Complex Numbers Skill 12 Objectives Use the imaginary

Complex Numbers Skill #12

Objectives… • • Use the imaginary unit i to write complex numbers. Add, subtract, and multiply complex numbers. Use complex conjugates to write the quotient of two complex numbers in standard form. Find complex solutions of quadratic equations.

The Imaginary Unit i The imaginary unit i, defined as where i 2 = – 1. When adding real numbers to real multiples of this imaginary unit, the result is a set of complex numbers.

Complex numbers can be written in the standard form, a + bi. *Standard form of is…

Definition of a Complex Number

To add (or subtract) two complex numbers, you add (or subtract) the real and imaginary parts of the numbers separately. *Combine like terms.

Example–Adding and Subtracting Complex Numbers

Example–Solution

Example–Solution

Many of the properties of real numbers are valid for complex numbers as well. Associative Properties Commutative Properties Distributive Property

Example– Multiplying Complex Numbers Perform the operation(s) and write the result in standard form. a. 5(– 2 + 3 i ) b. (2 – i )(4 + 3 i ) c. (3 + 2 i )(3 – 2 i ) d. 4 i(– 1 + 5 i ) e. (3 + 2 i )2

Example–Solution

Example–Solution

Example 2 – Solution

Complex Conjugates

Example–Multiplying Conjugates Multiply each complex number by its complex conjugate. a. (1 + i ) b. 3 – 5 i

Example–Solution

Complex Conjugates To write the quotient of a + bi and c + di in standard form, where c and d are not both zero, multiply the numerator and denominator by the complex conjugate of the denominator to obtain Multiply numerator and denominator by complex conjugate of denominator. Standard form

Example–Writing a Quotient of Complex Numbers in Standard Form Write the quotient Solution: in standard form. Multiply numerator and denominator by complex conjugate of denominator.

Complex Solutions of Quadratic Equations

Example–Writing Complex Numbers in Standard Form a. b.

Example–Complex Solutions of a Quadratic Equation Solve (a) x 2 + 4 = 0 Solution: a. x 2 + 4 = 0 x 2 = – 4 x = 2 i

Example–Solution b. 3 x 2 – 2 x + 5 = 0 Quadratic Formula. Write in standard form.

Skill #12: Complex Numbers � Summarize � Questions? � Homework ◦ Worksheet � Quiz Notes