Download 5.4 Complex Numbers

WARM-UP
 Factor the following expressions:
 𝑓 𝑥 = 3𝑥 2 − 8𝑥 + 4
 𝑓 𝑥 = 15𝑥 2 − 27𝑥 − 6
4.4
COMPLEX NUMBERS
https://www.khanacademy.org/math/precalcul
us/imaginary-and-complex-numbers/theimaginary-numbers/v/introduction-to-i-andimaginary-numbers
As you watch the video please take notes on things that you may believe to be important!
NOTES FROM THE VIDEO
THE SQUARE ROOT OF A NEGATIVE NUMBER.
𝑖=
2
𝑖 =
3
𝑖 =
4
𝑖 =
The only
values youOF
need
two!
POWERS
I to remember are the first
You can always break down a high power into (𝑖 2 )𝑛 ∙ 𝑖
• 𝑖5 =
• 𝑖6 =
• 𝑖7 =
• 𝑖8 =
Simplify.
a) i 20
c) i 17
EXAMPLE 2
b) i 43
d) i 34
THE SQUARE ROOT OF A NEGATIVE NUMBER.
Property
Example
1. If r is a positive real number,
then r  i r .
5  i 5
2. By Property (1), it follows that
i r   r
2
i 5   i
2
2
 5  5
EXAMPLE 1
Simplify.
1) −180
WARM UP
2) i 85
HOMEWORK
 Simplifying Imaginary Number worksheet
HOMEWORK ANSWERS
1.6i
2.10i
3.9i
4.-14i
13. 𝑖 3
14. 𝑖 29
15. 3𝑖 11
16. -𝑖 10
HOMEWORK ANSWERS





5. i
6.-2i
19. 10i 3
20.10i 2
22.-5i 3
 24.4i







1
8
Back:
7.1
8.-1
12.i
13.not possible
14.-i
15.6i
 https://www.khanacademy.org/math/precalculus/imaginary and-complex-numbers/the-complex-numbers/v/complexnumber-intro
COMPLEX NUMBERS
complex number: a + bi
 a is the real part
 bi is the imaginary part
Real numbers: –1 , 3, 𝝅,
𝟓, 2/5
Complex Numbers: (a + bi, b ≠ 0), 2 + 3i, 5 – 5i
Pure Imaginary Numbers: (0 + bi, b ≠ 0), –4i, 6i
Add/subtract the real parts to get the real part,
add/subtract the imaginary parts to get the imaginary
part
 a. (–1 + 2i) + (3 + 3i)
 b. (2 – 3i) – (3 – 7i)
Add/subtract the real parts to get the real part,
add/subtract the imaginary parts to get the imaginary
part
 C.)(10-7i) + (6+9i)
 D. (-3+i) + (-4-i)
EXAMPLE 5
 Find the values of x and y that make the equation
3x + 4yi = 9 + 12i true.
Multiply the imaginary numbers.
 a. –i(3 + i)
 c. (5 + i)(3 – 7i)
Multiply the imaginary numbers.
 D. (1+2i)(11-4i)
 E. (4-i)(6-6i)
EXAMPLE 5
 Find the values of x and y that make the equation
2x + yi = –14 – 3i true.
Multiply the imaginary numbers.
 d. (2 + 3i)(2 – 3i)
Multiply the imaginary numbers.
 d. (4 - 6i)(4 + 6i)
a + b i and a – b i are conjugates.
When you multiply conjugates you get a real number
– no imaginary parts
EXAMPLE 4
 Divide the complex numbers. Answer should be in standard
form.
 a.
2  7i
1 i
EXAMPLE 4
 Divide the complex numbers. Answer should be in standard
form.
 b.
3  11i
1  2i
EXAMPLE 4
 Divide the complex numbers. Answer should be in standard
form.
 b.
3  i
3i
HOMEWORK
 Page 250: 26-31 and 43-47
HOMEWORK ANSWERS








26. 16+2i
27. -7
28. 3+7i
29. 9
30. 1
31 . 30+17i
43. x=-2, y=-3
46.
3x 2 = -48
2x 2 = -10
Solve 5y 2 + 20 = 0.
2x 2 + 26 = –10
4x 2 + 4 = 0
9x 2 + 4 = 0
EXAMPLE
9
2
3  x  4   7  31
EXAMPLE
8
1
2

2
x  1
5
WRITING QUADRATIC FUNCTIONS WITH
COMPLEX ROOTS
 imaginary solutions come in pairs
 if an imaginary number is a zero, then it’s conjugate is also a
zero
 so, if 4 + 3i is a solution, then 4 – 3i is also a solution
Write a quadratic function that has real coef ficients and a
leading coef ficient of 3 and has a complex root of 2 i.
Write a quadratic function that has real coef ficients and a
leading coef ficient of 1 and has a complex root of 4 + i.
TICKET OUT
Write a quadratic function that has real coef ficients and a
leading coef ficient of 1 and has a complex root of 3- i.