Download Day 8 - Introduction to Complex Numbers

Accelerated CCGPS…Math II
Name________________________
Intro to Complex Numbers
Period_____Date_______________
Review: If we want to solve an equation, then we must work with both sides to “undo” what was done
to the variable. For example, if x2 = 64, how do we “undo” squaring a number – we’ll take the
square root on both sides. So,
x2 
64 , and the square root of 64 is 8. However, -8 is
also a square root of 64, since (-8)(-8) = 64; hence, this equation actually has two solutions, which
are oftentimes written as ±8.
Example: (Sometimes we may need to use our rules for simplifying radicals as well.)
Solve:
5 x 2  4  64
5 x 2  4  4  64  4
Subtracting 4 from both sides to get:
5x 2
60

5
5
Dividing both sides by 5:
x2 
12
5 x 2  60
x 2  12
x 
Taking the square root on each side:
4 3  2 3
Solve the following equations by “undoing”:
1.)
x2 9
4.) x
7.)
2
 36  0
2x 2  2
2.)
x 2  144
3.) x
5.)
x 2 1  0
6.)
x 2 8  0
8.)
 4 x 2   36
9.)
1 2
x  32
2
x 2 2  7
12.) 16  x
10.)
x 2 3  1
11.)
13.)
3x 2  1  5
14.)
1 2
x  5  32
3
2
 128
15.) 2 x
2
2
 9
 11  x 2  5
From our science class, we know that the time it takes an object to hit the ground when it is
dropped from a height of s feet can be modeled by the equation h  16t
2
 s , where t is the
time in seconds. Find the time it takes when the object is dropped from the following heights:
16.) 80 feet
17.) 160 feet
18.) 320 feet
New Idea: Not all equations will have real number solutions. For instance, x2 = -1 has no real
number solutions because the square of any real number is never negative. To overcome this
situation, we can expand our system of numbers to include the imaginary unit, defined as
1  i .
(Note that i 2 = -1.) A complex number, written in standard form, is the number a + bi, wher a
and b are real numbers. The number a is the real part of the complex number, and the number bi
is the imaginary part.
Solve the following equations by “undoing”:
19.)
x 2   16
20.)
x 2   81
21.)
x 2  144  0
22.)
x 2 5  4
23.)
x 2 1  3
24.)
x 2  7  4x 2  5
Perform the indicated operations to write the expression as a complex number in standard form:
25.)
5  3i   2  4i 
26.)
3  2i   1  i 
27.)
 7  2i   3  3i 
28.)
5  i   3  8i 

29.) i  11  5i

31.) i 4  i



 
30.) i  6  i  4  2i

32.) 3i 1  2i


33.)
 4i 3  7i 
34.)
1  3i 1  i 
35.)
5  i 1  2i 
36.)
2  3i 3  4i 
37.)
3  2i    5  8i 
38.)
 2  4i   3  6i 
39.)
1

1  2
  i     2i 
3 2  3

40.)
 4  2i    1  5i 
41.)
5  8i   2  9i 
42.)
1
2  2
1 
  i   i 
2 3  3 4 
43.)
5  4i 3  6i 
44.)
2  5i 
45.)
 4  8i  4  8i 
46.)
1  i 
2
2