1. 4.5 Complex Numbers Objectives:
Write complex numbers in standard form.
Perform arithmetic operations on complex numbers.
Find the conjugate of a complex number.
Simplify square roots of negative numbers.
Find all solutions of polynomial equations.

2. Imaginary & Complex Numbers
The imaginary unit is defined as
Imaginary numbers can be written in the form bi where b is a real number.
A complex number is a sum of a real and imaginary number written in the form a + bi.
Any real number can be written as a complex number:
Example: 2 = 2 + 0i , −3 = −3 + 0i

3. Example #1 Equating Two Complex Numbers
Find x and y.
To solve make two equations equating the real parts and imaginary parts separately.

4. Example #2 Adding, Subtracting, & Multiplying Complex Numbers
Perform the indicated operation and write the result in the form a + bi.
Combine like terms.
Distribute the (-) and then combine like terms.

5. Example #2 Adding, Subtracting, & Multiplying Complex Numbers
Perform the indicated operation and write the result in the form a + bi.
Distribute & Simplify.
Remember:
Use FOIL, substitute and combine like terms.

6. Example #3 Products & Powers of Complex Numbers
Perform the indicated operation and write the result in the form a + bi.
Since these groups are the same but with opposite signs they are conjugates of each other. The middle terms always cancel with conjugates.

7. Powers of i
This pattern of {i, −1, −i, 1} will continue for even higher patterns.
A shortcut to evaluating higher powers requires you to memorize this pattern, but it is not necessary to evaluate them.

8. Example #4 Powers of i Find the following: Method 1:
If the exponent is odd, first “break off” an i from the original term.
Rewrite the even exponent as a power of i2 (divide it by 2).
Replace the i2 with −1.
Evaluate the power on −1. Even exponents make it positive and odd exponents keep it negative.
Multiply what is left back together.

9. Example #4 Powers of i Find the following: Method 2:
Use long division and divide the exponent by 4. Always use 4 which is the four possible values of the pattern {i, −1, −i, 1}.
The remainder represents the term number in the sequence.
For this problem the remainder is 1 which means the answer is the first term in the sequence {i, −1, −i, 1} which is i.

10. Example #4 Powers of i Find the following:
This time it isn’t necessary to “break off” any i because the exponent is already even.
If the remainder is 0 this indicates that the value is the 4th term in the sequence {i, −1, −i, 1} since you can’t have a remainder of 4 when dividing by 4.
Therefore, the answer is 1.

11. Example #4 Powers of i Find the following:
This time it is necessary to “break off” an i because the exponent is odd.
If the remainder is 3 this indicates that the value is the 3rd term in the sequence {i, −1, −i, 1}.
Therefore, the answer is −i.

12. Example #4 Powers of i Find the following:
If the remainder is 2 this indicates that the value is the 2nd term in the sequence {i, −1, −i, 1}.
Therefore, the answer is −1.

13. Example #5 Quotients of Two Complex Numbers
Express each quotient in standard form.
Multiply the top and bottom by the conjugate of the denominator, FOIL, and simplify.
Write your final answer as a complex number of the form a + bi.

14. Example #5 Quotients of Two Complex Numbers
Express each quotient in standard form.

15. Example #6 Square Roots of Negative Numbers
Write each of the following as a complex number.
After removing the i, make sure to place it out front as this can be confusing:
This time with the i off to the side there is no confusion.

16. Example #6 Square Roots of Negative Numbers
Write each of the following as a complex number.
Be sure to remove the i from each radical first!

17. Example #7 Complex Solutions to a Quadratic Equation
Find all solutions to the following:

18. Sum & Difference of Cubes

19. Example #8 Zeros of Unity
Find all solutions to the following:

20. Example #8 Zeros of Unity
Find all solutions to the following: