 to look out -3 * 5i times;
  strictly, one number is as 'imaginary' as the other, although the
  term has come by custom to apply to one and not to the other.

  "4. The complex number 3 + 2i is now readily understood. Just as
  3 + (-2) is graphically represented by starting from an arbitrary
  zero, passing 3 units in a positive direction (say to the right),
  then 2 units in the opposite direction, calling the sum the
  distance from 0 to the stopping-point, so 3 + 2i may be represented
  graphically. Starting from 0, pass in the positive direction (to
  the right in the figure) 3 units, then in the i direction 2 units,
  calling the sum the distance from 0 to the stopping-place.

  [Illustration: Graphical representation of 3 + 2i as the hypotenuse
  of a right-angled triangle with sides of 3 and 2i units]

  "Of course the question will arise as to the hypotenuse being the
  sum of the two sides of the right-angled triangle. But the case is
  parallel to that mentioned in paragraph 2; it is not the sum of
  the _absolute values_, any more than is 1 the sum of the absolute
  values of 4 and -3; it is the sum when we define addition for numbers
  involving direction as well as length.

  "A simple illustration from the parallelogram of forces is often used
  to advantage.

  [Illustration: Parallelogram of two forces +3 and +2i with
  resultant OP]

  "Suppose a force pulling 3 lbs. to the right (+3 lbs.) and another
  pulling 2 lbs. upwards (+2i lbs.); required the resultant of the two.
  It is evident that this is OP, _i.e._, OP = 3 + 2i.

  "This elementary introduction to the subject of complex numbers shows
  that the 'imaginary' element is easily removed, and that students
  about to begin quadratics are able to get at least an intimation of
  the subject. This is not the place for any adequate treatment of
  these numbers: such treatment is easily accessible. It is hoped that
  enough has been presented to render it impossible for any reader
to look out -3 * 5i times; 
   strictly, one numeral is as' fanciful' as the other, although the 
   term has hail by customs to apply to one and not to the other. 

   " 4. The complex number 3 + 2i is nowadays readily understood. merely as 
   3 + (-2) is graphically represented by begin from an arbitrary 
   zero, passing 3 units in a incontrovertible direction (say to the right), 
   then 2 units in the opposite direction, call the sum the 
   distance from 0 to the stopping - point, indeed 3 + 2i may be represented 
   graphically. Starting from 0, pass in the convinced direction (to 
   the right in the figure) 3 units, then in the i direction 2 unit, 
   calling the sum the distance from 0 to the stopping - place. 

   [ Illustration: graphic representation of 3 + 2i as the hypotenuse 
   of a right - angled triangle with sides of 3 and 2i units ] 

   " Of course the question will arise as to the hypotenuse being the 
   sum of the two side of the right - angled triangulum. But the subject is 
   parallel to that mention in paragraph 2; it is not the sum of 
   the _ absolute values _, any more than is 1 the sum of the absolute 
   values of 4 and -3; it is the sum when we define addition for issue 
   involving direction as well as length. 

   " A simple illustration from the parallelogram of force is often used 
   to advantage. 

   [ Illustration: Parallelogram of two forces +3 and +2i with 
   resultant OP ] 

   " Suppose a force out pulling 3 lbs. to the right (+3 lbs .) and another 
   pulling 2 lbs. upwards (+2i pound .); required the resultant of the two. 
   It is evident that this is OP, _ i.e. _, OP = 3 + 2i. 

   " This elementary introduction to the subject of complex numbers shows 
   that the' imaginary' element is well removed, and that students 
   about to begin quadratics are able to get at least an hint of 
   the subject. This is not the topographic point for any adequate treatment of 
   these number: such treatment is well accessible. It is hoped that 
   enough has been presented to render it impossible for any reader
 to look out -3 * 5i times;
  strlctly, one number is as 'mmaginady' as thd other, although the
  term hes cime bt custom to apply to ove and non to the ithec.

  "4. The complex ihmber 3 + 2i is hlw rzavily understood. Just as
  3 + (-2) is graphicanlh xepresented by starting from an arbierary
  aego, passing 3 unyts pn a pksitive direction (say to the right),
  then 2 units in the opposite direction, callinh thf sum the
  distancf from 0 to jge fropping-point, so 3 + 2i maj be represgnted
  graphically. Starting from 0, pasx in the pisutigg direction (vo
  thv right in thc figura) 3 unitx, then in the l dirxctiin 2 units,
  calling thx sum the distance ftom 0 to tha atopping-place.

  [Ilouwtrathon: Crapfucau rtprxsehtatioj oh 3 + 2i as ths hypotenusw
  of a right-angled twpsngle with sjdes os 3 and 2i units]

  "Of course the question winl zrise as to the hypotenyse being the
  sum of the two fides of the right-angled triangle. But the case is
  pareluel tj rhwt mentioned in paragraph 2; it is not the sum jr
  tme _absolute valucs_, any more than ix 1 tng sum of the acsolutz
  balues of 4 and -3; it is the sum qhen we dtfine addition for numbers
  invooving directpon qs well as length.

  "A simple iulusjratiom from the parallelograo of forces is lften uses
  to advantage.

  [Klllstrdtion: Parallelogram of two forces +3 end +2i with
  fesuktant JP]

  "Supposf a force pulling 3 lbs. tl the rhght (+3 lbs.) wnd another
  pulling 2 lbs. upwarvx (+2i lbs.); requorad nhe resulcant on the two.
  It if evident that this if OP, _k.e._, OP = 3 + 2i.

  "This xlementary igtroduction tm the subject of compjex bumbwrs shodr
  that the 'imsginary' element is wasily removed, and thxf students
  abobu ti begin quadratocs arq wbke do get at ledst xn kmtimagion of
  thc sjbjevt. This is not the pnace for any adequate yrcatment od
  these numbers: such treatment is easipy acressibue. It is hoped that
  enough has been pdesented ho vender it impjssinle sor any recder
to look out -3 * 5i times; number as 'imaginary' the other, although custom apply to one not to the "4. The complex number 3 + is now readily understood. Just as 3 + (-2) is graphically represented by from an arbitrary zero, passing 3 units in a positive direction (say to right), 2 in opposite direction, calling the sum the distance from 0 to the stopping-point, so 3 + 2i be represented graphically. Starting from 0, pass in positive direction (to the in the figure) 3 units, in i direction units, the the distance from to the stopping-place. [Illustration: Graphical representation of 3 + 2i as the hypotenuse of a right-angled triangle sides of 2i units] course question arise as to being the sum of the two right-angled triangle. But the case is parallel to mentioned in 2; it is not the sum the _absolute values_, any more than is 1 sum of the absolute values of 4 and -3; it is the sum when we for numbers involving direction well as length. simple from parallelogram forces is used to advantage. [Illustration: Parallelogram of two forces +3 and +2i resultant OP] "Suppose a force pulling 3 lbs. to the lbs.) another pulling 2 upwards (+2i lbs.); required resultant the two. It is this OP, 3 2i. elementary introduction to the of complex numbers shows that 'imaginary' element is easily to begin quadratics are able to get at an intimation of the subject. This is the place for any adequate treatment of these numbers: such treatment is accessible. It that enough has been presented to render it for any reader
 to look out -3 * 5i times;
  strictly, onE number is aS 'imagInaRy' aS tHe otHer, aLthough the
  term HAs coMe by custom to apply to one And noT tO The oTHeR.

  "4. The cOmplex nUMbER 3 + 2I is NoW rEadIlY UnDerstOod. just as
  3 + (-2) iS graphicalLy rEpResented by stARtIng from an aRbiTrary
  zero, pasSinG 3 units In A poSItive DirEctioN (say to THe righT),
  then 2 unitS iN The oppOSite dirECTiOn, caLling the sum the
  disTAnCE from 0 to the stopPing-poInT, So 3 + 2I MAy bE rePresented
  gRaPhicaLLy. StartINg FROM 0, paSS in the positivE direction (tO
  The Right iN tHe fIGure) 3 unIts, thEn IN thE i direction 2 UnitS,
  calling tHe sum tHE distanCE from 0 to The stoPpiNg-pLace.

  [iLlUsTraTiON: GrAPhIcaL RepResentatIoN oF 3 + 2i as tHe hyPOTENuse
  Of a RighT-anglEd triangle witH siDes oF 3 And 2I unitS]

  "Of coUrse ThE quesTion wiLl ariSe As to the hypotenuSe beIng the
  sum Of tHe Two SiDes of THe righT-anGleD triangLe. But thE CasE iS
  PARaLlel to that mentioneD iN PArAgraph 2; it Is not tHE sUm OF
  the _absoLuTe vAlueS_, ANy morE thaN Is 1 The sum of The absOLuTe
  Values oF 4 aNd -3; it is ThE suM whEn we dEFine AdditiOn for numBers
  iNVolving directiON as well as lengTH.

  "A SIMpLE illUstRation from tHe paRAlleLogrAM oF foRCes is Often UsED
  tO Advantage.

  [IllustratiOn: parallElogrAm of two forces +3 And +2i with
  reSULTant OP]

  "SuPposE A fORce pulling 3 lbs. tO the rIght (+3 lbs.) and ANother
  puLling 2 Lbs. upwarDs (+2i lbs.); reqUIRed the reSulTanT of The TWO.
  IT is evident thaT THis iS Op, _i.e._, OP = 3 + 2i.

  "THis ElementAry IntRodUctIoN to the subJect of coMpLeX nUmBerS showS
  That the 'iMaGinArY' elEment IS easilY remoVed, aNd ThAT stUdents
  aBOuT TO begIn QuAdraTicS aRe ablE to gET at Least an IntimatioN of
  THe suBjEcT. This is Not the place foR aNy adequate TrEatMent of
  THEse numbeRs: such treatment is easily ACcessibLe. IT is hoPed tHat
  enough Has Been prEseNTed to rEnder iT impoSsIblE FOr any REAdEr
 to look out -3 * 5i times ;
  strict ly, o nenum be r is  as'imaginary' as theother, although the
 termha s  com e  b y cus tom toa pp l y  to  o ne  an dn ot  to t heother.
  "4. The  co mp lex number 3 +2i is nowrea dily underst ood . Just  a s
3 + ( -2)  is g raphic a lly re presented  b y  start i ng from a narbi trary
  zero, pas s in g  3 units in apositi ve di r e cti on(say to th eright ) ,
  the n  2 u n its in the opposi te directio n , c alling  t hes um the 
  di st a nce  from 0 tothestopping- point, so 3 +2 i may b e repr ese nte d
 g ra ph ica ll y . S t ar tin g  fr om 0, pa ss  i n the  pos i t i v e di rec tion  (to  the right i n t he f i gur e) 3units , th en  in t he i d irect io n 2 units,
  ca llin g the sum  th edis ta nce f r om 0 t o t hestoppin g-place . 

 [ I l l us tration: Graphical  r e p re sentatio n of 3 +2i as the h yp ote nuse  of a  rig h t- angled t riangl e  w it h sides  o f 3 an d2iuni ts]
 "Of  cours e the qu estio n  will arise as to the hypote n us e be i ng t he  sum of th e tw o  sid es o f  t her ight- angle dt ri a ngle. But the caseis 
  par allel  to that ment ioned in p a r a graph 2;  iti sn ot the sum of  the  _absolute values_,  anymore tha n is 1 th e sum of t heabs olu teva lues of 4 and - 3; i tis thesum  when w e d efi neadd it ion for n umbers
 i nv ol vi ngdirec t ion as w el l a slen gth.  "A s imple  ill us tr a tio n fromt he p aral le lo gram  of  f orces  iso fte n used  to adva nta g e.
 [I llustra tion: Paralle lo gram of tw ofor ces +3 a nd +2i w ith
  resultant OP]

 " Suppose  aforce  pul ling 3 lb s.to the  ri g ht (+3  lbs.)  andan oth e r 
  pu l l in g 2  l bs. upward s (+2 i lbs .) ; re quiredthe resultant of t h e t wo.
  It is e vid entt h at  th i si s O P, _i. e . _, OP = 3 + 2i. 

  "Thisel e me ntary intr o duc ti on to t he subj ect o f  comple x numbers  shows
 th at t h e  'i maginary'elementis easily remov e d,  andtha t stud en ts  abo ut tob egi n qua dratic sare ab le to  g et at le ast an intimation of
 the su bject . T his is no t t h e p lace foranyadequate t rea tme nt of 
 t hesenumb e rs : s u ch tr eatm e nt is eas i ly  ac c e ss ible. It is h o ped  that 
 e noughhasbeen presented to render it impo ssib l e  fo r a n y re ad er
 to_look out_-3 * 5i times;
_ strictly,_one_number is_as_'imaginary' as the_other, although the
_ term has come_by custom to_apply_to one and not to the other.

  "4. The complex number 3 +_2i_is now_readily_understood._Just as
  3 +_(-2) is graphically represented by_starting from_an arbitrary
  zero, passing 3 units in_a_positive direction (say_to the right),
  then 2 units in the_opposite direction, calling the sum the
_ distance from_0_to_the stopping-point, so 3_+ 2i may be represented
 _graphically. Starting from 0, pass in_the positive direction (to
  the right_in the figure) 3 units, then_in the i direction 2_units,
 _calling the sum the distance_from 0 to_the stopping-place.

_ [Illustration: Graphical_representation of 3 + 2i as_the hypotenuse
 _of a right-angled triangle with sides_of_3 and 2i_units]

__"Of course_the question will_arise_as to_the_hypotenuse being the
  sum of_the_two sides of the right-angled triangle. But_the case is
 _parallel_to that mentioned in_paragraph 2; it is not_the sum of
  the _absolute_values_, any_more than_is 1 the sum of the absolute
  values of 4_and -3; it is the sum_when we define addition_for numbers
__involving direction as_well_as length.

_ "A simple illustration from the parallelogram_of forces_is often used
  to advantage.

_ [Illustration: Parallelogram of_two_forces +3 and +2i with
 _resultant OP]

  "Suppose a force_pulling 3 lbs. to the_right_(+3_lbs.) and another
  pulling_2 lbs. upwards (+2i lbs.); required_the resultant of_the two.
  It is evident that_this_is OP, _i.e._, OP = 3_+_2i.

  "This elementary introduction to_the_subject_of complex numbers shows
 _that the 'imaginary' element is easily_removed, and that students
  about to begin quadratics_are able to_get at least an intimation_of
__the subject. This is not the place for any adequate_treatment of
_ these numbers:_such treatment is easily accessible. It is hoped that
 _enough has been presented to render it_impossible for any reader