Mr. Marchez draws a triangle on the board. He labels the height (2 + 3
*
i
*
) and the base (2 - 4
*
i
*
). "Find the area of the triangle," he says. (Recall that the area of a triangle is
,
*
b
*
is the length of the base and
*
h
*
is the length of the height.)

### Watch This

First watch this video.

Khan Academy: Multiplying Complex Numbers

Then watch this video.

Khan Academy: Dividing Complex Numbers

### Guidance

When multiplying complex numbers, FOIL the two numbers together (see
*
Factoring When the Leading Coefficient Equals 1
*
concept) and then combine like terms. At the end, there will be an
term. Recall that
and continue to simplify.

#### Example A

Simplify:

a)

b)

**
Solution:
**

a) Distribute the to both parts inside the parenthesis.

Substitute and simplify further.

Remember to always put the real part first.

b) FOIL the two terms together.

Substitute and simplify further.

### More Guidance

Dividing complex numbers is a bit more complicated. Similar to irrational numbers, complex numbers cannot be in the denominator of a fraction. To get rid of the complex number in the denominator, we need to multiply by the
**
complex conjugate
**
. If a complex number has the form
, then its complex conjugate is
. For example, the complex conjugate of
would be
. Therefore, rather than dividing complex numbers, we multiply by the complex conjugate.

#### Example B

Simplify .

**
Solution:
**
In the case of dividing by a pure imaginary number, you only need to multiply the top and bottom by that number. Then, use multiplication to simplify.

*
When the complex number contains fractions, write the number in standard form, keeping the real and imaginary parts separate. Reduce both fractions separately.
*

#### Example C

Simplify .

**
Solution:
**
Now we are dividing by
, so we will need to multiply the top and bottom by the complex conjugate,
.

Notice, by multiplying by the complex conjugate, the denominator becomes a real number and you can split the fraction into its real and imaginary parts.

In both Examples B and C, substitute
to simplify the fraction further.
**
Your final answer should never have any power of
**

*greater than 1.*
**
Intro Problem Revisit
**
The area of the triangle is
so FOIL the two terms together and divide by 2.

Substitute and simplify further.

Now divide this product by 2.

Therefore the area of the triangle is .

### Guided Practice

1. What is the complex conjugate of ?

Simplify the following complex expressions.

2.

3.

4.

#### Answers

1.

2. FOIL the two expressions.

3. Multiply the numerator and denominator by .

4. Multiply the numerator and denominator by the complex conjugate, .

### Vocabulary

- Complex Conjugate

The “opposite” of a complex number. If a complex number has the form , its complex conjugate is . When multiplied, these two complex numbers will produce a real number.

### Explore More

Simplify the following expressions. Write your answers in standard form.