1. Complex Numbers
Introduction

2. What is a complex number?
A complex number is made up of a real number and an imaginary number. The standard form is a+bi where a is the real term and bi is the imaginary term.

3. Powers of i i = i2 = i3 = i4 = √-1 i  i = √-1  √-1 = -1

4. Practice Simplify i5 = i6 = i7 = i8 =
Any power of i that is a multiple of 4 is also one. We can find any power of i by writing it as the product of the highest power that is a multiple of 4 and i, i2, and i3.

5. More Practice
Simplify i25 = i36 = i11 = i19 =

6. Solve quadratic equations with complex roots

7. Solve quadratic equations with complex roots

8. Solve quadratic equations with complex roots

9. Practice
Page 369 # 17-24, o, o, o, o, 81-92

10. Operations with complex numbers
Ex. 4 Multiply or divide. Simplify. a.)  b.)  c.) d.)

11. Operations with complex numbers
Ex. 5 Write in standard form a+bi.

12. Operations with complex numbers
Addition & Subtraction
(a+bi) + (c+di)
= (a+c) + (b+d)i
(a+bi) – (c+di)
= (a+c) – (b+d)i

13. Operations with complex numbers
Ex. 6 Find each sum or difference.
a.) (4 – 5i) + (-5 + 8i) =
b.) (-6 + 3i) + (12 – 9i) =
c.) ( i) – (5 – 3i) =
d.) (15 – 8i) – ( i) + ( i) =

14. Operations with complex numbers
Multiplication
(a+bi)(c+di)
= (ac – bd) + (ad + bc)i
FOIL

15. Operations with complex numbers
Ex. 7 Find each product. a.) (5 + 3i)(2 – 7i) b.) (4 – 5i)2 c.) (3 – i)(-3 + i) d.) (9 – 8i)(9 + 8i)

16. Operations with complex numbers

17. Operations with complex numbers

18. Practice
Page 369 # , 109, 110