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Chapter 1: Problem 47

Explain how to multiply complex numbers and give an example.

Short Answer

Expert verified
To multiply complex numbers, expand the product just like a normal algebraic equation and keep track of the \( i \) terms that represent imaginary numbers. An example of multiplication of complex numbers would be \( (2 + 3i) \times (1 - 4i) = 14 - 5i \).

Step by step solution

01

Understand Complex Numbers

A complex number has the form \( a + bi \), where \( a \) and \( b \) are real numbers. \( a \) is the real part of the complex number and \( bi \) is the imaginary part. Here, \( i \) is the imaginary unit that has the property \( i^2 = -1 \).
02

Express the Complex Numbers to be Multiplied

Let's take two complex numbers for example, \( z1 = a + bi \) and \( z2 = c + di \). Now we need to multiply these two complex numbers.
03

Use the Distributive Property

To multiply the two complex numbers, we use the distributive property just as we would do when expanding a product of binomials in algebra. So, \( z1 \times z2 = (a + bi)(c + di) = ac + adi + bci - bd \). Remember that the last term is \( -bd \) because \( bdi^2 = bd(-1) \).
04

Group Real and Imaginary Parts

Now, we group the real part and the imaginary part together. So, the product is \( (ac - bd) + (ad + bc)i \).
05

Example

As an example, let's multiply \( (2 + 3i) \) and \( (1 - 4i) \). Following the above steps, we get the real part as \( (2*1 - 3*(-4)) = 2 + 12 = 14 \). The imaginary part is \( (2*(-4) + 1*3) = -8 + 3 = -5 \). So the product is \( 14 - 5i \).

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Most popular questions from this chapter

In Exercises \(21-28,\) divide and express the result in standard form. $$\frac{5 i}{2-i}$$

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$(-3-\sqrt{-7})^{2}$$

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x+4}{2 x-1} \leq 3 $$

Solve each quadratic inequality in Exercises \(1-28\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ 2 x^{2}+3 x>0 $$

What is the complex conjugate of \(2+3 i ?\) What happens when you multiply this complex number by its complex conjugate?
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