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5.4: Complex Numbers

5.4: Complex Numbers. Objective: Students will be able to… Solve quadratic equations with complex solutions Perform operations with complex numbers. To take the square root of a negative number we have…. The Imaginary Unit, i Definition: ***Therefore… ****.

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5.4: Complex Numbers

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  1. 5.4: Complex Numbers Objective: Students will be able to… Solve quadratic equations with complex solutions Perform operations with complex numbers

  2. To take the square root of a negative number we have…. The Imaginary Unit, i Definition: ***Therefore… ****

  3. The square root of a negative number… • If r is a positive, real # then Ex.) Ex.)

  4. Examples: Simplify

  5. Fill in the chart…notice any patterns??

  6. Based on the pattern, what would the following be?

  7. Solving a Quadratic Equation: Isolate squared expression: Take square root of both sides: Simplify using properties of radicals and imaginary numbers:

  8. Examples: 1. 2.

  9. Complex Numbers STANDARD FORM: a + bi If a = 0, b ≠ 0, a + bi is a pure imaginary number. Real part Imaginary part

  10. Complex Plane • Vertical axis: imaginary axis • Horizontal axis: real axis • Example: Graph • 2 + 3i • -1- 2i • 2 - 4i • 3i

  11. When adding or subtracting complex numbers, add or subtract their real parts and their imaginary parts separately. (its like combining like terms!!) Examples: Perform the indicated operation. • (-1 + 2i) + (3 + 3i) • (2 – 3i) – (3 – 7i) • 2i – (3 + i) + (2 – 3i)

  12. Multiplying Complex Numbers Remember…..i2 = - 1!!!! (Whenever multiplying, if you get i2, substitute -1) Examples:Perform the indicated operation. 1. 2. 3.

  13. Extra Examples

  14. Dividing Complex Numbers Complex Conjugates—Differ only in the sign a + bi, a – bi To write a quotient of 2 complex #s in standard form: Multiply numerate and denominator by complex conjugate of denominator.

  15. Examples: Write in standard form.

  16. Examples: • 2.

  17. Absolute Value of Complex Numbers z = a + bi | a + bi | = Geometrically, the absolute value of complex #s is the distance from the origin.

  18. Find the absolute value. Which number is closest to the origin in the complex plane? a.) - 2 + 5i b.) 5 – 3i

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