Problem 47 - College Algebra

Question

Explain how to multiply complex numbers and give an example.

Step-by-Step Solution

Answer

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 1: 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 \( i^2 = -1 \).

Step 2: 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.

Step 3: 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 \( bdi^2 = bd(-1) \).

Step 4: Group Real and Imaginary Parts

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

Step 5: 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 \( 14 - 5i \).