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Chapter 10: Problem 128

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

Short Answer

Expert verified
To divide complex numbers, we use the conjugate of the divisor, which helps us to eliminate the imaginary part from the denominator. Our example gives a result of -1 - 0.4i.

Step by step solution

01

Understanding Complex Numbers

Complex numbers are in the form \(a + bi\) where \(a\) is the real part and \(bi\) is the imaginary part. For instance, if we have \(3 + 4i\) and \(1 + 2i\), the real parts are 3 and 1 respectively while the imaginary parts are 4 and 2 respectively.
02

Finding the conjugate

The conjugate of a complex number is found by changing the sign of the imaginary part. In our case, the conjugate of \(1 + 2i\) is \(1 - 2i\).
03

Multiplying by the conjugate

We multiply both the numerator and the denominator by the conjugate. This won't change the value of the fraction, but it will allow us to get rid of the imaginary part in the denominator. So, \((3 + 4i) *(1 - 2i)\) will be in the numerator and \((1 + 2i) * (1 - 2i)\) in the denominator.
04

Expanding the brackets

We need to multiply out the brackets using the distributive property. This gives us (-5 - 2i) in the numerator, and (5) in the denominator.
05

Simplifying the fraction

Lastly, divide the real and imaginary parts of the numerator by the real number in the denominator to get the final result: \(-5/5 - 2i/5 = -1 - 0.4i\)

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Most popular questions from this chapter

In Exercises \(39-64,\) rationalize each denominator. $$\sqrt[3]{\frac{2}{x y^{2}}}$$

Explain how to perform this multiplication: \((2+\sqrt{3})(4+\sqrt{3})\)

What are conjugates? Give an example with your explanation.

In Exercises \(75-92,\) rationalize each denominator. Simplify, if possible. $$\frac{17}{\sqrt{10}-2}$$

In Exercises \(105-112,\) add or subtract as indicated. Begin by rationalizing denominators for all terms in which denominators contain radicals. $$\frac{5}{\sqrt{2}+\sqrt{7}}-2 \sqrt{32}+\sqrt{28}$$
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