A Review of Complex Arithmetic

Transcription

1 /0/005 Rviw of omplx Arithmti.do /9 A Rviw of omplx Arithmti A omplx valu has both a ral ad imagiary ompot: { } ad Im{ } a R b so that w a xprss this omplx valu as: whr. a + b Just as a ral valu a b xprssd as a poit o th ral li, a omplx valu a b xprssd as a poit o th omplx pla: Im{ } a a + b b R{ }

2 /0/005 Rviw of omplx Arithmti.do /9 Th valus (a,b) ar a artsia rprstatio of a poit o th omplx pla. Rall that w a altrativly dot a poit o a -dimsioal pla usig polar oordiats: dista from th origi to th poit φ rotatio agl from th horizotal ( R{ } ) axis i.., Im{ } a + b b φ a R{ } Usig our kowldg of trigoomtry, w a dtrmi th rlatioship btw th artsia (a,b) ad polar ( φ ) rprstatios. From th Pythagora thorm, w fid that:, a + b

3 /0/005 Rviw of omplx Arithmti.do 3/9 Likwis, from th dfiitio of si (opposit ovr hypotus), w fid: b b siφ a + b or, usig th dfiitio of osi (adat ovr hypotus): a a osφ a + b ombiig ths rsults, w a dtrmi th tagt (opposit ovr adat) of φ : siφ b taφ osφ a Thus, w a writ th polar oordiats i trms of th artsia oordiats: a + b ( ) ta b a b φ os si a a + b a + b

4 /0/005 Rviw of omplx Arithmti.do 4/9 Likwis, w a us trigoomtry to writ th artsia oordiats i trms of th polar oordiats. For xampl, w a us th dfiitio of si to dtrmi b: b siφ ad th dfiitio of osi to dtrmi a: Summarizig: a o φ s a osφ b siφ Not that w a xpliitly writ th omplx valu i trms of its magitud ad phas agl φ : a + b osφ + siφ ( osφ siφ ) + Hy! w a us Eulr s quatio to simplify this furthr!

5 /0/005 Rviw of omplx Arithmti.do 5/9 Rall that Eulr s quatio stats: osφ + siφ so omplx valu is: a + b ( os si ) + φ φ φ Now w hav two ways of xprssig a omplx valu! a + b ad/or φ Not that both rprstatios ar qually valid mathmatially ithr o a b sussfully usd i omplx aalysis ad omputatio. Typially, w fid that th artsia rprstatio is asist to us if w ar doig arithmti alulatios (.g., additio ad subtratio).

6 /0/005 Rviw of omplx Arithmti.do 6/9 For xampl, if: a + b ad a + b th: ( ) ( ) + a + a + b + b ( ) ( ) a a + b b ovrsly, for gomtri alulatios (multipliatio ad divisio), it is asir to us th polar rprstatio: For xampl, if: ad φ th: ad: ( φ + φ ) ( φ φ )

7 /0/005 Rviw of omplx Arithmti.do 7/9 Not i th abov alulatios w hav usd th gral fats: y y z x x x x ad x z x ( y + z ) ( y z ) Additioally, w ot that powrs ad roots ar most asily aomplishd usig th polar form of : ad ( ) ( ) ( ) ( ) φ Thrfor: ad: ( ) ( ) φ φ ( ) φ

8 /0/005 Rviw of omplx Arithmti.do 8/9 Fially, w dfi th omplx ougat ( * ) of a omplx valu : omplx ougat of a b φ A vry importat appliatio of th omplx ougat is for dtrmiig th magitud of a omplx valu: Typially, th proof of this rlatioship is giv as: ( )( ) a ( a b ) b ( a b ) a+ b a b + a + ab ba b a + b Howvr, it is mor asily show as:

9 /0/005 Rviw of omplx Arithmti.do 9/9 ( )( ) 0 ( φ φ ) Aothr importat rlatioship ivolvig omplx ougat is: ( ) ( ) ( a a) ( b b) + a + b + a b + + a Thus, th sum of a omplx valu ad its omplx ougat is a purly ral valu. Additioally, th diffr of omplx valu ad its omplx ougat rsults i a purly imagiary valu: ( ) ( ) ( a a) ( b b) a + b a b + + b Not from ths rsults w a driv th rlatioships: + a R{ } b Im{ }