ar b A a a th e 88 . nd a ic . ssoci ted thus intim tely with
da o f a a a four fun mental operations rithmetic, the ch r cter o f the numeral notation determines the simplicity or com l xi f An i n p e ty O all reckoni ngs with numbers . unusual t erest a a n b , therefore, tt ches to the origi of the eautifully cle ar and simple notation which we are fortunate enough
a a o i s b e a re c i to possess . Wh t boon that n tation will pp ated by one who attempts an exercise in division with th e
R o m an o f all G a . or, worst , with the later reek numer ls Th e system of not ation i n current use t o day may be A b characteri zed as the positional decimal system . num er is resolved into the s um
a l 0" 1 O d n a 1 o,
“ 1 0 o f 1 0 a and where is the highest power which it cont ins, a a are all b a 1 0 an d re re n, o num ers less th n ; then p — d n - i sente by the mere se quence of umbers a O t b t o o si ti o n eing left the p Of any number a , in this se quence to indic ate the power of 1 0with whi ch it is to b e ass o ci
Fo r f o b e — b e ated . a system o this s rt to complete to capable o f representing all numbers unambiguously — a sym b o l which will indicate the absence of any particular ” o f 1 0 sum a 1 0 a 1 0 ( t power from the + 1 o, T 0 1 01 and 1 1 b b e b . is indispensa le hus without , must oth
Bu o at a d an b m a b e 1 1 . t b written this sym l h n , y num er y xpre sse d unambiguously in terms of it and symbols fo r 1 2 , , 9 .
h Th e a Bab an T e positional idea is very old . ncient yloni s c o mmonly employed a decimal not ation similar to that Of the Egyptians ; but their astronomers h ad besid es this a
b a sexa es ima l a . very remarka le not tion , a g position l system
MB R —S S M F A B 88 N U E Y TE O LGE RA .
a a 400A . D . and d a cert inly l ter than , there is no evi ence th t Th a A . D e a it w as e arlier th n 5 00 . earliest known inst nce of a date written in the new notation i s 73 8 By the a 0 a d v d time th t came in, the other char cters had e elope
so - c D a b n into the alled evanagari numer ls (ta le, colum a o f d the cl ssical numerals the In ians . Th e perfected India n system probably passed over to th e
A b a 773 A . D . a ra i ns in , long with certain astronomical writings .
H ma be x o f owever that y , it was e pounded in the early part
9 th b Alkhw ari zmi and a o n ad the century y , from th t time spre ad a A ab a d a gr u lly throughout the r i n worl , the numer ls tak n ing different forms in the E ast a d in the West . Europe in turn derived the system fro m the Arabians i n ” 1 2th Go ba a ab o 5 the century, the r numer ls (t le, c lumn ) o f the Arabians of Spain being the pattern forms of the Euro " a a ab Th e d d pe n numer ls (t le , column arithmetic foun e on th e new system was at first called a lg o ri thm (after Alkh wari zmi d o f ) , to istinguish it from the arithmetic the abacus which it came to replace . A word must be said with reference to this arithmetic o n
b o r b . v b the a acus In the primiti e a acus, reckoning ta le , b b th e unit counters were used, and a num er represented y appropriate number o f these counters in the appropriat e
. 21 b columns o f the instrument ; e g . 3 y 3 counters in the ’ ’ o f l 00s 2 o f l 0s and 1 the column , in the column , in Th e n an aba c o lumn Of units . R oma s employed such cus but a was i n in all the most element ry reckonings, it use
an d d a a . in Greece , is in use to y in Chin
B a l o r ithm efore the introduction of g , however, reckon ing o n the abacus h ad been improved by the use in its c o lumns of separate characters (called api ces ) for each o f
n b 1 2 9 d o f . the um ers , , , instea the primitive unit counters This improved abacus reckoning was prob ably invented by G erbe rt (Pope Sylvester and certainly used by hi m at ab 9 70— 9 80 and b Rheims out , ecame generally known in the following century . S YS M S OF O A N 9 HI S TORI C TE N T TIO . 8 R — S YS M OF A B RA 9 0 N UM B E TE L GE .
N o w a but b l these apices are not R oman numer ls, sym o s which do not differ gre atly from the Gobar numerals and f r e a . b o a cle rly, like them , of Indian origin In the a sence p o sitive e vidence a great controv ersy h as sprung up among historians o f mathematics over the immediate origin Of the
Th e a o apices . only e rlier menti n of them occurs in a pas
a Of was s ge the g eometry which, if genuine,
A D . t a written about 5 00 . Basing his argumen on this p s
r i a an - . C n r r a ea l r i s ge, the histori a to u g es th t the s I nd an numerals found their w ay to Alexandria before her inter r E was b a b end cou se with the ast roken off, th t is , efore the
o b B o o f 4th and. a the century, were tr nsf rmed y oetius int B a . On a the pices the other hand, the p ssage in oetius is
a b be and quite gener lly elieved to spurious, it is maintained Gerbert g o t a i d n that u his p ces irectly or i directly from the A ab a o f 0 b a di d r i ns Spain, not taking the , either ec use he o f b b b h e not learn it, or ecause , eing an a acist, did not
appreciate its value .
At all a a - A ab events , it is cert in th t the Indo r ic numerals, 1 2 9 a d C a E u , , (not appe re in hristi n rope more than a b century efore the complete positional system and a lg o ri thm. Th e Indi ans are the inventors not only of the positional a but Of v decim l system itself, of most the processes in olved A in elementary reckoning with the system . ddition and subtraction they performed quite as they are performed a a a f now d ys ; multiplic tion they ef ected in many ways, ours but di b . among them, vision cum rously
THE A T . III . FR C ION
m f a v o s. O o f 89 . Pri iti e Fracti n the artifici l forms num ber — as w e m a a a y call the fraction , the irr tion l , the nega to tive , and the imaginary in contradistinction the positive
9 2 UMB E R—S YS TE M OF AL B R A N GE .
Th sums o f such unit fractions . e Oldest mathematical
a — a a d D tre tise known, p pyrus roll entitle irections for ” A i K d o f All D T ttain ng to the nowle ge ark hings , written by a scribe named Ahm es in the reig n o f R a - a- us (therefore b 1 700 a d as a o f efore fter the mo el, he s ys , a more a — a ab ncient work, opens with t le which expresses in this
a o f 2 b o dd b m nner the quotient y each num er from 5 to 9 9 . T 2 b 5 3 1 5 b hus the quotient of y is written , y which is
—— a l 2 b 1 3 8 5 2 1 04. me nt é é ; and the quotient Of y ,
O n nly 2, among the fractions having umerators which ff 1 d di er from , gets recognition as a istinct fraction and receives a symbol o f its o wn .
2 B b o i o r x s m r c o . Th e 9 . a yl n an Se ag e i al F a ti ns fractional notation o f the Babylonian astronomers is o f gre at interest n L intrinsically a d historically . ike their notation o f i n r x m a Th n t eg e s it is a se agesi al position l not ation . e de o mi n ator is always 60 o r some power of 60 i ndicated by th e a Th position of the numerator, which lone is written . e 22 fraction , for instance, which is equal to , would in 2 60 £82 T this n otation be written 22 30. hus the ability to represent
a b o r o f fr ctions y a single integer a sequence integers, which the E gyptians secured by the use o f fractions having a c o m mo n a 1 ab n numer tor, , the B ylonia s found in fractions having f Th c o mmon denominators an d the principle o position . e Egyptian system is superior in that it gives an e xact expres
o f v b n sion e ery quotient , which the Ba ylonia can in general A a x . s do only appro imately regards pr ctical usefulness ,
Bab a i s b n th e b however, the yloni n eyond compariso etter
O- b b 1 0 60 and system . Supply the sym ol and su stitute for ,
i t Th e Bhind papyr us o f th e British Museum ; translated by A .
E o L z 1 877. isenl hr , eip ig , I 9 THE FR A CT ON. 3
b a n this notation ecomes th t of the modern decimal fractio , in whose di stinctive merits it thus shares . s o i n b As in their origin, also their su sequent history, the se xagesimal fractions are intimately associated with Th e a G d a and A ab astronomy . stronomers of reece , In i , r ia
o o f all empl y them in reckonings any complexity, in those involving the lengths of lines as well as in those involving
a o f . G a P the me sures angles So the reek stronomer, tolemy (1 5 0 in the Almag est (li g zah , o ut/ r a fts ) measures chords d and — de as well as arcs in egrees , minutes , seconds the gree of chord being the 60th part Of the radius as the degree o f arc is the 60th part of the arc subtended by a chord equal t o d the ra ius . Th e sexagesimal fraction held its o w n as the fra ction pa r excellence a 1 6th for scientific . comput tion until the century, when it was displaced by the d ecimal fraction in all uses ex cept the measurement o f angles .
k F F a o G ee o s. G 9 3 . r racti n r cti ns occur in reek writings both m athematical and non - mathematical - much earlier than P u * Th b t a . e G as a tolemy, not in rithmetic reeks drew sh rp
’ a d b a t 9 t x and art istinction etween pure rithmetic, otp um fi, the o f Ao t a r- t x as b and reckoning, y fi, etween pure metrical geom
Th e d t r a n . T et y . fr ction was relegate to Ao y o mi here is no a a a i a no a pl ce in pure science for rt fici l concepts , pl ce, there
’ o a ct t 9 r t m w as G f re, for the fr ction in p un j; such the reek posi T a — A . as d tion hus , while the metric l geometers rchime es “ ’ 25 0 h i s M a o f C mk ko v er a t ( in e sure the ircle ( n pn g ) , n f and H o 1 20B . C . a o a er ( ) employ fr ctions , either the tre
n b D a A . tises o Greek arithmetic efore ioph ntus (300 D . ) which
’ if Th e usual metho d o f expressing fracti o ns was to write th e numer a o an a and af th e d o i a o i a d o t r with ccent, ter it en m n t r tw ce with uble
” 1 7 o o e " ” B fo a i i m a : e . R a K a a a o a to ccent g . t g e re sex ges m l fr ct ns c me 2 1 o u a a o i a o f d u a o s v g e ctu l reck n ngs with fr cti ns were ef ecte by nit fr cti n , o f hi o l th e o a o d o a d w w ch n y den min t rs ( ubly ccente ) were ritten. 4 UMB R —S YS OF A B R A 9 N E TEM LGE .
’ av — 7th 8 9 th b E d h e come down to us the , th, ooks of ucli s ” n Elements (3 00B . C . ) a d the Introduction to Arithmetic — w i e u m .Oi Ni m h e o a r t c o a u 1 A . D z ( i y y) m/ m ?) c s ( 00 . ) recogni es
. T z a the fraction hey do, it is true, recogni e the fr ctional
a . E d x d a an rel tion ucli , for instance, e pressly eclares th t y b a o r a i . e . num er is either a multiple , a p rt, p rts o f f o b Euc . multiple a part, every other num er ( VII, and he demonstrates such theorems as these
I A be the same a rts o B tha t O i s o D then the sum r f p f f , o
f ' di fierenc e of A a nd O i s the same pa rts of the sum o r difierence o B a nd D tha t A i s o B 6 f f (VII, and
I A be the sa me a rts o B tha t C is o D then a lter natel f p f f , , y,
A i s the sa me arts o C that B i s o p f f D (VII,
But a x d b the rel tion is e presse y two integers , that which a a indicates the p rt and that which indic tes the multiple .
a and E d h as o f e x It is a r tio, ucli no more thought pressing it except by two numbers than he h as o f e xpressing the ratio of two geometric magnitudes e xcept by two magni f d . T o b tu es here is no conception a single num er, the o ne b fraction proper, the quotient of Of thes e integers y the other .
d t 9 r t xd D a o n In the p un of ioph ntus , the other hand, the last and transcendently the greatest a chievement o f the b a Greeks in the science Of num er, the fr ction is granted the position in elementary arithmetic which it h as held ever since .
OF THE AT IV . ORIGIN IRR IONAL .
Th The D ove o f L . e G b 9 4. isc ry Irrational ines reeks attri uted the discovery o f the Irrational to the mathematician an d philosopher Pythagoras (5 25
This is m mi c it declarati o n o f th e mo st reliable d o cument ext ant o n th e o o f o t o E i d a o i o f th e a hist ry ge me ry bef re ucl , chr n cle ncient
- B R A 9 6 N UMB ER S YS TE M OF ALGE .
o ther two s i des — was suggested to him by the fact that in connection with the fact that the triangle — are 3 4 5 - d b a whose sides , , , is right angle , for oth lmost k d E — h e cert ainly fell within the nowle ge Of the gyptians ,
a a a h ad d would n tur lly h ve sought , after he succeede in
a a b d emonstr ting the geometric theorem gener lly, for num er triplets corresponding to the sides of any right triangle as
do 3 4 5 o f a n . , , to the sides the particul r tria gle e Th e search of cours proved fruitless , fruitless even in a the case which is geometrically the simplest, th t Of the
To d a necessa ril isosceles right triangle . iscover th t it was y fruitless ; in the face of preconceived ideas and the app ar o f n x t ent testimony the senses, to co ceive that lines may e is
a o f a which h ve no common unit measure , however small th t unit be taken ; to demonstrate that the hypothenuse and side of the isosceles right triangle actually ar e such a pair f w * o as o f P a . lines, the great achievement ythagor s
f s D s e Co s s o ov G h m s. 9 5 . n equence thi i c ry in reek M at e atic One must know the antecedents and follow the consequences
z Of this discovery to reali e its great significance . It was
Hi s d o a o ma a a th e o o w n was em nstr ti n y e sily h ve been f ll i g, which ’ Old o A o m 340 C to ad o f a en ugh in rist tle s ti e ( R . ) be m e the subject o a f and to b e o d at th e end o f th e l oth o o p pul r re erence, which is f un b k ’ in all o ld edi tio ns o f Eucli d s Elements If there b e any lin e whi ch the side and di ago nal o f a square bo th o a an a o f let rm o f i c nt in ex ct number times , their lengths in te s th s 2 2 i he a and b 6 2 a l ne respectively then .
Th e a and b ma a a c o o a o so a a : numbers y h ve mm n f ct r , 7 ; th t cry — nd b z a and are to a a o . Th s y, where B prime e ch ther e equati o n
' 2 2 ‘ b 2 a then reduc es o n o a o f fa o fi mm , the rem v l the ct r y co o n t o b o th to 2 2 its members , 3 2 F o i a o i t fo l o a 3 and o an r m th s equ ti n l ws th t , theref re B, is even and a a hi i to o dd number, hence th t w ch is pr me B is . ’ ’ 2 2 But s et B a th e a o : 2 at , where B is integr l , in equ ti n 3 it 2 2 o 4 2 a D r 2 a and o a bec mes , whence , theref re , is even .
a h as u o to o O dd nnd and f th s been pr ven be b th . even, is there o re no t a number . OR I I OF TH E IRRA TI ON AL 9 G N . 7 the first recognition of the fundamental difference between
a d and b A fo r the geometric m gnitu es num er, which ristotle mulated brilliantly 200 ye ars later in his famous di sti nc
b and d and tion etween the continuous the iscrete, as such w as potent i n bringing abo ut that complete banishment o f numerical reckoning from geometry which is so character i sti c a G a a b of this dep rtment Of reek m them tics in its est, its creativ e period . N o o ne before Pythagoras h ad questioned the possibility o f expressing all siz e relations am o ng lines and surfaces in
b — a a b o d d terms Of num er, r tion l num er Of c urse . In ee , e xcept that it recorded a few facts regarding congruence Of
a d b b a E a w figures g there y o serv tion, the gypti n geometry as nothing else tha n a meagre collection o f formula s for com h w a . T e a as puting are s e rliest geometry metric al . But to the severely logical Greek no alternative seemed
b was a x o possi le , when once it known th t lines e ist wh se lengths — whatever unit be chosen fo r measuring them a b b e a d n b c nnot oth integers , than to h ve o e with num er and me asurement in geometry altogether . Congruence be
a Fo r c ame not only the final but th e s o le test of equ lity .
o f z a a a d a the study . si e rel tions among unequ l m gnitu es
o f a d pure geometric theory proportion was cre te , in which
o n no t a a a d o f proporti , r tio, was the prim ry ide , the metho exhaustio ns making the theory available fo r figures bounded b y curved lines and surfaces .
Th e o w as o f i E d utcome the system , geometry wh ch ucli e xpounds in his Elements an d of which Apoll o nius makes
nd hi s C a ab o sple id use in onics , system s lutely free from
x a o r e tr neous concepts methods, yet, within its limits, Of a gre t power .
d a be a d d a d It nee h rdly d e th t it never, occurre to the ’ Greeks to meet the diffi culty which Pythago ra s discovery h ad b b i rra ti o na l number rought to light y inventing an ,
Fo r itself incommensurable with rational numbers . arti
fic i al n r concepts such as that they had neither t alent o liking . MB E R - S YS TE M F A 9 8 N U O LGEB RA .
On a d di d d the other h n , they evelop the theory of irra ti o nal a as a d a o m gnitudes ep rtment of their ge metry, the
a a a d b o ne a irr tion l line, surf ce , or soli eing incommensur ble i a a a o d . a with some chosen (r tion l) line, surf ce, s l Such theory forms the content Of the most elaborate boo k of E d’ E h 1 0t . ucli s lements , the
m A ox e s of r o s. a 9 6 . ppr i at Value I rati nal In the pr ctical or metric al geometry which grew up after the pure geometry h ad a and a a d re ched its culmination, which tt ine in the works o f H er o the Surveyor almost the proportions of o u r modern ‘ elementary m ensurati o nfi appr o xi ma te va lues of irrational
N o r o numbers played a very import ant role . d such approx i m i n H A d ’ at o s appear for the first time in ero . In rchime es ” “ M C b o f x a ro x ima easure of the ircle a num er e cellent . pp 22 i a o xi a i fo r t ons occur, among them the f mous appr m t on 7
W o f a a , the ratio of the circumference circle to its di m
' Th e a x a I / 2 d be o ld eter . ppro im tion for x is repute to as 5 P as l ato . It is no t certain how these approxim ation s were effec ted j' They involve the use of some method for extracting square Th e a a o f roots . e rliest explicit st tement the method in common use tod ay for e xtracti ng square roots o f numbers (whether e xactly o r approximately) occurs in the com ’ f T f A x A . D P mentary o heon o le andria (380 . ) on tolemy s
The fo rmul a V s (s a ) (s b) (s c ) fo r th e area o f a triangle in o f d due to H o terms its si es is er . TMany attempts have been made to disco ver th e metho d s o f appro ximatio n used by Archi medes and Hero fro m an examinatio n
‘Z h l a / a 4 b I a m1 o f r but . T e fo x thei results , with little success rmu 2 a
o r o m Of th e a o a o but no n will acco unt f s e simpler ppr xim ti ns , si gle metho d o r s et o f meth o ds have been fo un d which will a cc o unt fo r th e
ff S ee u : Die ad a Irrati o nalitaten d er mo re di icult . G nther qu r tischen 2 A m o d L z 1 88 . o E i . Alten und deren ntw cklungs eth en eip ig , ls in
- d r A W af l l ter . Ha a d . Han buch d e klassischen ltertums issensch t, lbb n
1 00 M R - S S M OF A B R A N U BE Y TE LGE .
b a d a b solves three geometric pro lems which, st te lge raically, 2 2 are but r a x a x = b the th ee forms of the quadr tic ; + , 2 2 2 2 * x a x b x b ax. And C A o , the onics of p llonius, so astonishing i f regarded as a product o f the pure geo
d d a metric method use in its emonstr tions, when stated in b a b the language Of alge r , as recently it has een stated by Z euth en ' o f u f , j almost convicts its author the se o algebra Of i v as his instrument n estigation .
Hero . But i 1 00. in the writ ngs of H ero o f Alexandr ia 1 ( 20 R C . ) the equation first comes clearly into the light i H aga n . ero was a man Of practical geni us whose aim w as to make the rich pure geometry o f his pred ecessors available
him o f o ld for the sur veyor . With the rigor the geometric x th e method is rela ed ; proportions, even e quations, among mea sures Of magnitudes are permitted where the earlier geometers allow only proportions among the m agnitudes
a a themselves ; the theorems Of geometry are st ted metric lly,
all i a b in formulas ; and more than th s, the equ tion ecomes
z a recogni ed geometric instrum ent .
H fo r d a o f o f s th e ero gives the i meter a circle in terms , sum i and a : of d ameter, circumference , are , the formula 1
H e could have reached this formula onl y by so lvi ng a
uadrati c e ua ti o n a — a q q , and th t not geometrically, the n ture Of the Oddly constituted quantity 3 precludes that suppo u — but b a b i sitio , y a purely lge raic reckon ng like the following
Th e area of a circle in terms of its di ameter being
E VI 29 28 Da a 84 85 . lements , , , t , ,
Di e L r vo n d en K ni im A m . o a I eh e egelsch tten ltertu C penh gen, 1 88 6 . 34 1 . See a o i d er Ma a . 1 C nt r Gesch chte them tik , p R K A B R A 1 G EE LGE . 01
7rd 7r a the length of its circumference , and ccording to
’ 22 Archi medes approximation - we have the equation 7
2 cl 1 1 29
b 1 1 and m Clearing of fractions, multiplying y , co pleting the square,
r 1 21 o 638 a 841 1 5 4 3 841 , whence 1 1 a 29 841
V 1 5 4 s 841 29 1 1
E x a a b mb cept that he l cked an lge raic sy olism , therefore, H w as an b b a Of w ero alge raist, an alge r ist po er enough to solve an affected quadratic equation .
Th e a o f 1 D o s 300 A D . G 01 . i phantu ( l st the reek a D o f A x nd w as a mathematici ns , iophantus le a ria, a gre t algebraist . Th e period between him and Hero w as not rich in cre a but d d tive mathematicians , it must have witnesse a gra ual development of algebraic ideas and Of an algebrai c symbolism .
At all v a t9 r u< oi D th e e e ents , in the p un of iophantus alg brai c equation h as been supplied with a symbol for th e n a and o f unk own qu ntity, its powers the powers its recip A d o c al 6 and a b fo r . d i s r to the th, sym ol equality ition d b a but represente y mere juxt position, there is a special
1 n d b b o . O sym ol, 4, for su tracti n the other han , there are b t — b no general sym ols for known quanti ies, sym ols to
a ab serve the purpose which the first letters of , the lph et ad i n n a b ada — are m e to serve eleme t ry alge ra now ys , there n fore o literal coefficients and no general formulas .
' With the symbolism h ad grown up many of the formal
D o rules o f algebraic reck o ning als o . i phantus prefaces th e 1 02 N MB R - s YS TE M F A U E O LGEB R A .
a t 9 u< d b p nm with rules for the addition, su traction , and mul ti li c ati o n o f o H p p lynomials . e st ates e xpressly that the d b a pro uct of two su tr ctive terms is a dditive . Th e apt 9unr u< oi itself is a collection o f problems concern
b are d b d a ing num ers , some of which solve y etermin te b b d a . alge raic e quations , some y in etermin te Determinate equations are solved which have given posi
as an o f tive integers coefficients , and are of y the forms
“ " 2 ? 2 a re = bx a x bx c a x c bx a x bx c a a , , , ; lso 3 2 b a x x 4x 4. d single cu ic equ tion, In re ucing equa b tions to these forms, equal quantities in Opposite mem ers are cancelled an d subtractiv e terms in either member are rendered additive by transposition to the other member . 2 2 Th e indeterminate equations are of the form y = a x
bx c D a r ati o na l , ioph ntus regarding any pair of positive n b as a substi um ers (integers or fractions) solution which, n x * T tuted a d a . a for y , s tisfy the equation hese equ tions are a v x N handled with m r ellous de terity in the apt 9umtxoi . O d effort is ma e to develop general comprehensive methods , but e ach e x ercise is solved by some clever device suggested M b d a a . y its indivi u l peculi rities oreover, the discussion is n a o n e suffic i n b ever exh ustive, solution g when the possi le f number is infinite . Yet until some trace o indeterminate
n a t 9 u< oi d d Di O h an e quatio s earlier than the p um is iscovere , p tus must rank as the originator of thi s department Of mathematics . Th e determinate quadratic is solved by the method which Th we have already seen used by H ero . e equation is first multiplied throughout by a number which renders the c o 2 “ ” effici ent o f x a d perfect square , the square is complete , th e a b b o f a and s qu re root of oth mem ers the equ tion taken,
i t Th e d a o D o a a o o o a d to esign ti n i ph ntine equ ti ns , c mm nly pplie indetermin ate equatio ns o f th e first deg ree when investigated fo r inte D o a d a o o a n s o . o o gr l s luti ns , is striki g mi n mer i ph ntus n where c nsi ers a o and o n th e o a d a o f a o a o o o f such equ ti ns , , ther h n , ll ws r cti n l s luti ns f th e o d d indeterminate equatio ns o sec n egree .
B R - M OF A 1 04 N UM E S YS TE L GEBRA.
An e x amination of the writings of these mathematicians and of the remaining mathematical literature of India leaves little room for doubt that the Indian geometry w as taken b H and b — ma odily from ero, the alge ra whatever there y h ave been o f it before Aryabh atta — at le ast powerfully N o r affected by Diophantus . is there occasion for surprise r abh att a D in this . A y lived two centuries after iophantus and si x H and E a after ero, during those centuries the st had frequent communication With the West through various ’ I n ul mT a a a an channels . partic ar, fro r j n s reign till l ter th
A . D v 300 . an acti e comm erce was kept up between India o f E w f and the east coast gypt by ay o the Indian Oce an . Greek geometry and Greek algebra met very difi erent Th e f fates in India . Indians lacked the endowments o the geometer . So far from enriching the science with new d a e iscoveries , they seem with difficulty to have kept liv ’ e ven a proper understanding o f H ero s metrical formulas . H But algebra flourished among them wonderfully . ere the fine talent for reckoning which could create a perfect nu
a b a a u fo r meral not tion , supported y t lent eq ally fine sym
o li l a d a and ad b c a re soning, foun a gre t opportunity m e D b gre at achievements . With iophantus alge ra is no more than an art by which disconnect ed numerical problems ar e o f a w solved ; in India it rises to the dignity science, ith n general methods and concepts o f its o w .
F Its A br i c mbol sm. 1 03 . lg e a Sy i irst of all, the Indians an d de vised a complete , in most respects adequate, sym A d was by D a b holism . d ition represented, as ioph ntus, y
a x as d x mere juxtaposition ; subtr ction, e actly a dition, e cept that a dot was written over the coefficient o f the subtra Th b bha a a hend . e sylla le written after the f ctors indic ted d a uo a product ; the divisor written under the dividen , q
ab 7rd b b a tient ; a syll le , , written efore a num er, its (irr tional) square root ; one member o f an equation placed over the Th e o . ther, their equality equation was also provided with A IN DIAN AL GEB R . 5 symbols for any number o f unkn own quantities and their powers .
Th Its ve o of the e v . e 1 04. In nti n N g ati e most note w o rthy feature o f this symbo lism is its representation o f
To b a b o be subtraction . remove the su tr ctive sym ol fr m tween minuend and subtrahend (where Di o phantus h ad
a d b a a to b a d pl ce his sym ol m) , to tt ch it wholly the su tr hen an d then connect this modified subtrahend with the minuend add l d d a sub itively, is, formal y consi ere , to tr nsform the t ra ction o f a positive quantity into the addition of the d corresponding negative . It suggests what other evi ence
a a a a l ebra o ne s to Indi a the i mmensel m kes cert in, th t g y useful c o ncept of the a bso lute neg a ti ve . Thus o ne o f these dotted numbers is allowed to st and
B a z by itself as a member o f an equation . h skara recogni es d b a as as i m o ssi the ou le sign of the squ re root , well the p bility o f the square root o f a negative number (which is as b d a di ima very interesting, eing the first ictum reg r ng the g i n r n o a a d ad . y ) , no l nger ignores either root of the qu ratic M o a h ad a di re th n this , recourse is to the s me expe ents fo r n a fo r a a interpreti g the neg tive , tt ching a concrete phys
a d a o - Th a t as da . e ic l i e it, are in common use to y prim ry a a a w as d ebt as me ning of the very n me given the neg tive , a ‘ w as Th th t given the positive mea ns . e opposition between the two was also pictur ed by lines described in opposite d irections .
Its Use f r . In 1 05 . o Ze o But the contributions of the d ians to the fund of algebraic concepts di d not stop with th e b a solute negative . T um b o f O hey made a n er , and though some of their B a a at s h ad reckonings with it are childish, h skar , lea t,
” sufficient understanding o f the nature of the “ quotient g “ sa f no an (infinity) to y it suf ers ch ge , however much it ” a d o r . H e a D is incre sed iminished associ tes it with eity . 1 06 N UM R - S S E M OF A B RA BE Y T LGE .
f r A a Its Use o r o mb rs. d a 1 06 . I ati nal Nu e g in, the In i ns were the first to reckon with irration al square roots as with numbers ; Bhaskara e xtracting s quare roots of binomial surds and rationalizing irration al denominators of fractions
r f even when these a e polynomial . O course they were as little able rigorously to j ustify such a procedure as the G ab a f reeks ; less le , in f ct, since they had no equivalent o x B the method o f e haustions . ut it probably never occurred to them that justification w as necess ary ; they seem to have been unconscious o f the gulf fixed between the di screte and
. And as i n a O and a continuous here , the c se of the neg tive,
o f a t and with the confidence p successful reckoners , they were re ady to pass immediately from numerical to purely b a ad sym olical re soning, re y to trust their processes even where form al d emonstration o f the right to apply them
a be b . T ce sed to attaina le heir skill was too great, their f ar . instinct too true , to allow them to go wrong
1 D rm e and d rm e E o s 07 . ete inat In ete inat quati n in Indian A — Alg ebra . s regards e qu ations the only changes which the Indian algebraists m ade in the treatment of determin ate
as f f equations were such grew out o the use o the negative . This brought the triple classification o f the quadratic to an n an d n f e d d secure recognitio for both roots o the quadratic . Brahmagupta sol ves the quadratic by the rule of Hero and D a o f an x ioph ntus, which he gives e plicit and gen
h r a a . rid a a a a a o f er l st tement C , a m them tici n some dis tin ction belonging to the period between Brahm agupta and
B a a m d o f d h sk ra, a e the improvement this metho which consists in first multiplyin g the equation throughout by four times the coefficient of the square o f the unknown quantity and so preventing the occurrence o f fractions under the radical si g nfi“ Bhask ara also solves a few cubic and biquadratic equa tions by speci al devices .
” This meth o d still go es under th e name Hind o o meth o d .
08 N UMB R —S YS M F A A 1 E TE O L GEBR .
x o f b astronomers put full e positions their reckoning, alge ra, ’ a Alfaz ari s a a and geometry into their tre tises, tr nsl tion laid open to his countrymen a rich tre asure of mathematical ideas and methods . It is impossible to set a date to the entrance of Greek T . v d Da ideas hey must ha e ma e themselves felt at mascus,
O d a h a the residence of the later mayya caliphs, for th t city d Bu numerous inhabitants of Greek origin and culture . t the first translations o f Greek mathematical writings were made ’ in the reign of H arli n Arraschi d (786—809 ) when Euclid s E ’ A b lements and Ptolemy s Almagest were put into ra ic .
La a o f A A ter on, translations were m de rchimedes , pollo ’ ni H o f o f D a b Abu a a us , ero, and last all, ioph ntus ( y l W f , 9 40
Th e earliest mathematical author o f the Arabians is Alkh w ari zmi f d a o f 9 th , who lourishe in the first qu rter the B a . a ab e century esides astronomic l t les, he wrote a tre tis b on alge ra and one o n reckoning (elementary arithmetic) .
Th e h as ad - x latter alre y been mentioned . It is an e position o f o f d a the positional reckoning In i , the reckoning which m aev E edi al urope named after him Alg o rithm . Th e tre atise o n algebra bears a title in which the word
Al ebra a fo r : v i z . Ald ebr wa lma g ppears the first time , j
k . aba la Ald e r i . d j b ( e . re uction) signifies the making o f all terms o f an equation positive by transferring negative terms b k a l n l i . to the opposite mem er of the e qu tion ; a m aba a ( e. o f Opposition) , the cancelling equal terms in opposite mem o f bers an equation . ’ Alkhwarizmi s classification of equations o f the 1 st and 2 d degrees i s that to which these processes would naturally vi z lead,
2 2 a x = bx bx = c bcc = , , c,
2 2 w bx = c = + , x bcc + c .
T v a u hese equations he sol es sep rately, following p the solution in each case with a geometric demonstration of its AR AB IAN A B A LGE R . 1 09
H e z b correctness . recogni es oth roots of the quadratic when they are positive . In this respect he is Indian ; in all — da a o f others the avoi nce of neg tives, the use geometric — h demonstrati o n e is Greek .
B de Alkhwari z mi a f esi s , the most f mous algebraists o the
A ab a Alka rchi Alcha ami bo o f r i ns were and yy , th whom lived in the i l th century Alkarch i gave the solution o f equations of the forms
? " p ” ” ? ” a re bcc 0 a m + c bx bx c a x . , \ ,
H e d a at also reckone with irration ls , the equ ions
b o f eing pretty j ust illustrations his success in this field . Alch ayyami was the first m athematician to make a system a o f b H e atic investig tion the cu ic equation . classified the various forms which this e quation takes when all it s terms
and o o a — b th e are positive, s lved each f rm geometric lly y f intersections of c o ni c sfi A pure algebraic s o lution o f the b b cu ic he elieved impossible .
L Alkh wari z mi Alk arc h i Alc h a ami ‘E ike , and yy were astern But a A b Arabians . e rly in the 8th century the ra ians con An A ab quered a gre at part o f Spain . r ian realm was established there which became independent o f the Bagdad Th a 747 an d d d 3 00 a . e caliph te in , en ure for ye rs inter course o f these Western Arabians with the East was no t
3 u o th e a o x bx a i . Thus s pp se equ ti n , g ven 2 3 ? 2 : — Fo r b th e a and fo r a r . h x r x . substitute qu ntity p , , p T en p ( ) N o w this equatio n is th e result o f eliminating y fro m between th e 2 2 two a o x z x r x th e o f th e a equ ti ns , py, y ( ) first which is equ
o o f a a a o a th e o d o f a r . ti n p r b l , sec n , ci cle Let these tw o curves be co nstructed they wi lli nt ersec t in o ne real o d n f o th e o and th e a i a o f o i i s a o o p int isti ct r m rig in , bsc ss this p nt r t 3 z: 2 9 H d er Ma a . 7 o f x bx a . S ee a . nkel , Geschichte them tik , p This metho d is o f g reater interest in th e histo ry o f geo metry than in a o f a a It o an a a o o f o o f th e o th t lgebr . inv lves nticip ti n s me m st ’ impo rtant ideas o f Descartes Gé o mé tri e (see p . 1 1 0 UMB R - S S M OF A B R A N E Y TE LGE . frequent enough to e xercise a controlling influence on their
o r d T aesthetic scientific evelopment . heir mathem atical productions ar e of a later date than those o f the E ast
n a x arithmeti o - b T a d lmost e clusively c alge raic . hey con structed a formal algebraic notation which went over into the Latin transl ations o f their writings and rendered the path o f the Europeans to a knowledge of the doctrine o f ‘ a a 1 a a b h ad A ab a e qu tions e s er th n it would h ve een, the r i ns f E h o f o the ast been their only instructors . T e b est known
a a a I bn A a en d 1 1 th their m them tici ns are fi h ( of century) , f h h I bn Alba nna end o 1 3t Alha sadi 1 5 t . ( century) , ( century)
T Ar b A ebr r k r e h . . G 1 09 a ian lg a ee ath r t an Indian hus, o f a d a o f A b a a a the three gre ter ep rtments the ra i n m them tics , the India n influence gained the m astery in reckoning only . n Th e Arabian geometry i s Greek through a d through . a b b G While the lge ra contains oth elements , the reek pre f d a d a . d x a b o a omin tes In eed, e cept th t oth roots the qu r tic
z d d a are recogni ed, the octrine of the etermin te equation is
a n a a as a altogether Greek . It voids the eg tive lmost c re fully as Diophantus does ; and in its use o f the geometric method o f demonstration it is actuated by a Spirit less modern still — the spirit in which Euclid may have con c ei ved o f algebra when he solved his geometric quadratics . Th e theory o f indeterminate equations seldom goes beyond
D o a d d a . i ph ntus ; where it oes, it is In i n ’ Th e A ab ba d o n P but r ian trigonometry is se tolemy s , is
o i ts superior in two important p articulars . It empl ys the sine wh ere Ptolemy employs the ch o rd (being in this re d a an d an al b a ad o f a spect In i n) , has ge r ic inste geometric d f x a d form . Some of the metho s o appro im tion use in n reckoning o ut trigonometric tables show great clever ess .
d A ab a - a d In eed, the r i ns make some amends for their ill dvise return to geometric algebra by this excellent achievement in algebra ic geometry . Th e preference o f the Arabians for Greek algebra was
- 1 1 2 N UMB ER S YS TEM OF ALGEB RA .
tr ba d o n Co d ex A rcer i a nus and a d e y se the , metho for effect a ing division on the abacus with apices . Yet these chieve
ments are eno ugh to place him far above hi s contemporaries . H i s influence gave a str o ng impulse to m athematical studies
e H e i s where inter st in them had long been dead . the fore runner o f the intellectual activity ushered in by the trans
lati o ns A ab fo r b from the r ic, he rought to life the feeling o f the need for mathematics which these translations were mad e to s atisfy .
E 1 1 2 e o f m . Ar b M e s eo rd . . ntranc the a ian ath atic L na o It w a s the elementary branch o f the Arabian m athematics which took root quickest in Christendom — reckoning with n nin e digits a d 0.
' Leo nardo o f P s — Fi bo na cci as h e w as o a d i a , als c lle di d great service in the diffusion o f the new learning through his Li ber Aba c i (1 202 and a remarkable a o a t and a b o f A ab a present ti n of the ri hmetic lge ra the r i ns, which remained fo r centuries the fund from which reckoners and algebraists drew and is indeed the foundation of the d mo ern science . Th e four fundamental operations o n integers and frac tions are taught after the Arabian method ; the extraction o f the squ are root and the doctrine o f irrationals are presented in their pure algebraic form ; quadratic equations are solved an d applied to quite complicated problems ; neg a ti ves a re a cce ted when th r i n p ey a dmi t of i nte pr eta t o a s d ebt. Th e last fact ill ustrates e xcellently the character o f the
i r n L be Aba i . a a a but a d c It is not mere tr nsl tion, in e pendent and masterly treatise in o ne department o f th e new mathematics .
B d Li ber Aba c i L o P ra cti ca esi es the , e nardo wrote the
Geo metri a e a i s b o f E d , which contains much th t est ucli ,
A d H and o f a rchime es , ero, the elements trigonometry ; lso
Liber a a dra to rnm o f a b the Q , a collection origin l alge raic
o b pr lems most skilfully handl ed . AR Y UR O A AL B R A E L E PE N GE . 1 1 3
M m the A e o f m L . eo 1 1 3 . athe atics during g Scholasticis ’e na d was a a math emati c i an but as wa r o gre t fi fine his work s, bo end 1 5 th it re no fruit until the of the century . In him
h ad b a b a to A ab a there een rilli nt response the r i n impulse . But the awakening was onl y mome ntary ; it quickly yielded ” a a “ da to the he vy leth rgy of the rk ages .
Th e a e a d g of schol sticism , the age of evotion to the forms
f and d a a e o f a d ul o thought, logic i lectics , is the g gre test r1 ess and confusion in mathematical th i nk i ng T Algebra owes the entire period b ut a single contribution ; the Its a w as N o concept of the fractio nal power . uthor ic le Oresme (died who also gave a symbol fo r it an d the rules by which reckoning with it is governed .
Besides Leo nard o there flo urished in th e first quarter o f th e 1 3 th
o r d a n V mo ra n us H e an a G i a ma a a J as l e . century ble e m n them tici n , was th e a o o f a a d De nu mer zs da ti s o uth r tre tise entitle , in which kn wn
a are fo r th e d and o f o ne De qu ntities first time represente by letters , tra ng uli s which is a rich tho ugh rather systemless co llectio n o f theo rems nd o f nd A a a o S ee u a pr blems principally o Greek a r bi n rigin . G nther d es math emathi sch en U im d M a Geschichte nterrichts eutschen ittel lter ,
. 1 5 6 p .
M k 3 49 - 2 o a a d er a a . 35 To TC mp re H nkel , Geschichte them ti , pp . the unfruitfulness o f these centuries th e S u mma o f Luc a Pa c io li bears h i . o o h as th e d o o f t e a w tness This b k , which istincti n being e rliest o o o n a a d a a d 1 49 4 and o d th e a b k lgebr printe , ppe re in , emb ies rith a a and o o f th e m d R a metic , lgebr , ge metry ti e just prece ing the en is It no t n d o r m o d no t a ad d s ance . c o ntains a i ea eth lre y presente by E t o a a m o r a th e Leo nard o . ven in respect lgebr ic sy b lism it su p sses Li ber Aba c i o nly to th e extent o f using abbreviatio ns fo r a few fre “ “ ” d fo r and R . fo r res u o as . quently rec rring w r s , p plus , (the o a And i no t to b e a d d as o a unkn wn qu ntity) . th s is reg r e rigin l with ’ Pac i o li fo r th e Arabians o f Leo nard o s time made a simil ar use o f a a o In a a a o ad a d o f C o a 1 2th bbrevi ti ns . tr nsl ti n m e by Gerh r rem n ( u f o an o A a o r a t h e r ad cent ry) r m unkn wn r bic igin l letters (r ix) , c d d a ma are d to th e o a (census) , ( r g ) use represent unkn wn qu ntity ,
a and th e a o . its squ re , bs lute term respectively Th m o f Bac i o li h as a m o a d its a e S u ma gre t erits , n twithst n ing l ck ma ma a d o f th e It a It a d th e . o f o rigin lity . s tisfie the tic l nee s time s o a f and c o is very co mprehen ive , c nt ining ull ex ellent instructi n in the 1 1 4 UM R- S S M OF AL R A N BE Y TE GEB .
~ 1 1 4 The e s . o o o f the C b Bi . R nais ance S luti n u ic and Th e quadratic Equati ons . first achievement in algebra by the mathematici a ns o f the Ren aiss anc e w as the algebraic solution of the cubic equation : a fine beginning of a new
o o f era in the hist ry the science . 3 Th e cubic x + mx z nw as solved by Ferro of Bo log na 1 5 05 an d a d and d d 1 5 35 b in , secon time in epen ently, in , y ’ F a Ta rta li a b a trans fo r erro s countrym n, g , who y help of g 2 x x = n o m ation made his method apply to zt m rt als . Cardan o f M a w as b u But il n the first to pu lish the sol tion,
rs Ill a n ‘e 1 4 in his A g afi 5 5 . Th e Ars M ag na records another brilliant discovery : the solution — after a general method — o f the biquadratic 4 2 x 6 x 36 60x b Ferrar i Ca n. y , a pupil of rda T i n a b hus Italy, within fifty ye rs of the new irth of a b a a a o f at ad lge ra, fter p use sixteen centuries the qu a o o f b a a a b a r ti , the limits possi le tt inment in the lge r ic solution o f equation s were reached ; for the algebraic solution o f the general e quation o f a degree higher than
4 1 s i b as w as d b Abel ' mpossi le , first demonstrate y j Th e general solution of higher equati o ns proving an b b m th e o stinate pro le , nothing was left the searchers for
art o f o a r th e m o d o f L o a d o fo r th e a reck ning fte eth s e n r , merch nt man and a a a o f a o f a o a , gre t v riety m tter purely the retic l interest a o — th e a o r o f a a ls representing element ry the y numbers , lgebr , o and th e a a o o f a a t o o o a ge metry , pplic ti n lgebr ge metry . C mp re a o der Ma a II 308 C nt r , Geschichte them tik , , p . .
' It sho uld b e added that the 1 5 th century pro duced a math emati clan wh o deserves a disting uished pl ace in th e general histo ry o f a a o n a o o f o o to o o m them tics cc unt his c ntributi ns trig n metry , the a o o Re i o mo nta nus 1 436 L Je rd anus h e w as str n mer g ( ike , a a Germ n . i s “ Th e o o f o : A a a d e re uli s pr per title this _w rk is rtis m gn e sive g Al r i i ’ “ g eb a c s liber unus . It h as sto len th e title o f Card an s A rs ” a a Arithmeti c ae d at B 1 a 5 70. m gn , publishe sel , ‘ M m r a a 1 82 A é o su les E a o A é : 6 . o r ire qu ti ns lg briques Christi ni , ls ll ’ r 6 . C e e s Jo a I . 5 in urn l , , p
1 1 6 N MB R — S YS M OF A B RA U E TE LGE .
n d O the other hand, neither Vieta nor his istinguished E a H a rri o t 1 5 60 a follower, the nglishm n ( ccept even negative ro o ts ; though H arriot does not hesitate t o perform
b a a an d a a alge r ic reckonings on neg tives, even llows a neg o ne tive to constitute member o f an equation .
A b m H . 1 1 bo sm. e o 6 . lg e raic Sy li Vi ta and arri t Vieta H a t w di d d d and rrio , ho ever, istinguishe service in perfect in b b b m u se g the sym olism of alge ra ; Vieta, y the syste atic o f let ters to represent known quantities - algebra first “ ” “ * b a a o r a ec me liter l universal rithmetic in his hands ,
— H a b d b a a o f rriot, y ri ding alge r ic st tements every non b v but sym olic element, of e erything the letters which rep
n as o b o f resent quantities know as well unkn wn , sym ols H ’ i a b a . a Art s Ana l Oper tion , and sym ols of rel tion rriot s y ti ed e Praxi s (1 631 ) has quite the appe arance of a modern alg ebra j'
There are is o lated instances o f thi s us e o f letters much earli er than
a th e De numeri s d a tis o f Jo rdanus Nemo rarius and Viet in , in the B ut th e d t Of a Alg o r i thmus d emo nstra tus Of th e s ame a utho r . cre i m king it th e general practice o f algebraists belo ngs to Vieta . 1 One h as o nly to reflect h o w much o f th e po wer o f algebra is due
’ t o its admirable symbo lism to appreciate the imp o rtance o f th e Artzs d B ut na l tic a e Pra x is o m a a . A y , in which this symb lis is fin lly est blishe ad to a and o ne addi tio n o f co nsequence h as since been m e it , integ r l W fractio nal expo nents intro duced by Descartes (1 6 37) and allis th e a a d a b ut Harrio t substituted small letters fo r c pit ls use by Viet , d fo llo wed Vieta in representing kno wn quantities by co nso nants an o Th o o o f o n un kno wn by v wels . e present c nventi n representing kn w o f th e a a n o th e a quantities by th e earlier letters lph bet, u kn wn by l ter, is due t o Descartes . ’ Vieta s no tatio n is unwieldy and ill adapted to purp o ses o f alge ’ r I ad o f as a o d o to b ai c recko ning . nste restricting itself, H rri t s es , d d o o a m o a o th e us e o f brief and easily apprehen e c nventi n l sy b ls , it ls l fo r A3 3 B A Z emplo ys wo rds subject to th e rules o f syntax . Thus o r a a a — 3 bba z as H a o o d a a ( , rri t w ul h ve written it) , Viet writes
— lo In a A rubus B qua d 3 i n A a eg ua tur Z so li c . this respect Viet is t to a o but to a o f d o and inferio r no o nly H rri t, sever l his pre ecess rs A U O AN AL B R A 1 1 E R L Y E R PE GE . 7
f H r 1 m l T eo rem o A eb . ri o and 1 7 . Funda enta h lg ra a t H a o h as b d d d Girard . rri t een cre ite with the iscovery of the “ fundamental theorem o f algebra — the theorem that the number o f roots of an algebraic equation is the s ame as
Th e A rti s Ana l tica e P ra xi s its degree . y contains no mention
f — d d b o a an d a o this theorem in ee , y ign ring neg tive im gi
a H a d o nary roots, le ves no place for it ; yet rriot evel ps
a a a d d far ad system tic lly metho which, if carrie enough, le s to the discovery of this the o rem as well as to the relations n n holding between the roots of a e quation a d its coefficients . By multiplying together binomial factors which involve a and d a the unknown qu ntity, setting their pro uct equ l to ” 0 b “ a o and a th e , he uilds canonic l equati ns, shows th t
o f a — a — are roots these equ tions the only roots, he s ys the positive values o f the unknown quantity which render these
T h e b d d a ba ca be b a a o 0. inomi l f ct rs hus uil s ,
a a o ut o f a o in which is the unknown qu ntity , the f ct rs a b a c and at b a o and , , proves th is root of this e quati n
a 0 b o a d . the only root, the neg tive root eing t t lly ig nore While no attempt is made to sho w that if the terms o f
o mo a be d o ne b c an a c m n e qu tion collecte in mem er , this
o a to o o a th e D a m a a S n t bly his c ntemp r ry , utch m the tici n tevinus 1 5 48 wh o o l d fo r a a x2 3 x 8 as ( w u , inst nce , h ve written ’ 1 ® S Q) Th e geo metric affil iatio ns o f Vieta s no tatio n are
o i o . It th e a bv us suggests Greek rithmetic . It is sur prising that algebraic symbo lism sho uld o we so little to the I o h 1 u L Pac i o li a a a a a f t e 6 c . s ee o gre t t li n lgebr ists th ent ry ike ( n te ,
. 1 1 3 o a few a a o fo r o d a p ) they were c ntent with bbrevi ti ns w r s , o a d o a o as h as a d and an o o ne sync p te n t ti n , it been c lle , inc mplete at th at. Th e current symbo ls o f o peratio n and relatio n are chiefly o f English and a o a n d o r ad o d as o o : vi z Germ n rigin, h vi g been invente pte f ll ws . R ec o rd e 1 5 5 6 R ud o l 1 5 25 th e vi n ulum Vi et by in by f in c , by a 1 5 9 1 brack ets B o mbelli 1 5 72 R ah n 1 6 5 9 in ; , by , ; by in ; by arri o t 1 631 and o a 1 5 h H in . The sig ns ccur in t century manu
d s o d a d at a o a o d 9 script i c vere by Gerh r t Vienn . The n t ti ns a b an b fo r f a o ad o d o m A a a the r cti n were pte fr the r bi ns . — A B R A N UMBER S YS TEM OF LGE .
a d b a a be separ te into inomial factors, the case of c nonic l equations raised a stron g presumption for the s o undness o f this view o f the structure of an equation . Th e first statement of the fundamental theorem and of the rel ations between coefficients and roots occurs i n
a ab d n b Inven a rem rk ly clever and mo er little ook, the ’ i n o uvelle en l A br e o f Albert Gi ra rd b d t o N d , pu lishe in A da 1 629 a a mster m in , two years e rlier, therefore , th n the
a d d i ma Arti s A na lyti ca e Pra xi s . Gir r stan s in no fear of g i nar but o n d o f z y roots, rather insists the wis om recogni ing T a o them . hey never occur, he s ys , except when real r ots b o ut are lacking, and then in num er just sufficient to fill
’ the entire number o f roots to equality with the degree o f h ' u ti o n t e eq a . Girard also anticipated Descartes in the geometrical i n But I nventi o n N o uvelle d t erpretati o n o f negatives . the oes a a d and not seem to have ttr cte much notice, the genius and authority o f Descartes were needed to give the interpreta tion general currency .
T T . ACCE A CE OF HE EGAT E T VI P N N IV , HE GENERAL AT ONAL AND THE A N IRR I , IM GI ARY AS NUMBERS .
’ De e é mé Th mé 1 1 G o e v . e Gé o 8 . scart s trie and the N g ati e Th me of Descartes appeared in 1 6 3 7 . is famous little tre a tise enrich ed geometry with a general and at the s ame time simple and natural method of investigation : the method of a v b an a n as representing geometric cur e y e qu tio , which ,
D a n a a o f esc rtes puts it, expresses ge er lly the rel tion its ale To points to those of some chosen line of reference . form a n D a b such equ tio s esc rtes represents line segments y letters , — b a b 0 . b x and . H e the known y , , , etc , the unknown y y
’ ’ S o In d e c é mé L II . o o o f D a o G trie , ivre C usin s e iti n esc rtes w rks ,
Vo l. . V , p 33 7 .
1 20 M R - S S OF A N U BE Y TEM L GEBRA . symbols which there represent numbers here represent line
. N o t a a b x segments only is this the c se with the letters , , ,
. are a n o ms o f y, etc , which mere n mes ( ) line segments , not a a but their numeric l me sures, with the algebraic c o mbina o f tions these letters . a b an d a b are respectively
sum and ff a and b a b the di erence of the line segments ; , the
an a and b fourth proportional to assumed unit line, , ; the
o b a and / d fourth pr portional to , , the unit line ; and x ,
. etc , the first, second, etc . , mean proportionals to the unit * line and a . ’ Desc artes j ustification o f this use o f the symbols o f
“ numerical algebra is that the geometric constructions o f
b a a b . which he makes , , etc , represent the results are “ ” as a add o b a o n the same numeric l iti n, su tr cti , multiplica nd M a . n tion, division, evolution, respectively oreover, si ce all geometric constructions which determine line segments may be resolved into combinations o f these constructions as the operations of numerical algebra into the fundament al b Operations , the correspondence which holds etween these fundamental constructions and operations holds equally x and between the more comple constructions operations . Th e entire system of the geometric constructions under consideration may therefore be regarded as formally iden
a o f a b a and be tic l with the system lge raic Oper tions, represented by the s ame symbolism . “ In what sense his fundamental constructions are the ” as a a Des same the fund mental Operations of rithmetic, The a c artes does no t e xplain . true re son of their formal b are b a identity is that oth controlled y the commut tive, T a o f a a and d b a . ssoci tive , istri utive l ws hus in the c se the
a b = ba and a bc a bc fo r former as of the latter, , ( ) ; the
a a and b a as fourth proportion l to the unit line, , is the s me
a b a the fourth proportion l to the unit line , , and ; and the
a t o a and be a as fourth proportion l the unit line , , is the s me
3 1 3—3 1 4 é o é L 1 . I d . . . G m trie , ivre bi pp TH E GE N ERAL I R RA TI ON AL . 1 21
But o o o a a b and 0. the fourth pr p rti n l to the unit line , , this
a o w as a o f D a da re s n not within the re ch esc rtes , in whose y the fundamental la ws o f numerical algebra had not yet been discovered .
. E . The Co o b e ew o . 1 20. ntinu us Varia l N t n uler It is
’ ’ customary to credit the Geo metr i e with having intro duced
co nti nuo u s va r i a ble a but o s u ffi the into mathem tics , with ut
D a w a c o n cient re ason . esc rtes prepared the y for this
’ ’ Th e x but a use o f Geo metri e . cept, he m kes no it in the and y which enter in the equation of a c urve he regards no t as a ab but as d a a o f w o v ri les in etermin te quantities , a p ir h se * - n Th e a a o f values correspond to e ach curve poi t . re l uthor this concept is N ewton (1 642— 1 727) o f whose great i nven ” n m d o f fluxi o ns “ flo w tio , the etho , continuous variation, , is the fundamental idea . ’ ’ But N a D a b a ewton s c lculus , like esc rtes alge r , is geo a an a and metric r ther th purely numeric l, his followers in
E a d a x o f a ngl n , as also, to less e tent , the followers his gre t
a L b z o n a ul riv l , ei nit , the continent , in employing the c lc us ,
o o f a ab for the most part c nceive v ri les as lines, not num Th bers . e geometric form again thre atened to become para i n a m a mount m thematics , and geo etry to ench in the new “ ” analysis as it h ad formerly enchained the Greek arith — metic . It is the great service of E uler (1 707 1 783 ) to have b a d c o n roken these fetters once for all, to h ve accepte the ti nuo usl v ari a ble number and y in its purity, therewith to
F r n have created the pure analysis . o the relations of c o ti nuo usly variable numbers constitute the field o f the pure
a o uncti o n b but analysis ; its centr l c ncept, the f , eing a device for representing their interdependence .
Th 1 21 e G r r o . n . ene al Ir ati nal While its concer with variables puts analysis in a cert ain opposition to elementa ry a b a ab lge ra, concerned as this is with const nts , its est lish
— G o L II. é é I d . 33 . 7 338. m trie , ivre bi pp 2 MB R - S S M OF A B R A 1 2 N U E Y TE LGE . ment of the continuously variable number in m athem atics brought about a rich addition to the number - system of alge
- i i H th e a bra the g enera l rra t o na l. itherto only irr tional ” b h ad b d b o f a num ers een sur s , impossi le roots r tional numbers ; henceforth their domain is as wide as that of all possible lines incommensurable with any assumed unit line .
T m r z m he o e A . 1 22 . I ag ina y, a Rec g ni d nalytical Instru ent Out o f the ex cellent results of the use o f the negative grew a spirit of toleration for the imaginary . Increased atten L b z tion was paid to its properties . ei nit noticed the real sum o f conj ugate imaginaries (1 676— 7 Demoivre di s covered (1 730) th e famous theorem
(cos 6 i si n cos nd i si n nd ; and Euler (1 748) the equation
i c o s 9 i 0 e " sin , which plays so g reat a role in the modern theory o f functions . E a x x uler also, pr ctising the method of e pressing comple b o f and r num ers in terms modulus angle , formed thei prod u ct s an d a b , quotients, powers , roots , log rithms, and y many brilli ant discoveries multiplied proofs o f th e power o f the imaginary as an analytical instrument .
’ A s Geom r o of m . 1 2 3 . rg and etric Rep esentati n the I ag inary But the imaginary w as never regarded as anything better
a an a b a fic ti o n — be a d b th n lge r ic to voide , where possi le , by the mathematician who priz ed purity of method until a method was discovered for representing it geometrically . if A N a IVessel b u 1 79 7 orwegi n , , pu lished s ch a method in ,
F n Ar and and a renchma , g , the same method independently 1 in 806 .
at W B a d S e e W . . em n in Pro cee ings o f the American Asso ciatio n fo r th e Ad a o f S 1 89 7 v ncement cience , .
- A E M F A R . 1 24 N UMB ER s YS T O . L GEB a s um of multiples of the units 1 and i ; his also is the name “ complex number and the concept o f complex num b b a i b ers in general, where y + secures a footing in the n theory of numbers as well as i algebra . H e to o an d A be d , not rgand, must credite with really breaking down the opposition o f m athematicians to the ’ a A a E ss i w a d a im ginary . rg nd s a s little notice when it p eared and but was d p , soon forgotten ; there no withstan ing o f G hi s the great authority auss , and precise and masterly f ”f presentation o this doctrine .
ECOGN T ON OF THE E Y Y O C VII . R I I PUR L S MB LI ‘ E A CHARACTER OI ALG BR .
QUAT ERN IO NS . AUS DEHN UNGS LEHRE.
Th r o f m e . T o ne 1 25 . e P inciple Per an nce hus, after
a a and a another, the fr ction, irr tional, negative, imagin ry, gained entrance to the number - system of algebra: N o t one of them w as accepted until its correspondence to some x b actually e isting thing had een shown , the fraction and
a a d a ex irr tion l, which originate in relations mong actually i i n a d st g things, natur lly making goo their position earlier
a a d a th n the negative and imagin ry, which grew imme i tely ” ut n “ o of the equatio , and for which a real interpretation h ad to be sought . Inasmuch as this correspondence of the artificial numb ers
x - a and to things e tra arithmetic l, though most interesting o f b h as the reason the practical usefulness of these num ers, not the least bearing on the n ature o f their position in pure arithmetic o r algebra ; a fter all of them h ad been accepted b d h as num ers, the necessity remaine of justifying t is
ee a o 1 4. S Wo 1 1 . 7 G uss , C mplete rks , , p S YM O I AR A R OF A E B R A 1 2 B L C CH CTE LG . 5
a a b a b a d Th w as ccept nce y purely lge r ic consi erations . is a d b E first ccomplishe , though incompletely, y the nglish “ a a P ea c o ckfi m them tician , Pe acock begins with a valuable distinction between a r ith
i a l and s lic a l a L met c ymbo lgebra . etters are employed in
but o v an d a the former, only to represent p siti e integers fr c
b a b d as d a tions , su tr ction eing limite , in or in ry arithmetic , b a d to the case where su tr hen is less than minuend . In the
a a d b ar e l tter, on the other h n , the sym ols left altogether
a general, untrammelled at the outset with any particul r meanings whatsoe ver . It is then a ssumed that the rules of operation applying to the symbols o f arithmetical algebra apply without altera tio n in symbolical algebra ; the mea ni ng s of the oper a ti o ns themselves and thei r r esults being d eri ved fro m these r ules of
er ti Op a o n . This assumption Peacock names the Pr inciple of Per ma nenc e o ui va lent Fo rms and a as n , illustr tes its use follows T
a a b a a b c d a i In arithmetic l lge r , when , , it may re d ly b e demonstrated that — — = — — (a b) (c d ) a c a d bc + bd .
at By the principle of permanence, it follows th 0— b 0— d = 0 0— 0 d — b x 0 ba ( ) ( ) X x + ,
b) d ) bd .
m " m ” a a a b a a a t m r . O again In rithmetic l lge ra , when A and n are positive integers . pplying the principle of permanence, p i ? (d ) ai a i m to q factors P P + + mq terms a q q
p a , whence
n m o a A a 1 830 and 1 845 a Arithmetical a d Sy b lic l lgebr , ; especi lly
B A o a o R o 1 833 . h d o . A o t e l ater e iti n ls ritish ss ci ti n ep rts ,
A a d o o f 1 845 63 1 5 69 639 . 1 lgebr , e iti n , , , 1 26 UMB R —S YS M OF A N E TE L GEBRA .
H ere the meanings of the product b) d ) and o f the P “ symbol a ? are both derived from certain rules o f Operation
in arithmetical algebra . Peacock notices that the symbol also has a wider m ean ing in symbolic al than in arithmetical algebra ; for in the “ fo rmer means that the expression which e xists on o ne side o f it is the result o f an operation which i s indicated ” d and if on the other si e of it not performed . ” H e also points o ut that the terms “ real and “ imagi ” “ ” o r b a nary impossi le are rel tive , depending solely o n th e meanings attaching to the symbols i n any particular
application of algebra . Fo r a quantity is real when it can b e shown to correspond to any real o r possible e xistence ; otherwise it is i mag i naryflr Th e solution of the problem : to
5 3 a divide a group of men into equ l groups , is imaginary ’ a a A a though positive fr ction, while in rg nd s geometry the
- a so c lled imaginary is re al . Th e principle o f perm anence is a fine st atement o f the assumption o n which the reckoning with artificial numbers d d o f n epen s , and the statement the ature of this d ependence a d d as i s e x cellent . R eg r e an attempt at a complete presen t ati o n o f d o f a a b P the octrine rtifici l num ers , however, e a ’ cock s Algebra is at fault in classing the positive fraction with the positive integer and not with the negative and
m a b i agin ry , where it elongs, in ignoring the most difficult o f all a b a no t d rtificial num ers , the irr tional, in efining arti
fic i al b as b o f but l ‘ ln num ers sym olic results operations, p Cl p ally in not subj ecting th e operati ons themselves to a final analysis .
The me ws o f A br ] “ 1 26 . Funda ntal La lg e a Symbolical ” Of n d a Alg ebras . the fu ament l laws to which this analysis d two a and d b lea s , , the commut tive istri utive, had been noticed years before Peacock by the inventors o f symbolic
A a A di 63 1 . I d . 5 5 7 lgebr , ppen x , 1 bi .
- S YS TE M OF A B 1 28 N UMBER LGE RA . the d ictum that the laws regulating its operations consti a a a a b tut ed the. essenti l ch r cter of lge ra might have been ’ long delayed h ad no t Grego ry s paper been quickly followed “ ” d o f a b a ua terni o ns by the iscovery two lge r s , the q of
i lto n and Ausd ehnun slehre Gra ssma nn H a m the g of , in which
ne o f a o f a b a o f b a o the l ws the lge r num er, the commut tive
law a n h ad o a d . for multiplic tio , l st its v li ity
A o f 1 2 7 . Quaternions . ccording to his own account the “ di sc o v eryfi H amilton came upon qua terni o ns in a se arch fo r a second im aginary unit to correspo nd to the perpe ndi c u n lar which may b e drawn in space to the lines 1 a d i . a o f d a d x In pursu nce this i e he forme the e pressions , a i b c x i z a b c x su + +j , + y + j , in which , , , , y, z were p
be b an d a posed to real num ers , j the new imagin ry unit an d d sought, set their pro uct
a x — by
Th e n a . question the was, wh t interpretation to give if ’ ' ’ o do a i b c th e It w uld not to set it equal to + j , for then theorem that the modulus of a product is equal to the
d o f a d product of the mo uli its f ctors , which it seeme indis
b a a a d i n pensa le to m int in, would lose its v li ity ; unless , ' ’ ’ d d a b 0 i 0 a a ee , c , and therefore j , a very unn tur l as 1 i d f supposition, inasmuch is if erent from 0. N o w as course left for destroying the if term, therefore , but a bz c a a to m ke its coefficient , y, v nish, which was t nta
” b c z are mount to supposing, since , , y, perfectly general, that ji ii .
A den ial o the co mmuta tive la w ccepting this hypothesis , f
was H a o a as it , milton was driven to the conclusi n th t the system upo n which he h ad fallen cont ained at le ast three
a a u b i . H e a d im gin ry nits , the third eing the product j c lle
o o a M i II V o l 2 a az 5 1 844. Phil s phic l g ne , , . , S MB O I AR A R OF A B R A 1 2 Y L C CH CTE LGE . 9
It o o as a o b o f this , t k gener l c mplex num ers the system, a ib o hd x i z lew ua ter ni o ns b r + + j + , + y +j + , q , uilt thei d pro ucts, and assuming
2 Q 2 i =J = k
hi — i k = j,
found that the modulus law was fulfilled . A geometrical interpretation w as found fo r the imag ” i nar tri let i b c ltd b b c d y p + j , y making its coefficients , , , , the rectangular c o - ordinates of a po int in space ; the lin e drawn to this point from the origin picturing the triplet by n its length a d direction . Such directed lines H amilton named vec to rs .
To a a o f i o interpret geometric lly the multiplic tion int j, it w as then only necess ary to co nceive o f the j a xis as
d d i a an d turned b i t o h rigi ly connecte with the xis , y thr ug
t a d h: a right angle in the fi pl ne , into coinci ence with the
h n o f o f axis . T e geometrical meani gs other perations o l lowed readily .
a d b d o o f In secon paper, pu lishe in the same v lume the
P a Ma az H a o a d a hilosophic l g ine , milton c mp res in et il the
a a o ua terni o ns and th e a b o f b l ws of oper ti n in q lge ra num er, for the first time e xplicitly st ating and naming the a ss o i ti l w c a ve a .
’
Grassmann s A n s e . Ausd eh 1 2 8 . usdehnu g l hre In the nun slehre as G a d g , r ssmann first presente it, the elementary magnitudes are vect o rs . Th e fact that the e quation AB B 0: A O always holds
a i s among the segments of a line , when ccount taken of
o as d their directi ns well as their lengths , suggeste the o f d d a and le d probable usefulness irecte lengths in gener l ,
G a a A o f r ssm nn , like rgand, to make trial of this definition 1 30 UMB R - S S M OF A B R N E Y TE LGE A .
a d a o f o A B 0 no t d ition for the general c se three p ints , , , , in the same right line . But the outcome w as not gre at until he added to this his d u definition of the pro ct of two vectors . H e took as the
d a b a an d b a pro uct , of two vectors , , the p rallelogram gen crated by a when its initial p o int is carrie d along b fro m initial to final extremity . This definition m akes a product vanish no t only when o ne a a but o of the vector f ctors v nishes , als when the two
a . a d l w are p rallel It cle rly conforms to the istributive a . On the other hand, since
a b ba = 0 o r ba — a b + , ,
v l aw a the commutati e for multiplic tion has lost its validity, and as a o f a b , in quaternions , an interch nge f ctors rings about a change in the sign of the product . ’ Th e opening chapter of Grassmann s first tre atise on the Ausdehnung slehre (1 844) presents with admirable cle ar ness and from the general standpoint o f what he c alls ” “ F d a a ormenlehre (the octrine of forms ) , the fund ment l laws to which operations ar e subject as well in the Aus d ehnung slehre as in common algebra .
The Do r e o Art N mbe l Deve 1 29 . ct in f the ificial u rs ful y l Th e d o f a and Ausd ehn un s o ped . iscovery qu ternions the g ’ lehre m ade the algebra of number i n re ality what Gregory s d h ad ad a b a efinition m e it in theory, no longer the sole lge r ,
A and but merely one of a class of algebras . higher st point w as a b a d be cre ted, from which the laws of this alge r coul o f aw e seen in proper perspective . Which these l s wer d and a w as a o f e istinctive , wh t the signific nce each, cam o ut cle arly enough when numerical algebra could ' b e com pared with other algebra s whose characteristic laws were not the same as its characteristic laws .
— 1 32 N UMBER S YS TEM OF ALGEB RA . extends it to infinite gro ups and assemblages in his no w famous f e Memo irs o n the theo ry o infinite assembl ages . S e p articularly
‘
a a A a XL I . 489 . M them tische nn len, V , p Very recently much attentio n h as been given to the questi o n Wh at is th e s i mplest system o f c o nsistent and independent laws — o r “ ” a o as are a d —w h th e d a a o a o xi ms , they c lle y which fun ment l per ti ns o f th e o rdi nary algebra may be defined ? A very c omplete résumé d i n - L z o f th e literature may b e fo und in a paper by O. Ho l er eip iger
E . a a B 1 9 01 . S ee a o . o o o f erichte , ls V Hunting t n in Tr ns cti ns the
A ca Ma a ca So c Vo l . 1 1 1 . 264. meri n them ti l iety , , p
U N IVER SIT Y O F C ALIFO R N IA LIBRAR Y Lo s Ang eles
T h i s b o o k i s DUE o n th e l as d t stam t a e ped b elo w.
— Fo r m L9 3 2 m ( 5 8 7 6 8 4 ) 444