Chap 1 The Number System

# We have shown earlier that the cartesian coordinates

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

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 02/01/2009 for the course MATH 2343124 taught by Professor Staff during the Fall '08 term at UCSD.

Ask a homework question - tutors are online