Download U2Day4

Bellwork
1.
2.
(6 x3  10 x 2  x  8)  (2 x 2  1)
(3x 3  10 x 2  12 x  22)  ( x  4)
Last Nights Homework
7.
1.
2x  4
23
6..
1
x2
2 x  11
x2  2x  4  2
x  2x  3
248
6 x 2  25 x  74 
x3
9 x 2  16
25
7..
x 2  8 x  64
27
8..
35
9..
4 x 2  14 x  30
a )1 b)4 c. 4
10
2..
15
3..
5.
21.
45a.
10.
45b.
x 3  3x 2  1 
d. 1954
Yes ( x  2)and ( x  1)are
45c.
(2 x  1)
( x  2)( x  1)( 2 x  1)
45d .
x  2, x  1, x  1 / 2
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)
-5 + 7i
• b) (3 - 4i) – (-3 - 5i)
6+i
• c) 3 - (-2 + 3i) + (-5 + i)
-2i
• d) (3 + 2i) + (4 – i) – (7 + i)
0
Example 2: Multiply
• a) 6 (3 – 4i)
• b) 2i (2 – 3i)
18-24i
6 + 4i
• c) i (-3i)
3
• d) √-4●√-16
-8
Example 2: Multiply
• e) (2 – i)(4 + 3i)
11 + 2i
• f) (3 – 4i)( 2 + i)
10-5i
• g) (3 + 2i)(3 – 2i)
• h) (3 + 2i)2
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.
a)
7
2  3i
b)
6i
 2i
conjugate 2  3i
conjugate i
14  21i 14 21i
 
13
13 13
 1  6i  1

 3i
2
2
Example 3: Divide
1
c)
1 i
conjugate 1  i
1 i 1 1
  i
2
2 2
d)
2  3i
4  2i
conjugate 4  2i
2  16i 1 4
  i
20
10 5
Example 4: Simplify
ii
.25
i 2  1
i 3  i
i4  1
.5
.75
1
Divide each exponent by 4 and determine the decimal,
which will in turn tell you what it equals
a) i 6  i 2  1
b) i176
c) i 13
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.
z  z2  c
• 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.
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