Download Algebra 2 unit 5

Algebra 2 Unit 5
Imaginary and Complex Numbers
Content to be Learned:
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Definition of the imaginary number, i
Simplifying negative radical expressions
Quadratic formula ~ Imaginary numbers
Complex numbers
Operations on complex numbers
The Imaginary Number, i
Recall that if you graph a quadratic function, you get one of three results:
Two Real Zeros
One Real Zero
No Real Zeros
(means that the zeros are
imaginary)
We have looked at (and solved) quadratic equations that either have 2 solutions (and
cross the x-axis twice) or 1 solution (cross the x-axis once). Now, what would the
quadratic formula look like for a graph that has no real solutions, meaning that the
graph never crosses the x-axis and the solutions are imaginary?
First, we must understand what the imaginary number is. By definition,
i=
The way it is used is as follows:
Recall:
25  5
 1 and i2 = -1
Now,
 25  25  1  25   1  5i
Try simplifying the following:
 100
 36
 24
 50
What would 3i2 simplify to? __________________ How about (3i)2_________________
Now that you can simplify radicals of negative numbers, you can solve quadratic
equations whose graphs do not cross the x-axis, meaning that the roots (zeros) are
imaginary)!!
Example: Use the quadratic formula to solve 3x2 + x + 2 = 0.
Practice
Directions: Simplify each expression.
9
2)  24
 300
4)  75
1)
3)
 63
6)
 225
7)  112
8)
 99
5)
Practice
Complex Numbers and Operations
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A complex number is ______________________________________________
________________________________________________________________
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We can perform operations on complex numbers very much like how we perform
operations on algebraic expressions.
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Remember: i2 = -1 and
1 = i
Adding and Subtracting Complex Numbers
Example 1: Add (7 + 5i) + (8 - 3i)
Answer: 15 + 2i
Example 2: Add (-3 + 4i ) + (7i + 5)
Answer:
Example 3: Add (4 – 3i ) + (-2 – i)
Answer:
Example 4: Subtract: (5 + 6i ) – (2 + 3i )
Answer: 3 + 3i
Example 5: Subtract: (-4i – 3) – (5 + 2i )
Answer:
Multiplying Complex Numbers- Think FOIL
Example 1: Multiply (3 + 2i )(1 + 4i )
Solution: 3 + 12i + 2i + 8i 2
3 + 14i + 8(-1)
3 + 14i – 8
-5 + 14i
Example 3: Multiply (3 + 7i )(4i – 8)
Example 4:
i(2 + i)(3 – 4i) + 2i
Example 2: Multiply (-2 – 3i )(-5 + i )
Practice
Division of Complex numbers
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In order to divide complex numbers, we need to know about conjugates.
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The conjugate of a + bi is a – bi
 The conjugate of a – bi is a + bi
Examples:
What is the conjugate of 2 -3i ?_________________________
What is the conjugate of 2 + 3i ?_________________________
What is the conjugate of -2 -3i ?_________________________
What is the conjugate of -2 + 3i ?_________________________
Example: Simplify (-3 – 4i)  (2 + 3i)
Practice