Chap 1 The Number System

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online