Imaginary and Complex
Numbers
1
Warm­Up!
Simplify each of the following.
1. √36
2.
25
3. √24
9
√
4. √3 (√15)
5. Solve for x.
x2 ­ 10x + 34 = 0
????
2
Unit Imaginary Number
i is a number whose square equals ­1
i2 = ­1
i=√­1
Example.
√-24
√­36
3
If x is a non­negative real number, then
√ ­x = i √ x
Example.
x=
10 ±√ ­36
2
4
A complex number is a number of the form a + bi, where the
real number a is called the real part and the real number b is
called the imaginary part and i is √­1 .
e.g. 5 + 3i
real
imaginary
Imaginary Number Line
­3i ­2i
­i
0i
i
2i
3i
5
Back to the Warm­up...
5. Solve for x.
x2 ­ 10x + 34 = 0
6
imaginary
5 + 3i
real
7
Powers of i
erase to reveal
i0 = 1
i1 = i
i2 = ­1
i3 = ­i
i4 = 1
i5 = i
i6 = ­1
i7 = ­i
i8 = 1
i9 = i
...
What about i1066?
erase to reveal
1066
i
= ­1
8
a + bi and a ­ bi are called complex conjugates of each other.
Let's take a look at what happens when we multiply complex
conjugates...
Find the complex conjugates of the following and multiply
them together.
5 + 3i
4 ­ 2i
6 ­ 9i
Do you see a pattern emerging?
9
Product of Complex Conjugates
(a + bi) (a ­ bi)
2
2 2
a ­abi + abi ­b i
2 2
a ­b (­1)
2
2
a + b
(a + bi) (a ­ bi) = a2 + b2
10
last examples
4+3i 5­2i
11