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• |z | ≥ 0 and |z | = 0 if and only if z = 0
• |z | = Re(z )2 + Im(z )2 • Product Rule: |zw| = |z ||w|
• Power Rule: |z n | = |z |n for all natural numbers, n
• Quotient Rule: z
, provided w = 0
|w| To prove the ﬁrst three properties in Theorem 11.14, suppose z = a + bi where a and b are real
numbers. To determine |z |, we ﬁnd a polar representation (r, θ) with r ≥ 0 for the point (a, b).
From Section 11.4, we know r2 = a2 + b2 so that r = ± a2 + b2 . Since we require r ≥ 0, then it
must be that r = a2 + b2 , which means |z | = a2 + b2 . Using the distance formula, we ﬁnd the 11.7 Polar Form of Complex Numbers 845 √
distance from (0, 0) to (a, b) is also a2 + b2 , establishing the ﬁrst property.7 The second property
follows from the ﬁrst. Since |z | is a distance, |z | ≥ 0. Furthermore, |z | = 0 if and only if the
distance from z to 0 is 0, and the latter happens if and only if z = 0.8 For the third property, we
note that, by deﬁnition, a = Re(z ) and b = Im(z ), so z = a2 + b2 = Re(z )2 + Im(z )2 .
To prove the product rule, suppose z = a + bi and...
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