Download 1.1-1.4---imaginary-numbers

Daily Check
For each equation find the discriminant
and the number of solutions.
1. 3x  4 x  2  0
2
2. x  6 x  9  0
2
Launched Object
h(t) = -16t2 + 64t + 80
a) How many seconds until the max height
2 sec.
is reached?
144 ft.
b) What will be the max height?
c) How many seconds until the object hits
5 sec.
the ground?
Math I
Day 10 (8-24-09)
UNIT QUESTION: What is a
quadratic function?
Standard: MM2A3, MM2A4
Today’s Question:
How do we take the square root of
negative numbers?
Standard: MM2N1.a, b, c, d
2
2
2
i
i  1
• You can't take the square root of a negative
number, right?
• When we were young and still in Math I, no
numbers that, when multiplied by
themselves, gave us a negative answer.
• Squaring a negative number always gives
you a positive. (-1)² = 1. (-2)² = 4 (-3)² = 9
So here’s what the math people did:
They used the letter “i” to represent the
square root of (-1). “i” stands for
“imaginary”
1
So, does
really exist?
i  1
Examples of how we use
16  16  1
 4i
 4i
i  1
81  81  1
9i
 9i
Examples of how we use
i  1
45  45  1
 3  3  5  1
 3 5  1
 3 5 i
 3i 5
200  200  1
 2  2  2  5  5  1
 2  5 2  1
 10 2  i
 10i 2
i 
2
i 
3
i 
4
i 
5
1st
Ex: Solve x2+ 6x +10 = 0
 b  b  4ac 6  6  4 110

x
2 1
2a
nd
a=
6  36  4 1 10 6  236

40


b=
2 1
2
2
2
c=
6  4 6  2i  6  2i and 6  2i


2
2
2
2
 3  i and  3  i
Complex Numbers
A complex number has a real part & an
imaginary part.
Standard form is:
a  bi
Real part
Example: 5+4i
Imaginary part
The Complex Plane
Real Axis
Imaginary Axis
Graphing in the complex plane
.
 2  5i
2  2i
4  3i
 4  3i
.
.
.
Adding and Subtracting
(add or subtract the real parts, then add or subtract the
imaginary parts)
Ex: (1  2i)  (3  3i)
 (1  3)  (2i  3i)
 2  5i
Ex: (2  3i)  (3  7i)
 (2  3i )  (3  7i )
 1 4i
Ex: 2i  (3  i )  (2  3i )
 (3  2)  (2i  i  3i )
 1 2i
Graphing in the complex plane
.
 2  5i
2  2i
4  3i
 4  3i
.
.
.
Absolute Value of a Complex Number
The distance the complex number is from
the origin on the complex plane.
If you have a complex number (a  bi )
the absolute value can be found using:
a b
2
2
Graphing in the complex plane
 2  5i
.
5
2
Examples
1.  2  5i
 (2) 2  (5) 2
 4  25
 29
2.  6i
 (0) 2  (6) 2
 0  36
 36
6
Which of these 2 complex numbers is
closest to the origin?
-2+5i
Try These!!!
1.
4i
2.
3i
 (4) 2  ( 1) 2
 (0) 2  (3) 2
 16  1
 17
 09
 9
3
Which of these 2 complex numbers is
closest to the origin?
3i
Practice
Coach Workbook Page 21
Assignment
Page 4 #16-21
Page 8 #6-10
Page 19 #1-10