Download Complex Numbers - Harrison High School

CCGPS Geometry
4 – Operations and Rules
4.3 – Notes and Practice
Name: ________________________________________________ Date: _______________________
Complex Numbers
UNIT QUESTION: In what ways can algebraic methods be used in problems solving?
MCC9-12.N.RN.1-3, N.CN.1-3, A.APR.1
Today’s Question: How do we take the square root of negative numbers?
MCC9-12.N.CN.1-3
  1
Examples:
1.
16
2.
81
3.
45
4.
200
Powers of 
Always divide the exponent by 4.

If it divides evenly, then the answer is 1.

If you get a remainder of 1 or 0.25, then the answer is

If you get a remainder of 2 or 0.50, then the answer is 1

If you get a remainder of 3 or 0.75, then the answer is 

Examples:
5.  13
6.  27
7.  54
8.  72
Add and Subtract Complex Numbers
 Add or subtract the real parts, and then, add or subtract the imaginary parts.
9.
 3  2   7  6 
10.
6  5   1 2 
11.
9  4    2  3 
CCGPS Geometry
12. 9  10  2   5
4 – Operations and Rules
13.
11
4

4.3 – Notes and Practice
 4 3  (2 4  6 3 )
Multiplying Complex Numbers
 Treat the  ’s like variables, then change any that are not to the first power.
Examples:
14.   3   
17.
 4  3  4  3 
15.
 2  3  6  2 
18. 2 1 4 
16.
 3    8  5 
19.
 3  2  5  9 
Conjugates
 Two complex numbers of the form a + bi and a – bi are complex conjugates.
 The product is always a real number.
Example
 2  4  2  4 
Dividing Complex Numbers
 Multiply the numerator and denominator by the conjugate of the denominator.
 Simplify completely.
Examples:
Write each expression as a complex number in standard form.
20.
5  2i
3  8i
21.
3  11i
1 2i
22.
23.
8  3i
1 2i
24.
6  3i
2i
25.
5
1 i
5  6i
3i
CCGPS Geometry
4 – Operations and Rules
4.3 – Notes and Practice
Complex Numbers – Practice
Write each expression in standard form.
1. i 2
2. i14
3. i 7
4. i 23
5. i71
6. i 204
7. i167
8. i 312
9. (3  i 2 )  i 4
10. (2  i)  (i 4  i 3 )
11. (5  i 3 )  (3  i 3 )
12. (2i)2(3i)3
13. (5  6i)(7  2i)
14. (2  3i)(2  3i)  4  i
15.
2
3i
16.
4i
3  3i
17.
4  2i
2  4i
18. Which has the same value as i 5  i 3 ?
a. -2i
b. -2
c. 2
d. 2i
19. Let r = (4 + i) and s = 1 – i. What is the value of r2 – s?
a. 14 + i
b. 15 + I
c. 15 + 7i
d.14 + 9i