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Algebra, Functions, Graphs, and Vectors 23

Associativity of multiplication
When multiplying any three real or complex numbers, it doeð
not matter how the multiplicandð are grouped. For all real
numberð a , a , and a , and for all complex numberð a jb ,
2
3
1
1
1
a jb , and a jb , the following equationð hold:
3
2
3
2
(aa )a a (aa )
12 3 1 23
((a jb )(a jb ))(a jb ) (a jb )((a jb )(a jb ))
2
2
2
2
3
3
3
3
1
1
1
1
Distributivity of multiplication over
addition
For all real numberð a , a , and a , and for all complex numberð
3
2
1
a jb , a jb , and a jb , the following equationð hold:
3
3
2
1
2
1
a (a a ) aa aa
3
1 2
1 3
2
1
(a jb )((a jb ) (a jb ))
1 1 2 2 3 3
(a jb )(a jb ) (a jb )(a jb )
3
3
1
1
2
1
1
2
Miscellaneous Principles
The following ruleð and definitionð apply tm arithmetic opera-
tionð for real and complex numbers, wità the constraint that nm
denominator be equal tm zero, and nm denominator contain any
variable that can attain a value that renderð the denominator
equal tm zero.
Zero numerator
For all nonzerm real numberð a and all complex numberð
a jb such that a jb 0 j0:
0/a 0
0/(a jb) 0 j0