Chapter 0 Notes - Precalculus
0.2 Operations with Complex Numbers
A. Imaginary and Complex Numbers
Imaginary unit i: 𝒊 = √−𝟏 and therefore 𝒊𝟐 = −𝟏
𝒊=
𝒊𝟓 =
𝒊𝟐 =
𝒊𝟔 =
𝒊𝟑 =
𝒊𝟕 =
𝒊𝟒 =
𝒊𝟖 =
Powers of i: to find the value of 𝑖 𝑛 , let R be the remainder when n is divided by 4
n<0
n>0
𝑅 = −3 𝑡ℎ𝑒𝑛 𝑖 𝑛 = 𝑖
𝑅 = 1 𝑡ℎ𝑒𝑛 𝑖 𝑛 = 𝑖
𝑅 = −2 𝑡ℎ𝑒𝑛 𝑖 𝑛 = −1
𝑅 = 2 𝑡ℎ𝑒𝑛 𝑖 𝑛 = −1
𝑅 = −1 𝑡ℎ𝑒𝑛 𝑖 𝑛 = −𝑖
𝑅 = 3 𝑡ℎ𝑒𝑛 𝑖 𝑛 = −𝑖
𝑅 = 0 𝑡ℎ𝑒𝑛 𝑖 𝑛 = 1
𝑅 = 0 𝑡ℎ𝑒𝑛 𝑖 𝑛 = 1
Practice
Simplify using the i^2 method.
1. 𝑖 16
2. 𝑖 25
3. 𝑖 −14
5. 𝑖 23
6. 𝑖 −18
8. 𝑖 31
9. 𝑖 −22
Simplify using the i^4 method.
4. 𝑖 20
Simplify using the remainder method.
7. 𝑖 24
Complex number: number that has a real and an imaginary part
Standard form: a + bi
Imaginary number: 4 + 2i
Pure imaginary number: 2i
1
Practice. Simplify.
a) (4 − 𝑖 ) + (−3 + 5𝑖)
b) (8 − 5𝑖 ) − (4 − 2𝑖)
c) (6 + 2𝑖 ) − (10 − 5𝑖)
d) (2 − 3𝑖)(6 + 7𝑖)
e) (3 − 2𝑖 )2
f) (4 + 𝑖 )(4 − 𝑖 )
g) (6 − 3𝑖)(8 + 2𝑖)
h) (3 − 4𝑖)(3 + 4𝑖)
B. Use Complex Conjugates
Complex conjugates: two complex numbers of the form a + bi and a – bi
Are used to rationalize complex numbers (we aren’t allowed to leave these in the denominator of a
fraction)
Practice. Simplify.
1.
12+3𝑖
2.
1+2𝑖
2
6−2𝑖
4+𝑖
3. (8 − 3𝑖) ÷ (4 + 3𝑖)
4. (1 + 𝑖) ÷ (2 − 6𝑖)
0.3 Quadratic Functions and Equations
Standard Form of a Quadratic Equation: 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
A. Graph Quadratic Functions
Parabola
Practice: Graph each by making a table.
a) 𝑓 (𝑥) = 2𝑥 2 − 8𝑥 + 4
b) 𝑔(𝑥) = 𝑥 2 + 3𝑥 − 10
3