4.5 Complex Numbers

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

4. Example #2Adding, 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 #2Adding, 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 #3Products & 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 #4Powers of i Method 1: • Find the following: 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 #4Powers of i Method 2: • Find the following: 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 #4Powers 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 #4Powers 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 #4Powers 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 #5Quotients 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 #5Quotients of Two Complex Numbers • Express each quotient in standard form.

15. Example #6Square 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 #6Square 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 #7Complex Solutions to a Quadratic Equation • Find all solutions to the following:

18. Sum & Difference of Cubes

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

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