1. Section 2.4 Complex Numbers

2. What you should learn
How to use the imaginary unit i to write complex numbers
How to add, subtract, and multiply complex numbers
How to use complex conjugates to write the quotient of two complex numbers in standard form
How to find complex solutions to quadratic equations

3. {1, 2, 3, 4,…} How many natural numbers are there? Real Number System

4. {0, 1, 2, 3, 4,…} Real Number System How many whole numbers are there?
Natural
Whole

5. How many integers numbers are there?
Real Number System
Natural
{...-3, -2, -1, 0, 1, 2, 3, …}
How many integers numbers are there?
Whole
Integers

6. Real Number System Fractions How many rational numbers are there?
Natural
Whole
Integers
Rational

7. Real Number System How many irrational numbers are there? Natural
Whole
Integers
Irrational
Rational

8. Real Number System Each set is a subset of the Real Number System.
Natural
Each set is a subset of the Real Number System.
The union of all these sets forms the real number system.
The number line is our model for the real number system.
Whole
Integers
Irrational
Rational
Real
Numbers

9. Definition of Square Root
If a2 = n then a is a square root of n.
42 = (4)(4) = 16
 4 is a square root of 16
(-4)2 = (-4)(-4) = 16
 -4 is a square root of 16

10. Whatever it is it is not on the real number line.
What square root of -16?
Whatever it is it is not on the real number line.

11. Definition of i
The number i is such that
Imaginary Unit

12. Complex Numbers
REAL
Imaginary
Complex

13. Definition of a Complex Number
If a and b are real numbers, the number a + bi is a complex number, and it is said to be written in standard form.
If b = 0 then the number a + bi = a is a real number.
If b ≠ 0, then the number a + bi is called an imaginary number.
A number of the form bi, where b ≠ 0 is called a pure imaginary number.

14. Examples

15. If you square a radical you get the radicand2 2
Whenever you have i2 the next turn you will have -1 and no i.

16. Equality of Complex numbers
If a + bi = c + di, then a = c and b = d.

17. Is a negative times a negative always positive?
Trick question. This is not a negative times a negative.

18. Example

19. Example

20. Example

21. Example
Cancel the i
factor

22. Add
Collect like terms.

23. First distribute the negative sign.
Subtract
Now collect like terms.

24. Multiplication F O I L

25. Simplify each expression. Express your answer in form.
F-O-I-L
Recall
i2=-1
Combine like terms.
Combine like terms.

26. Multiply by the conjugate factor.
Write in the form 2
Multiply by the conjugate factor.

27. Powers of i Anything other than 0 raised to the 0 is 1.
Anything raised to the 1 is itself.

28. Simplify as much as possible.

29. Use the Quadratic Formula

30. Homework Section , 83 odd