Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 1 - Section 1.6 - Complex Numbers - 1.6 Exercises: 77

Answer

$z=a+bi$, $w=c+di$ So $\frac{}{z} +\frac{}{w}=\frac{}{z+w}$

Work Step by Step

$z=a+bi$, $w=c+di$, prove that $\frac{}{z} +\frac{}{w}=\frac{}{z+w}$ Find the conjugate of the two complex numbers by changing the sign of their imaginary part: $\frac{}{z}=a-bi$ $\frac{}{w}=c-di$ So $\frac{}{z} +\frac{}{w}=a+c-bi-di$ $(1)$ Evaluate $z+w=a+bi+c+di=a+c+bi+di$ Find the conjugate by changing the sign of the imaginary part of the complex number: $\frac{}{z+w}=a+c-bi-di$ $(2)$ From $(1)$ and $(2)$, we have: $\frac{}{z} +\frac{}{w}=\frac{}{z+w}$
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