1. Operations with Complex Numbers
Unit 1 Lesson 2

2. Make Copies of:
Comparing Polynomials and Complex Numbers Graphic Organizer
Kuta-Operations with Complex Numbers WS

3. GPS Standard MM2N1b- Write complex numbers in the form a + bi
MM2N1c- Add, subtract, multiply, and divide complex numbers
MM2N1d- Simplify expressions involving complex numbers

4. Essential Questions How do I add and subtract complex numbers?
How do I multiply complex numbers?

5. COMPLEX NUMBERS Imaginary Real Numbers Numbers Rational Numbers
Irrational
Numbers

6. Standard Form of a Complex Number
a + bi
IMAGINARY
PART
REAL PART

7. Adding/Subtracting Complex Numbers
Adding and subtracting complex numbers is just like any adding/subtracting you have ever done with variables.
Simply combine like terms.
(6 + 8i) + (2 – 12i) = 8 – 4i
(7 + 4i) – (10 + 9i) = 7 + 4i – 10 – 9i = – 5i
When adding or subtracting, treat the i as if it were any other variable.
Make certain answers are always given in a + bi form. Never put the imaginary part in front of the real part.

8. To Add Complex Numbers (a + bi) + (a + bi) Drop the parentheses
Combine like terms
Remember: the real number comes first, then the imaginary number

9. Examples (3 + 5i) + (2 – 7i) (12 – 3i) + (2 + 4i) (13 +24i) + (17+ 5i)

10. Test Prep Example
What is (5 – 2i) + (6 + 4i)?
-3i 3i
11 + 2i
11 + 6i

11. Test Prep Example
Perform the indicated operation (2 + 3i) +(13 – 2i) =
A) i B) 15 + i C) 11 – 5i D) -11 – i

12. To Subtract Complex Numbers:
(a + bi) – (a + bi)
Change the minus sign to plus
Change the sign of each term in the second set of parentheses
Drop parentheses
Combine like terms
Remember: real number comes first, then imaginary number

13. Examples (6 + 7i) – (4 + 3i) (8 + 2i) – (3 – 7i) (12 – 7i) – (2 + 6i)

14. Test Prep Example Perform the indicated operation.
(-9 + 2i) – ( i) =
-21 – 6i
-3 + 6i
3 – 2i
21 + 2i

15. Multiplying Complex Numbers
This will be FOIL method with a slight twist at the end.
An i2 will ALWAYS show up. You will have to adjust for this.
(4 + 9i)(2 + 3i) = i + 18i + 27i2 = i – 27 = i
(7 – 3i)(6 + 8i) = i – 18i – 24i2 = i + 24 = i
Multiplication with complex numbers is just like any other multiplication with binomials. You FOIL the same way. You SQUARE or do the FIVE-STEP SHORTCUT the same way. The only difference is the having to adjust the i2 out of the problem.
(Don’t forget that you can never leave a power of i greater than the first power in any answer.)
The net effect of the i2 showing up is that it changes the sign of the last term when simplified. Watch the examples we do.
+6i2 will yield: – 6.
-3i2 will yield: + 3.

16. Examples

17. Test Prep Examples 1. (5 – 3i)(6 + 2i) Multiply and simplify.
A) 24 – 8i
B) 36 – 8i
C) i
D) i

18. Binomial Squares and Complex Numbers
You can still do the five-step shortcut, or you can continue to do FOIL.
You will still have to adjust for the i2 that will show up.
(7 + 3i)2 = i + 9i2 = i – 9 = i
(8 – 9i)2 = 64 – 144i + 81i2 = – 144i – 81= -17 – 144i
Don’t forget the five-step shortcut:
1.) Square the first term.
2.) Put the sign from the binomial.
3.) Take twice the product of the first and last terms.
4.) Put a plus sign.
5.) Square the last term.
The only thing that is new is that there will be an i2 on the last term that must be adjusted out of the problem. See the notes for the previous slide.

19. Example

20. Test Prep Example Which has the same value as (4 + 3i)2 ? 7 7 + 24i 25

21. D2S and Complex Numbers
Situations that in the real numbers would have been differences of two squares (D2S) demonstrate in the complex numbers what are known as conjugates.
(3 + 4i)(3 – 4i) = (3)2 – (4i)2 = 9 – 16i2 = = 25
When conjugates are used, there will be no i in the answer.

22. Examples

23. Test Prep Example 2.) Perform the indicated operation.
(4 – 7i)(4 + 7i) =
-33
16 – 49i
16 – 105i 65

24. Test Prep Example What is the square of 4 – 7i? A) 33 – 56i
B) -33 – 56i
C) i
D) i

25. Test Prep Example Which is equivalent to (3 + 2i)(2 + 5i)? A) -4 + 19i
B) i
C) i

26. Test Prep Example What is a if a + bi = (2 – i)2 A) a = 3 B) a = 5
C) a = 2
D) a = 1

27. Test Prep Example Simplify: -10 + √-16 2 A) -5 + 2i B) -5 – 4i
C) i
D) i

28. Test Prep Example Perform the indicated operation. (3 – 8i)(4 + i) =
A) 4
B) 20 – 29i
C) 12 – 8i
D) i

29. Test Prep Example Multiply 2i(i – 2) over the set of complex numbers.
B) 2 – 4i
C) -2 – 4i
D) 2 + 4i

30. Graphic Organizer
Comparing Polynomials and Complex Numbers.doc

31. Assignment
Kuta-Operations with Complex Numbers.pdf

32. Support Assignment
Pg 8: 1-27
Pg. 13: