Unformatted text preview: w = c + di for real numbers a, b, c and d. Then
zw = (a + bi)(c + di). After the usual arithmetic9 we get zw = (ac − bd) + (ad + bc)i. Therefore,
√ = a2 c2 − 2abcd + b2 d2 + a2 d2 + 2abcd + b2 c2 Expand √ a2 c2 + a2 d2 + b2 c2 + b2 d2 Rearrange terms a2 (c2 + d2 ) + b2 (c2 + d2 ) =
= (ac − bd)2 + (ad + bc)2 Factor (a2 + b2 ) (c2 + d2 )
a2 + b2 c2 + d2 Factor
Product Rule for Radicals = |z ||w| Deﬁnition of |z | and |w| Hence |zw| = |z ||w| as required.
Now that the Product Rule has been established, we use it and the Principle of Mathematical
Induction10 to prove the power rule. Let P (n) be the statement |z n | = |z |n . Then P (1) is true
since z 1 = |z | = |z |1 . Next, assume P (k ) is true. That is, assume z k = |z |k for some k ≥ 1.
Our job is to show that P (k + 1) is true, namely z k+1 = |z |k+1 . As is customary with induction
proofs, we ﬁrst try to reduce the problem in such a way as to use the Induction Hypothesis.
= zk z = z k+1 zk Properties of Exponents
|z | Product Rule |z |k |z | = = |z |k+1 Indu...
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