To parametrize x g y as y runs through some interval i

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Unformatted text preview: ction Hypothesis Properties of Exponents |z n | Hence, P (k + 1) is true, which means = |z |n is true for all natural numbers n. Like the Power Rule, the Quotient Rule can also be established with the help of the Product Rule. We assume w = 0 (so |w| = 0) and we get z w = 1 w (z ) = |z | 1 w Product Rule. 7 Since the absolute value |x| of a real number x can be viewed as the distance from x to 0 on the number line, this first property justifies the notation |z | for modulus. We leave it to the reader to show that if z is real, then the definition of modulus coincides with absolute value so the notation |z | is unambiguous. 8 This may be considered by some √ be a bit of a cheat, so we work through the underlying Algebra to see this is to true. We know |z | = 0 if and only if a2 + b2 = 0 if and only if a2 + b2 = 0, which is true if and only if a = b = 0. The latter happens if and only if z = a + bi = 0. There. 9 See Example 3.4.1 in Section 3.4 for a review of complex number arithmetic. 10 See Section 9.3 for a review of this techni...
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