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 .