Chap 1 The Number System

Proof a write z x y i where x y r then z z x

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Unformatted text preview: tion (8) will have a unique solution θ within this range. Definition. Suppose that z = x + y i = 0, where x, y ∈ R. Suppose further that the numbers r, θ ∈ R satisfy (6) and (7), and that r > 0 and −π < θ ≤ π . Then we say that the pair (r, θ) form the polar coordinates of z . Remarks. (1) In view of (7), we have z = r(cos θ + i sin θ). (2) Often, we write eiθ = cos θ + i sin θ. However, this is presupposing that we have understood the exponential function with complex exponents. √ Example 1.6.1. Suppose that z = 1 + i. Then |z | = 2 and arg z = π/4. Note also that √ π π z = 2 cos + i sin . 4 4 Try to draw the Argand diagram. Example 1.6.2. The polar coordinates (2, −2π/3) represent the complex number w = 2 cos − Try to draw the Argand diagram. The modulus has three very important properties that we often use. PROPOSITION 1D. (a) For every z ∈ C, we have |z |2 = z z . (b) For every z, w ∈ C, we have |zw| = |z ||w|. (c) For every z, w ∈ C, we have |z + w| ≤ |z | + |w|. Proof....
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