1. 4.8 COMPLEX NUMBERS
Part 1: Introduction to Complex and Imaginary
Numbers

2. REAL NUMBERS
See Page 12 in Textbook

3. COMPLEX NUMBERS
 The set of Real Numbers is a subset of a larger set
of numbers called Complex Numbers
 The complex numbers are based on a number whose
square is –1
 The imaginary unit i is the complex number whose
square is –1 .



4. SQUARE ROOT OF A NEGATIVE REAL NUMBER
 For any real number a,

5. EXAMPLE: SIMPLIFY EACH NUMBER BY USING
THE IMAGINARY NUMBER

6. EXAMPLE: SIMPLIFY EACH NUMBER BY USING
THE IMAGINARY NUMBER

7. EXAMPLE: SIMPLIFY EACH NUMBER BY USING
THE IMAGINARY NUMBER

8. IMAGINARY NUMBERS
 An imaginary number is any number of the form a
+ bi, where a and b are real numbers and b ≠ 0.
a + bi
↑ ↑
Real Imaginary
Part Part
 If b = 0, then the number is a real number
 If a = 0 and b ≠ 0, then the number is a pure
imaginary number

9. COMPLEX NUMBERS
 Imaginary numbers and real numbers make up the
set of complex numbers

10. THE QUADRATIC FORMULA
 Every quadratic equation has complex number
solutions (that are sometimes real numbers).
 We can use and the quadratic formula to
solve all quadratic equations.

11. FIND ALL SOLUTIONS TO EACH QUADRATIC
EQUATION

12. FIND ALL SOLUTIONS TO EACH QUADRATIC
EQUATION

13. GRAPHING COMPLEX NUMBERS
 In the complex number plane,
 The x – axis represents the real part
 The y – axis represents the imaginary part
 The point (a, b) represents the complex number a + bi
 The absolute value of a complex number is its
distance from the origin in the complex plane.

14. EXAMPLE: WHAT ARE THE GRAPH AND
ABSOLUTE VALUE OF EACH NUMBER?

15. EXAMPLE: WHAT ARE THE GRAPH AND
ABSOLUTE VALUE OF EACH NUMBER?

16. HOMEWORK
 P253 #1, 2, 8 – 17 all, 39 – 44 all