D in general z x y x 0 0 y x iy z x

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Unformatted text preview: write as iy). (d) In general, z = (x, y) = (x, 0) + (0, y) = x + iy. z = x + iy (e) Also, i2 = (0, 1)(0, 1) = (-1, 0) = -1. i2 = -1 (g) 4 - 3i = (4, -3). Note In view of (d) above, from now on we shall write the complex number (x, y) as x+iy. Definitions 1.4 The complex conjugate, z–, of the complex number z = x+iy given by z– = x - iy. The magnitude, |z| of z = x+iy is given by |z| = x2+y2 . Examples and Geometric Representation of Conjugation and Magnitude - in class. Notes 1. z + z– = (x+iy) + (x-iy) = 2x = 2Re(z). Therefore, 1 Re(z) = 2 (z+z–) 2 z - z– = (x+iy) - (x-iy) = 2iy = 2iIm(z). Therefore, 2. Note that zz– = (x+iy)(x-iy) = x2-i2y2 = x2+y2 = |z|2 Im(z) = 1 2i (z-z–) zz– = |z|2 3. If z ≠ 0, then z has a multiplicative inverse. Why? because: |z|2 z– zz– z· 2 = 2 = 2 = 1. Hence, |z| |z| |z| Examples 1 (a) = -i i 1 (c) 1 = (1+i) 2 (b) 1 2 (1-i) z-1 = z– |z|2 1 3-4i = 3+4i 25 1 (d) = cos(-ø) + isin(-ø) cosø + isinø 4. There is also the Triangle Inequality: |z1 + z2| ≤ |z1| + |z2|....
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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