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Another sleep deprived 45 minutes in your

Villa Walsh Pre-Calculus experience!

2.4 Complex Numbers

Homework for section 2.4

p164 #9-19, 25-29, 35-39, 43-45, 51-57, 63-77

The imaginary unit: i

Solve, using real numbers: x2 + 1 = 0

Mathematicians got tricky and created an expanded number system, which included: i.

i is defined as follows:

1i

2 1i

Real numbers combined with multiples of this imaginary unit are known as the:

Set of Complex Numbers

The STD form of a complex number is:

a + bi

Real part

Imaginary part

If b = 0, then a + bi = a and a therefore is a real

number.

If a = 0, then a + bi = bi and bi therefore is a pure imaginary number.

If b = 0, then a + bi is an imaginary number.

EVERY number can be written as a complex number.

Example: 4 = 4 + 0i

Sum and Difference of complex numbers. a bi c di a c b d i

a bi c di a c b d i

Examples

3 2 3i i 5 2 i

3 2 6 13i i 3 11i

Multiplication of complex numbers.

FO I La bi c di

Examples 3 2 3 2i i 13

22 3i 5 12 i

Note: after foiling, if you have an i2, you

must change it to a (-1) and multiply as needed.

Complex conjugates and division

and are:

complex conj

ugates

a bi a bi

a bi a bi c dic di c di c di

Doing this is along the same line as rationalizing the denominator - it cleans up the denominator so that the denominator is areal, rational number.

28 4 126

41 2 6

7 61i i ii

i

6 7 6 7 1 2

1 2 1 220 5

51 2

i i i

i ii

i

6 7 1 2

1

6 7

1 1 22 2

i i

i

i

i i

Examples

6 7

1 2

i

i

4 i

2 3

4 2

i

i

1 4

10 5i

Pull out the -1 first…THEN do whatever the math has you do.

5 5 1 5 i or 5i

Complex Solutions of Quadratic Functions

216 4 3 0 solve for x.x x

Example

216 4 3 0 solve for x.x x

4 16 4(16)(3)2(16)

x

4 11(16)

32

4 4 1132

i

1 118 8

i

NO T .125 .4145x i

The different powers of i

i

2i

3i

4i

5i

6i

7i

8i

1

1

i

1

1

1

i

1

…and so on.

2 34 2i i

3

i

6

2

31

2 i

6

i

Go! Do!