1. Imaginary Numbers
though they have real world applications!

2. Imaginary Numbers

3. Imaginary Numbers Imaginary numbers are not “make believe”.
Imaginary numbers have practical “real world” applications, such as electricity and fluid dynamics.
For many years people didn’t believe the solution to an equation could be less than zero. Today we call those numbers negative.

4. Imaginary Numbers
imaginary unit

5. -The imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i² = -1.
-The first four powers of i establish an important pattern and should be memorized.
Powers of i 5

6. Simplify .
Answer:
Example 9-1a

7. Simplify. a.
Answer:
Example 9-1c

8. Simplify .
Answer: = 6
Example 9-2a

9. Simplify .
Answer:
Example 9-2b

10. Simplify. a. b.
Answer: –15
Answer:
Example 9-2c

11. Simplify
Multiplying powers
Power of a Power
Answer:
Example 9-3a

12. Simplify .
Answer: i
Example 9-3b

13. Subtract 20 from each side.
Solve
Original equation
Subtract 20 from each side.
Divide each side by 5.
Take the square root of each side.
Answer:
Example 9-4a

14. Solve
Answer:
Example 9-4b

15. Complex Numbers
If a and b are real numbers, the number a + bi is a complex number, and it is said to be written in standard form. If b = 0, the number a + bi = a is a real number.
If b
0, the number a + bi is called
the imaginary number.
A number of the form bi, where b 0
is called a pure imaginary.

16. Commutative and Associative Properties
Simplify .
Commutative and Associative Properties
Answer:
Example 9-6a

17. Commutative and Associative Properties
Simplify .
Commutative and Associative Properties
Answer:
Example 9-6b

18. Simplify. a. b.
Answer:
Answer:
Example 9-6c

19. Conjugates
In order to find the conjugate of an expression simply change the sign between the two terms.
The point of multiplying by a conjugate is to get rid of the middle terms. (You end up with the difference of two squares)

20. COMPLEX CONJUGATES AND DIVISION.
***Remember, just like we can’t leave radicals in the denominator,
we can’t leave imaginary numbers in the denominator either. So, we
need to multiply both the numerator AND the denominator by the
conjugate of the denominator in order to get rid of all imaginary
numbers in the denominator.
WHAT ARE THE CONJUGATES???? 20

21. Simplify . and are conjugates. Multiply. Answer: Standard form
Example 9-8a

22. Simplify .
Multiply by
Multiply.
Answer:
Standard form
Example 9-8b

23. Simplify. a. b.
Answer:
Answer:
Example 9-8c