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.

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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!

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

Sum & Difference of Cubes

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

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