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Last Nights Homework

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2.4 Complex Numbers-How do you add, subtract, and multiply complex numbers?-How to use complex conjugates to divide complex numbers?-How do you plot complex numbers in the complex plain?

Quadratic Equations with a Negative Discriminant (b2 – 4ac < 0)• Complex Number: a + bi

• With the real number written first!• i = i• i2 = -1• i3 = -i• i4 = 1

Example 1: Add or Subtract• a) (3 + 5i) + (-8 + 2i)

• b) (3 - 4i) – (-3 - 5i)

• c) 3 - (-2 + 3i) + (-5 + i)

• d) (3 + 2i) + (4 – i) – (7 + i)

-5 + 7i

6 + i

-2i

0

Example 2: Multiply• a) 6 (3 – 4i)

• b) 2i (2 – 3i)

• c) i (-3i)

• d) √-4●√-16

18-24i

6 + 4i

3

-8

Example 2: Multiply• e) (2 – i)(4 + 3i)

• f) (3 – 4i)( 2 + i)

• g) (3 + 2i)(3 – 2i)

• h) (3 + 2i)2

11 + 2i

10-5i

13

5 + 12i

Example 3: Divide.-When there is a complex number in the denominator, then you must multiply the numerator and the denominator by the denominators conjugate.

ia

32

7)

i

ib

2

6)

iconjugate 32 iconjugate

13

21

13

14

13

2114 ii

i

i3

2

1

2

61

Example 3: Divide

ic

11

)i

id

24

32)

iconjugate 1 iconjugate 24

ii

2

1

2

1

2

1

ii

5

4

10

1

20

162

Example 4: Simplify

1) 26 iia 176) ib 13) ic

11

75.

5.1

25.

4

3

2

i

ii

i

ii

Divide each exponent by 4 and determine the decimal,which will in turn tell you what it equals

Plotting Complex Numbers

Example 5: Plot each complex number in the complex plane.• a) 2 + 3i

• b) -1 + 2i

• c) 4

• d)-3i

Fractal Geometry

• In 1980, a French mathematician named Benoit Mandelbrot started playing with graphing complex numbers in a computer.

• Here is the formula he was messing with • c is just some number like 3• z is a complex number z = a + bi• means it is a recursive formula.

• Some numbers you start with are going to get bigger and bigger. They’ll go off to infinity.

• Some numbers are going to get smaller and smaller. They go to zero.

czz 2

So, here’s what he Mandelbrot did:

• He told the computer to color the pixels on the computer screen for each number (point on the complex plane.)

• If the formula made the number go to zero, he told the computer to color it black. If the formula made the number shoot off to infinity, he told the computer to make it a color. The different colors meant how fast the number shot off.

Here’s the picture he got:

It’s called a fractal!

You can zoom in forever…and you always get some wild “complex” design!

Fractals in Art

Fractals in Nature

Tonight’s Homework• Pg180• #15, 17, 20, 30, 31, 34, 47, 49, 59, 66, • 71-74all