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Chapter 2: Problem 68

Explain how to divide complex numbers. Provide an example with your explanation.

Short Answer

Expert verified
The division of \(1 + 2i\) by \(3 + 4i\) results in \(-0.20 - 0.08i\).

Step by step solution

01

Understand the Concept

The first important thing to note is that any complex number \(a + bi\) has a conjugate given by \(a - bi\). We make use of this conjugate when dividing complex numbers.
02

Set Up the Problem

Let's say we have two complex numbers \(1 + 2i\) and \(3 + 4i\) and we want to divide the first by the second. So we have to solve: \(\frac{1 + 2i}{3 + 4i}\).
03

Multiply by the Conjugate

Now, get rid of the denominator which is a complex number itself. To do this, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(3 + 4i\) is \(3 - 4i\), so we multiply both top and bottom by this: \(\frac{1 + 2i}{3 + 4i} * \frac{3 - 4i}{3 - 4i}\)
04

Perform the Operations

Next, perform the operations. Use the formula for multiplication of complex numbers (FOIL method) for the numerator, and use the difference of squares for the denominator: \(\frac{(1*3 + 1*(-4i) + 2i*3 + 2i*(-4i))}{(3*3 + 3*(-4i) + 4i*3 + 4i*(-4i))} = \frac{-5 - 2i}{25}\)
05

Express in Standard Form

Finally, express the result in standard form, separating the real part from the imaginary part: \(-\frac{5}{25} - \frac{2i}{25} = -0.20 - 0.08i\)

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \((x+3)(x-1) \geq 0\) and \(\frac{x+3}{x-1} \geq 0\) have the same solution set.

If \(f\) is a polynomial or rational function, explain how the graph of \(f\) can be used to visualize the solution set of the inequality \(f(x)<0\).

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+x-6}{x-3}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can have three vertical asymptotes.

If you are given the equation of a rational function, explain how to find the horizontal asymptote, if there is one, of the function's graph.
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