Chap 1 The Number System

We have shown earlier that the cartesian coordinates

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Unformatted text preview: (a) Write z = x + y i, where x, y ∈ R. Then z z = (x + y i)(x − y i) = x2 + y 2 . (b) Write z = x + y i and w = u + v i, where x, y, u, x ∈ R. Then zw = (xu − yv ) + (xv + yu)i, so that |zw|2 = (xu − yv )2 + (xv + yu)2 = (x2 + y 2 )(u2 + v 2 ) = |z |2 |w|2 . The result follows on taking square roots. (c) Note that the result is trivial if z + w = 0. Suppose now that z + w = 0. Then |z | + |w| |z | |w| z w = + = + |z + w| |z + w| |z + w| z+w z+w z w z w ≥ Re + Re = Re + z+w z+w z+w z+w The result follows immediately. Remark. Proposition 1D(c) is known as the Triangle inequality. It can be understood easily from the diagram below: ii iiww ii w ii ii w ii w ii w ii w ii w w |w | w |z +w| ww w w w w w j w jj w jj w jj w jj w jj w jj |z | w jj wjj j 2π 3 + 2i sin − 2π 3 √ = −1 − i 3. = Re1 = 1. w z 0 Chapter 1 : The Number System page 10 of 19 First Year Calculus c W W L Chen, 1982, 2005 The inequality follows on noting that the sum of the lengths of two sides of a triangle is at...
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This note was uploaded on 02/01/2009 for the course MATH 2343124 taught by Professor Staff during the Fall '08 term at UCSD.

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