Problem 60 - College Algebra Essentials

Question

Explain how to multiply complex numbers and give an example.

Step-by-Step Solution

Answer

The product of two complex numbers \( a + bi \) and \( c + di \) is given by \( (ac - bd) + (ad + bc)i \). An example is the product of \( 3 + 2i \) and \( 1 - i \), which is \( 1 - i \).

Step 1: Introduction to Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part. They're usually represented in the form \( a + bi \), where \( a \) is the real part, \( b \) is the imaginary part, and \( i \) is the imaginary unit, defined as \( i = \sqrt{-1} \).

Step 2: Multiplication of Complex Numbers

When multiplying complex numbers, apply the distributive law, just as in the multiplication of binomials. If you have two complex numbers \( a + bi \) and \( c + di \), their product is given by \( (a+bi)(c+di)=ac + adi + bci - bd \), since \( i^2 = -1 \). This gives a real part of \( (ac - bd) \) and an imaginary part of \( (ad + bc) \). So the product is \( (ac - bd) + (ad + bc)i \).

Step 3: Example of Multiplication

Let's consider two complex numbers, \( 3 + 2i \) and \( 1 - i \). The product would therefore be \( (3*1 - 2*1) + (3*(-1) + 2*1)i = 1 - i \).