7.9 Complex Numbers

• Square roots of numbers are called numbers.

• Unit Imaginary Number• i is a number whose square is -1. That is,

• In other words, . i2 1

i 1

negativeimaginary

1. Find the pattern. • a. d.

• b. e. • c. f.

• d. g.

• Pattern?

i1

2i

3i

4i

5i

6i

7i

8i

• So, to calculate any high power of i, you can convert it to a lower power by taking the closest multiple of 4 that's no bigger than the exponent and subtracting this multiple from the exponent.

• Example: i99 i963 i 424 3 i3 i

Simplify.

2.

3.

4.

i17

i120

i64,002

Theorem• For every positive real number r,

r i r

Express in simplest form as a real number or a pure imaginary number.

• 5.

• 6.

• 7.

50

3 75

9

7

Express each of the following in simplest form.

8.

9.

10.

3 75

2 12 5 6

49 12

Express each of the following in simplest form.

11.

12.

3 7 4 175

300 192

• Remember, i is still a square root. So, we cannot have i left in the and we must rationalize!

denominator

Express each of the following in simplest form.

• 13.

• 14.

6

i3

5

2i

Express each of the following in simplest form.

• 15.

• 16.

7 63 4i

7

6