1. MTH 095 Intermediate Algebra Chapter 10 Complex Numbers and Quadratic Equations Section 10.1 Complex Numbers Copyright © 2011 by Ron Wallace, all rights reserved.

2. Review: Simplifying Square Roots of Positive Numbers a.Roots of “perfect” squares –Result is an integer b.Roots of multiples of perfect squares –Result is an integer multiple of a square root (i.e. simplified) c.Roots of all other positive numbers –Result is the square root (i.e. can’t be simplified)

3. Square Roots of Negatives Since the square of any number is positive, then you might concluded that the square root of a negative number is not “real” … but … Definition: Therefore … That is, in a sense, –1 is a perfect square.

4. Square Roots of Negatives So … since … for b > 0 …

5. Imaginary Numbers The set of numbers of the form … –That is, the product of a real number and i.

6. Imaginary Numbers Operations … –Addition –Subtraction –Multiplication –Division

7. Complex Numbers Real Numbers Imaginary Numbers What happens when you take a real number and an imaginary number and … Multiply? Divide? Add? Subtract?

8. Complex Numbers Real Numbers Imaginary Numbers Imaginary partReal part Complex Numbers

9. Complex Numbers Operations … –Addition

10. Complex Numbers Operations … –Subtraction

11. Complex Numbers Operations … –Multiplication

12. Complex Numbers Operations … –Division

13. Complex Numbers More examples with division …