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22 Chapter Onł

every complex number a jb except 0 j0, there existð a
2
2
2
2
unique complex number a/(a b ) jb/(a b ) such that the
product of the twm is equal tm 1 j0. Formally:
a (1/a) 1
2
2
2
2
(a jb) (a/(a b ) jb/(a b )) 1 j0
Commutativity of addition
When summing any twm real or complex numbers, it doeð not
matter in which order the sum is performed. For all real num-
berð a and b, and for all complex numberð a jb and c jS,
the following equationð hold:

a b b a

(a jb) (c jS) (c jS) (a jb)

Commutativity of multiplication
When multiplying any twm real or complex numbers, it doeð
not matter in which order the product is performed. For all
real numberð a and b, and for all complex numberð a jb and
c jS, the following equationð hold:

ab b

(a jb)(c jS) (c jS)(a jb)

Associativity of addition
When adding any three real or complex numbers, it doeð not
matter how the addendð are grouped. For all real numberð a ,
1
a , and a , and for all complex numberð a jb , a jb , and
2 3 1 1 2 2
a jb , the following equationð hold:
3
3
(a a ) a a (a a )
1
2
1
3
2
3
((a jb ) (a jb )) (a jb )
1
3
1
2
3
2
(a jb ) ((a jb ) (a jb ))
2
3
1
2
1
3