a + bi

When we take the square root of both sides of an equation or use the quadratic formula, sometimes we get a negative under the square root. Because of this, we'll introduce the set of complex numbers.

12 i

This is called the imaginary unit and its square is -1.

We write complex numbers in standard form and they look like:

bia This is called the real part This is called the imaginary part

We can add, subtract, multiply or divide complex numbers. After performing these operations if we’ve simplified everything correctly we should always again get a complex number (although the real or imaginary parts may be zero). Below is an example of each.

(3 – 2i) + (5 – 4i)ADDINGCombine real parts and combine imaginary parts= 8 – 6i

(3 – 2i) - (5 – 4i)SUBTRACTING

= -2 +2i

Be sure to distribute the negative through before combining real parts and imaginary parts3 – 2i - 5 + 4i

(3 – 2i) (5 – 4i)MULTIPLYINGFOIL and then combine like terms. Remember i 2 = -1

= 15 – 12i – 10i+8i2

=15 – 22i +8(-1) = 7 – 22i

Notice when I’m done simplifying that I only have two terms, a real term and an imaginary one. If I have more than that, I need to simplify more.

DIVIDING

i

i

45

23

To divide complex numbers, you multiply the top and bottom of the fraction by the conjugate of the bottom.

i

i

45

45

This means the same complex number, but with opposite sign on the imaginary term

FOIL

2

2

16202025

8101215

iii

iii

12 i

116202025

18101215

ii

ii

Combine like terms

41

223 i

We’ll put the 41 under each term so we can see the real part and the imaginary part

i41

2

41

23

Let’s solve a couple of equations that have complex solutions.

0252 x-25 -25

252 x

125125 x

01362 xxa

acbbx

2

42

Square root and don’t forget the

The negative 1 under the square root becomes i

Use the quadratic formula

12

131466 2 x

2

52366

2

166

2

166 i

2

46 i i23

i5

Powers of i

12 iii

iiiii )(123

111224 iii

iiiii 145

111246 iii

iiiii 1347

111448 iii

We could continue but notice that they repeat every group of 4. For every i 4

it will = 1

To simplify higher powers of i then, we'll group all the i 4ths and see what is left.

iiiii 88433 1

4 will go into 33 8 times with 1 left.

iiiii 320320483 1

4 will go into 83 20 times with 3 left.

a

acbbxcbxax

2

40

22

If we have a quadratic equation and are considering solutions from the complex number system, using the quadratic formula, one of three things can happen.

3. The "stuff" under the square root can be negative and we'd get two complex solutions that are conjugates of each other.

The "stuff" under the square root is called the discriminant.

This "discriminates" or tells us what type of solutions we'll have.

1. The "stuff" under the square root can be positive and we'd get two unequal real solutions 04 if 2 acb

2. The "stuff" under the square root can be zero and we'd get one solution (called a repeated or double root because it would factor into two equal factors, each giving us the same solution).04 if 2 acb

04 if 2 acb