1. 5.7 Complex Numbers
12/17/2012

2. Quick Review Exponent Rule:
If a number doesn’t show an exponent, it is understood that the number has an exponent of 1. Ex: 8 = 81 , x = x1 , -5 = -51 Also, any number raised to the Zero power is equal to 1 Ex: 30 = = 1
Exponent Rule:
When multiplying powers with the same base, you add the exponent.
x2 • x3 = x5
y • y7 = y8

3. The square of any real number x is never negative, so the equation x2 = -1 has no real number solution.
To solve this x2 = -1 , mathematicians created an expanded system of numbers using the IMAGINARY UNIT, i.

4. Simplifying i given any powers
The pattern repeats after every 4.
So you can find i raised to any power by dividing the exponent by 4 and see what the remainder is. Based on that remainder, you can determine it’s value.
Step 1. 22÷ 4 has a remainder of 2
Step 2. i22 = i2
Step ÷ 4 has a remainder of 3
Step 2. i51 = i3
Do you see the pattern yet?

5. Checkpoint
Find the value of
1. i 15
2. i 20
3. i 61 
4. i 122

6. Properties of Square Root of Negative Number

7. Example 1 Solve the equation. = 7x 2 49 – a. b. = 3x 2 5 – 29 SOLUTION
Solve a Quadratic Equation
Solve the equation. =
7x 2 49 – a. b. =
3x 2 5 – 29
SOLUTION
Write original equation. =
7x 2 49 – a.
Divide each side by 7. =
x 2 7 –
Take the square root of each side. = x + – 7
Write in terms of i. = x + – 7 i

8. Example 1 b. = 3x 2 29 – 5 = 3x 2 24 – = x 2 8 – = x + – 8 = x + – 8 i
Solve a Quadratic Equation b. =
3x 2 29 – 5
Write original equation.
Add 5 to each side. =
3x 2 24 –
Divide each side by 3. =
x 2 8 –
Take the square root of each side. = x + – 8
Write in terms of i. = x + – 8 i
Simplify the radical. = x + – 2 i 8

9. Checkpoint Solve the equation. 1. x 2 = – 3 ANSWER 3, i 3 – 2. = x 2 7
Solve a Quadratic Equation
Solve the equation. 1.
x 2 = – 3
ANSWER 3, i 3 – 2. =
x 2 7 –
ANSWER 7, i 7 – 3. =
x 2 20 –
ANSWER 5, 2 5 – i 4. =
x 2 3 2 + –
ANSWER 5, i 5 – 5. =
y 2 4 – 12
ANSWER 2, 2 – i

10. Adding and Subtracting Complex Numbers
Is a number written in the standard form a + bi where a is the real part and bi is the imaginary part. Add/Subtract the real parts, then add/subtract the imaginary parts
Complex Number
Adding and Subtracting Complex Numbers

11. Write as a complex number in standard form. ( ( 3 + 2i ( + 1 – i (
Example 2
Add Complex Numbers
Write as a complex number in standard form. ( ( 3 + 2i ( + 1 – i (
SOLUTION
Group real and
imaginary terms. 2i 3 ( + i 1 – = 2 i
Write in standard
form. = 4 + i 11

12. Write as a complex number in standard form. 2i 6 ( – 1
Example 3
Subtract Complex Numbers
Write as a complex number in standard form. 2i 6 ( – 1
SOLUTION
Group real and
imaginary terms. 2i 6 ( = 1 2 i + –
-1 + 2i
Simplify. = 5 + 0i
Write in standard
form. = 5 12

13. Write the expression as a complex number in standard form.
Checkpoint
Add and Subtract Complex Numbers
Write the expression as a complex number in standard
form. 6. ( 4 – ( 2i ( + 1 + 3i (
ANSWER i 5 + 7. i 3 ( – + 4i 2
ANSWER 3i 5 + 8. 6i 4 ( + 3i 2 –
ANSWER 3i 2 + 9. 4i 2 ( + 7i –
ANSWER 3i 4 –

14. Write the expression as a complex number in standard form.
Checkpoint
Add and Subtract Complex Numbers
Write the expression as a complex number in standard
form.
11. 2i 1 ( – + 5i 4
ANSWER 3i 5 +
12. i 2 ( – 4i 1
ANSWER 3i 3 +

15. Write the expression as a complex number in standard form.
Example 4
Multiply Complex Numbers
Write the expression as a complex number in standard form. a. 1 ( 3i + – 2i b. 3i 6 ( + 3i 4 ( –
SOLUTION
Multiply using distributive property. 1 ( 3i + – 2i =
6i 2 a. 1 ( – 2i 6 = +
Use i 6 2i – =
Write in standard form.

16. Example 4 b. 3i 6 ( + 4 – 24 18i 12i 9i 2 = 24 6i – 9i 2 = 24 6i – 1 (
Multiply Complex Numbers b. 3i 6 ( + 4 – 24
18i
12i
9i 2 =
Multiply using FOIL. 24 6i –
9i 2 =
Simplify. 24 6i – 1 ( 9 =
Use i 6i 33 – =
Write in standard form. 16

17. Complex Conjugates
Two complex numbers of the form a + bi and a - bi Their product is a real number because (3 + 2i)(3 – 2i) using FOIL 9 – 6i + 6i -4i2 9 – 4i2 i2 = -1 9 – 4(-1) = = 13 Is used to write quotient of 2 complex numbers in standard form (a + bi)

18. Write as a complex number in standard form. 2i 3 + 1 – a + bi SOLUTION
Example 5
Divide Complex Numbers
Write as a complex number in standard form. 2i 3 + 1 –
a + bi
SOLUTION 2i 3 + 1 – = •
Multiply the numerator and the denominator by i, the complex conjugate of i.
Multiply using FOIL. 1 2i 3 6i + –
4i 2 = 3 8i + 1 ( – 4 =
Simplify and use i 8i + – 1 5 =
Simplify. 5 1 – 8 i + =
Write in standard form. 18

19. Write the expression as a complex number in standard form.
Checkpoint
Multiply and Divide Complex Numbers
Write the expression as a complex number in standard form.
13. i 2 ( – 3i
ANSWER 6i 3 +
14. ( 2i 1 + i 2 –
ANSWER 3i 4 +
15. i 2 + 1 –
ANSWER 2 1 + 3 i

20. Graphing Complex Number
Imaginary axis
Real axis

21. Ex: Graph 3 – 2i
To plot, start at the origin, move 3 units to the right and 2 units down 3 2
3 – 2i

22. Ex: Name the complex number represented by the points.
Answers:
A is 1 + i
B is 0 + 2i = 2i
C is -2 – i
D is i D B A C

23. Homework
5.7 p.264 #17-20, 27/29, 33-35, 40, 43, 45, 46, 52-54, 64-71
5.6

24. Checkpoint Solve the equation. 1. x 2 = – 3 2. = x 2 7 – 3. = x 2 20 –
Solve a Quadratic Equation
Solve the equation. 1.
x 2 = – 3 2. =
x 2 7 – 3. =
x 2 20 – 4. =
x 2 3 2 + – 5. =
y 2 4 – 12

25. Write the expression as a complex number in standard form.
Checkpoint
Add and Subtract Complex Numbers
Write the expression as a complex number in standard form. 6. ( 4 – ( 2i ( + 1 + 3i ( 7. i 3 ( – + 4i 2 8. 6i 4 ( + 3i 2 – 9. 4i 2 ( + 7i –

26. Write the expression as a complex number in standard form.
Checkpoint
Add and Subtract Complex Numbers
Write the expression as a complex number in standard form.
10. 2i 1 ( – + 5i 4
11. i 2 ( – 4i 1
Write the expression as a complex number in standard form.
12. i 2 ( – 3i
14. i 2 + 1 –
13. ( 2i 1 + i 2 –