1. 1 What you will learn  Lots of vocabulary!  A new type of number!  How to add, subtract and multiply this new type of number  How to graph this new type of number  How to simplify expressions involving this new type of number

2. Objective: 5.4 Complex Numbers 2 Lions, Tigers and Complex Numbers…oh my! What is: Mathematicians created an “imaginary unit” to help us do math with negative square roots. i = i is called the imaginary unit.

3. Objective: 5.4 Complex Numbers 3 Simplifying with “i” Example: How do we simplify What is i 2 ? Any guesses for 5i 2 ?

4. Objective: 5.4 Complex Numbers 4 You Try!  Simplify: 1. 2. 3. 12i 2

5. Objective: 5.4 Complex Numbers 5 Solving a Quadratic Equation  Solve 3x 2 + 10 = -26

6. Objective: 5.4 Complex Numbers 6 You Try!  Solve 2x 2 + 26 = -10

7. Objective: 5.4 Complex Numbers 7 More Vocabulary  We are adding to our “types” of numbers. We had whole, integer, rational, irrational and real.  We are adding “complex” numbers.  A complex number is of the form a + bi where a and b are real numbers. This is called standard form.  “a” is the real part.  “bi” is the imaginary part.  If you have both an “a” term and a “bi” term, the number is an imaginary number.  If you have only a “bi” term, the number is a pure imaginary number.

8. Objective: 5.4 Complex Numbers 8 “Types” of Numbers – Expanded! Whole 1, 2, 3 Integer -3, -2, 0 Rational ¾, ½, -1/2 Real Numbers Irrational Imaginary Numbers 2+3i, 5-5i Pure Imaginary Numbers -4i, 6i Complex Numbers

9. Objective: 5.4 Complex Numbers 9 Graphing Complex Numbers  We graph complex numbers in the “complex plane”. Real axis Imaginary axis

10. Objective: 5.4 Complex Numbers 10 Graphing continued…  Plot the complex numbers in the complex plane. 2 – 3i-3 + 2i4i Real axis Imaginary axis 2i 4i -2i -4i 24 -4

11. Objective: 5.4 Complex Numbers 11 Adding and Subtracting Complex Numbers  Like adding and subtracting “like terms” 1. ( 4 – i) + (3 + 2i) 2. (7 – 5i) – (1 – 5i) 3. 6 – ( -2 + 9i) + (-8 + 4i)

12. Objective: 5.4 Complex Numbers 12 You Try!  Write the expression as a complex number in standard form. 1. ( -1 + 2i ) + (3 + 3i) 2. (2 – 3i) – (3 – 7i) 3. 2i – (3 + i) + (2 – 3i)

13. Objective: 5.4 Complex Numbers 13 Multiplying Complex Numbers  You will use the distributive property or FOIL. Write the expression as a complex number in standard form: 1. 5i(-2 + i) 2. (7 – 4i)(-1 + 2i)

14. Objective: 5.4 Complex Numbers 14 You Try Write the expression as a complex number in standard form: 1. -3i(3 + 2i) 2. (3 + 2i)(5 - 2i)

15. Objective: 5.4 Complex Numbers 15 Complex Conjugates  What happens when we multiply: (6 + 3i)(6 – 3i) (6 + 3i)(6 – 3i) are called complex conjugates.

16. Objective: 5.4 Complex Numbers 16 Simplifying “Division” Problems  We can use complex conjugates to simplify complex numbers of the form:

17. Objective: 5.4 Complex Numbers 17 You Try  Write the quotient in standard form.

18. Objective: 5.4 Complex Numbers 18 Homework  page 277, 18-34 even, 38-42 even, 48-52 even, 56-60 even, 80-85 all