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CCGPS Geometry
1 – Operations and Rules
1.2 – 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  i
Examples:
1.
16
2.
81
3.
45
4.
200
Powers of i
“I won, I won!” (Negatives in the middle)
Always divide the exponent by 4.

If you get a decimal of 0.25, then the answer is i.

If you get a decimal of 0.50, then the answer is – 1

If you get a decimal of 0.75, then the answer is –i.

If it divides evenly, then the answer is 1.
Examples:
5. i 75
6. i 29
7. i 251
8. i 9536
Add and Subtract Complex Numbers

Add or subtract the real parts, and then, add or subtract the imaginary parts.

Simplify. No powers of i higher than 1.

Write your answer in standard form (real 1st imaginary 2nd).
9.
 3  2i   7  6i 
12. 9  10  2i   5i
10.
6  5i   1 2i 
13.
11i
4
 
 4 i 3  2 i 4  6i 3
11.

9  4i    2  3i 
CCGPS Geometry
1 – Operations and Rules
1.2 – Notes and Practice
Multiplying Complex Numbers

Distribute.

Treat the i's like variables, then simplify any that are not to the first power.
14. i  3  i 
15.
 2  3i  6  2i 
Conjugates
16.

Two complex numbers of the form (a + bi) and (a – bi) are complex conjugates.

The product is always a real number.
Example

What is the conjugate of (2 – 4i)?
Dividing Complex Numbers

Multiply the numerator and denominator by the conjugate of the denominator.

Simplify completely.

No powers of i in the denominator of your answer!
Write each expression as a complex number in standard form.
17.
3  4i
2  4i
18.
5  2i
3  4i
 3  i  8  5i 
CCGPS Geometry
1 – Operations and Rules
1.2 – Notes and Practice
Name: ________________________________________________ Date: _______________________
Complex Numbers – CW/HW
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
20. Which of the following is not a real number?
a. 6
b.
36
c.
6
d. 6i2