1. 5.4 Review of Imaginary Numbers
Algebra 2

2. Learning Targets I can simplify radicals containing negative radicands
I can multiply pure imaginary numbers, and
I can solve quadratic equations that have pure imaginary solutions

3. And now a word from our local historian . . .
Until the 16th century, mathematicians were puzzled by square roots of negative numbers. As you know, some expression have irrational solutions. For example, the solutions to x2 – 5 = 0 are 5 and - 5 . But the equation x2 = -1 has no solution in the real numbers.

4. And now a word from our local historian . . .
This is because the square of a real number is nonnegative. However in 1545, the Italian mathematician Girolamo Cardano published Ars Magna in which he began working with what great mathematician René Descartes later called IMAGINARY NUMBERS.

5. Okay, so now we aren’t in the real world?
The number i is defined to be a solution to x2 = -1 and is NOT a real number. It is called the IMAGINARY UNIT. Using i as you would any constant, you can define square roots of negative numbers.

6. Since i = -1, it follows that i2 = -1
Oh yeah so . . .
Exactly,
To avoid 2(i) being read as 2i, write 2(i) as i2

7. Definition of a pure imaginary number
For any positive, real number b,
Where i is the imaginary unit, and bi is called a pure imaginary number.

8. Ex. 1a: Simplify

9. Ex. 1b: Simplify
-bi is also a pure imaginary number since -bi = -1(bi)

10. Ex. 2a: Simplify
Reminder: i2 = -1

11. Ex. 2b: Simplify
Reminder: i2 = -1

12. Okay, so you memorized squares, then you memorized cubes
Okay, so you memorized squares, then you memorized cubes. Time to memorize successive powers of i. See a pattern?!?!?!
NOW DO YOU SEE A PATTERN?

13. Ex. 3: Simplify i13 Method 2: i13= i12  i = (i2)6  i = (-1)6  i = ior

14. Ex. 4: Solve x2 +7 = 0

15. Ex. 38: Solve