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Honors Math 3
Unit 2, Lesson 3 Notes – Complex Numbers
Name ________________________________
What are Imaginary Numbers?
An imaginary number is a mathematical term for a number whose square is a negative real number. Imaginary numbers are
represented with the letter i, which stands for the square root of -1. This definition can be represented by the equation: i2 = 1. Any imaginary number can be represented by using i. For example, the square root of -4 is 2i.
When imaginary numbers were first defined by Rafael Bombelli in 1572, mathematicians believed that they did not really
exist, hence their name. Decartes coined the term imaginary in reference to these numbers in his 1637 book, La Geometrie.
However, imaginary numbers are as real as any other numbers and have gradually come to be accepted by the mathematical
community and the world at large. The work of mathematicians Leonhard Euler and Carl Friedrich Gauss in the 18th and
19th centuries was instrumental in this change.
While imaginary numbers are meaningless in the "real world" of most individuals, they are indispensable in such fields as
quantum mechanics, electrical engineering, computer programming, signal processing, and cartography. For perspective,
consider that negative numbers were also once considered fictitious, and that such concepts as fractions and square roots
might be considered meaningless to a person who does not need them in everyday life, though they are quite real to others.
(Taken from http://www.wisegeek.com/what-is-an-imaginary-number.htm)
Imaginary numbers occur when a quadratic function has no roots in the set of real numbers. This means that the
graph of this function would NOT cross the x-axis.
Graph y  x 2  4 on your calculator. Does it cross the x-axis? _______
Now solve the related quadratic equation x 2  4  0 .
Therefore, i 2  ( 1) 2  1
Definition of i: i  1
Simplify:
1.
81
2.
11
6.
52
3.
27
4.
You Try:
5.
36
7.
13
48
Complex Number: any number that can be written in the form a  bi where a and b are real numbers and i is the
imaginary part.
Up until now you have only learned about the real number system consisting of Rational numbers (which includes
Natural #’s, Whole #’s, and Integers) and Irrational numbers.
Let’s talk about the Complex Number System.
Write in the form a  bi .
8. 3  7
9.
12  12
10. 6  49
Operations with Complex Numbers
To add/subtract complex numbers combine the real parts and the imaginary parts separately.
11. (8  5i )  (2  i)
12. (4  7i)  (2  3i)
**Given the following statement, solve for x and y.
14. 2 x  3 yi  14  9i
13. 5i  5(6  2i )
Multiplying Complex Numbers
First, simplify each square root by pulling out the “i”. Then, multiply.
1.
6
10
3.
75
3
2.
10
15
5. 5i 2i
4. 8i 3i
If multiplying involves real and imaginary parts, we must FOIL.
6. (4  2i )(3  5i )
7. (8  5i )(2  3i )
8. (9  7i )2
9. (3  16)(4  1)
Simplifying Powers of i:
Your Turn:
Simplify each number by using the imaginary number i.
1.
49
2.
144
3.
7
4.
10
5.
8
6.
48
Simplify each expression.
7. (2 + 3i) + (5  2i)
8. (4  2i)  (1 + 3i)
9. (4  3i)(5 + 4i)
10. (5  3i)(5 + 3i)
11. (4  i)2
12.
13.
14. 3i(2 + 2i)
15.
16. (6 + 7i) + (6  7i)
17. (5 + 3i)  (8 + 2i)
18. (2  i)(3 + 6i)
19. (1 + 3i)2
20. (2i)(5i)(i)
21.
22.
23. 2(3  7i)  i(4 + 5i)
24.