COMPLEX NUMBERS§5.6

OBJECTIVES

By the end of today, you should be able to…

• Identify and graph complex numbers.

• Add, subtract, and multiply complex numbers.

Remember when you first learned to count?

Now, your number system has expanded. You use rational numbers, like ½, and irrational numbers, like .

Today, your number system is going to expand to include numbers such as .

2

2

INTRODUCING…

Hey.This is i.

i is defined as the number whose square is -1.

and

An imaginary number is a number in the form a + bi, where b≠0.

1i

12 i

You’re saying I’m not real?!

i

PROPERTY: SQUARE ROOT OF A NEGATIVE REAL NUMBER

For any positive real number a, . aia

4414 i

7 7i

x3 xi 3

EXAMPLE 1: SIMPLIFYING NUMBERS USING

36.1a

xb 9.1

236.1 xc

In your graphing calculator, type and choose enter.

What does your calculator say?

Choose MODE, and then go down to REAL. Move your cursor to the right once, to a + bi, and press ENTER. This mode allows your calculator to work with

Type and choose enter.

1

imaginary numbers!

1You

found me!

http://www.youtube.com/watch?v=_jxy9bx6SQ8&NR=1&feature=endscreen

COMPLEX NUMBERS

A complex number can be written in the form

Where a and b are real numbers, including 0.

Real part

Imaginary part

EXAMPLE 2: WRITE THE COMPLEX NUMBER IN THE FORM

COMPLEX NUMBER PLANE

The absolute value of a complex number is its distance from the

origin on the complex number plane.

You can find the absolute value of a complex number by using

the Pythagorean Theorem.

EXAMPLE 3: FINDING ABSOLUTE VALUE

OPERATIONS WITH COMPLEX NUMBERS

You can apply the operations of real numbers to complex numbers.

If the sum of two complex numbers is 0, then each number is the opposite, or additive inverse, of the

other.Find the opposite:

EXAMPLE 4: ADDITIVE INVERSE OF A COMPLEX NUMBER

ADDING COMPLEX NUMBERS

To add or subtract complex numbers, combine the real parts and the imaginary parts separately.

Combine “like” terms:

EXAMPLE 5: ADDING COMPLEX NUMBER

MULTIPLYING COMPLEX NUMBERS

For two imaginary numbers, bi and ci,

You can multiply two complex numbers of the form a + bi by using the procedure for multiplying

binomials.Multiply:

EXAMPLE 6: MULTIPLYING COMPLEX NUMBERS

FINDING COMPLEX SOLUTIONS

Some quadratic equations have solutions that are complex numbers.

EXAMPLE 7: FINDING COMPLEX SOLUTIONS