Algebra 2 Notes, Chapter 4 1

4.8 Complex Numbers

A. Imaginary Numbers

Consider: 25 Has no REAL solution because (?)2= -25

Imaginary Numbers were created to solve equations that did not have real solutions.

The imaginary unit “i” is used to represent: 1i and 2 1i Ex. Simplify the following

1. 36 2. 100 3. 7 4. 50

5. 90 6. 3 8 7. 2 6 8. 3 10 4 3

9. 2 3i i 10. 7 3 2i i i 11. 2

5i 12. 3

3i

Numbers like 3i , 97i , and 7i are called PURE IMAGINARY NUMBERS.

Algebra 2 Notes, Chapter 4 2

B: COMPLEX NUMBERS A COMPLEX NUMBER has a real part “a” and an imaginary part “bi”.

A complex number looks like: a bi Working with complex numbers: Treat i like any other variable… 1. (3 4 ) (7 5 )i i 2. (9 8 ) ( 7 10 )i i 3. (7 6 ) (6 10 )i i

4. (17 20 ) (12 30 )i i 5. (5 3 )(4 6 )i i 6. (8 7 )(2 5 )i i

7. 7 (9 3 )i i 8. 4 (20 30 )i i 9. 2(3 5 )i

10. (4 2 )(4 2 )i i 11. (6 5 )(6 5 )i i Numbers in the form

a bi and a bi are called complex conjugates.

Algebra 2 Notes, Chapter 4 3 Division is a little trickier— To simplify a division problem (a fraction with complex numbers), multiply the numerator and denominator by the COMPLEX CONJUGATE of the denominator.

12. 4

5i 13.

5 3

2

i

i

14. 7

3 5i 15.

11

3 2

i

i

16. 2 7

3 4

i

i

17.

1 5

5 9

i

i

Algebra 2 Notes, Chapter 4 4

C: Solving Equations Remember that a quadratic equation with no real solutions has a graph that has no x-intercepts. This type of quadratic equation does have SOLUTIONS, though… COMPLEX SOLUTIONS. SOLVE!

1. 2 9 0x 2. 25 20 0x 3. 2 15 0x

4. 23 2 0x x

5. 2 4 5 0x x

Quadratic Formula for 2 0ax bx c

2 4

2

b b acx

a