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)MULTIPLYING FOIL and then combine like terms. Remember i 2 = -1

= 15 – 12i – 10i+8i2

=15 – 22i +8(-1) = 7 – 22iNotice 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

ii

4523

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

ii

4545

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

FOIL

2

2

162020258101215

iiiiii

12 i

116202025

18101215

iiii

Combine like terms

41223 i

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

i412

4123

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

0252 x-25 -25

252 x

125125 x

01362 xxa

acbbx2

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

252366

2166

2166 i

246 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.

aacbbxcbxax

240

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 acb2. 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