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Chapter 1: Problem 22

Consider the complex number $$ z=\frac{a+i b}{c+i d} $$ where \(a, b, c\) and \(d\) are real. Determine \(\operatorname{Re}(z)\) and \(\operatorname{Im}(z)\).

Short Answer

Expert verified
Question: Given a complex number in the form \(z=\frac{a+i b}{c+i d}\), where \(a, b, c,\) and \(d\) are real numbers, find the real part (Re\((z)\)) and the imaginary part (Im\((z)\)) of the complex number. Solution: The real part of the complex number is Re\((z)=\frac{ac + b^2}{c^2 + d^2}\) and the imaginary part is Im\((z)=\frac{ad + bc}{c^2 + d^2}\).

Step by step solution

01

Find the complex conjugate of the denominator

First, find the complex conjugate of the denominator, which is in the form \(c + id\). The complex conjugate is given by: $$ c - id $$
02

Multiply the numerator and the denominator by the complex conjugate

Next, multiply both the numerator and the denominator by the complex conjugate of the denominator: $$ z=\frac{(a+ib)(c-id)}{(c+id)(c-id)} $$
03

Simplify the expression

Now, we will simplify the expression by expanding the numerator and the denominator: $$ z=\frac{ac + adi + ibc - (ib)^2}{c^2 - (id)^2} $$
04

Separate real and imaginary terms

Separate the real and imaginary terms in the expression: Real terms: $$ ac - (ib)^2 $$ Imaginary terms: $$ adi + ibc $$ Recall that \(i^2 = -1\). Then, we can rewrite the expression as: $$ z=\frac{(ac + b^2) + i(ad + bc)}{c^2 + d^2} $$
05

Identify the real and imaginary parts

Now that the expression is in standard form, we can identify the real and imaginary parts: Real part (Re(z)): $$ \operatorname{Re}(z)=\frac{ac + b^2}{c^2 + d^2} $$ Imaginary part (Im(z)): $$ \operatorname{Im}(z)=\frac{ad + bc}{c^2 + d^2} $$

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Most popular questions from this chapter

Consider the function $$ f(\theta)=\frac{1}{2}(a+i b) e^{i \theta}+\frac{1}{2}(a-i b) e^{-i \theta} $$ where \(a\) and \(b\) are real numbers. Show that \(f\) can be written in the form $$ f(\theta)=C_{1} \cos \theta+C_{2} \sin \theta $$ and determine the values of \(C_{1}\) and \(C_{2}\).
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