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A.6a Complex Numbers

A.6a Complex Numbers. A.6 Complex Numbers (continued). Objectives: Complex Numbers – Why? Definition of the imaginary unit Square Root of a Negative Number Adding/Subtracting Complex Numbers Multiplying Complex Numbers Complex Conjugate Product of a number and its conjugate

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A.6a Complex Numbers

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  1. A.6a Complex Numbers

  2. A.6 Complex Numbers (continued) • Objectives: • Complex Numbers – Why? • Definition of the imaginary unit • Square Root of a Negative Number • Adding/Subtracting Complex Numbers • Multiplying Complex Numbers • Complex Conjugate • Product of a number and its conjugate • Dividing Complex Numbers • Higher Powers • Simplifying complex exponents • Solving equations in the complex numbers • a) Determine the character of the solutions to a quadratic

  3. Complex Numbers Why Complex numbers ? Can you solve : x2 + 1 = 0 OR EXPAND We must our way of thinking about numbers!

  4. 2. Definition Complex Numbers Complex numbers are numbers of the form where a is called the real part and b is the imaginary part Now, we have a solution to It’s just . Cool huh?

  5. 3. Add/Subtract Complex Numbers Always write your solutions in standard form: Add complex numbers, add the real parts and add the imaginary parts together Subtract complex numbers

  6. 4. Multiply Complex Numbers Multiply complex numbers, follow the usual rules for multiplying two binomials. (think FOIL or Double Distribution!)

  7. Complex Numbers Given the following complex numbers, perform the indicated operations. 1) 2) 3) http://www.uncwil.edu/courses/mat111hb/Izs/complex/complex.html

  8. 5. Conjugate of Complex Numbers If is a complex number, then its conjugate, denoted by , is defined as

  9. Complex Numbers Let’s find the conjugate of the following complex numbers and then multiply by the conjugate. 1) 2) 3)

  10. 6. Divide Complex Numbers We can apply multiplying conjugates to quotients of complex numbers. Let’s write the following in standard form!

  11. Complex Numbers Given the following complex numbers, perform the indicated operations. 1) 2) 3) 4)

  12. 7. Solving Equations in theComplex Number System Example 1: Solve the following equation in the complex number system.

  13. Example 2. Solve the following equation in the complex number system

  14. Example 3. Solve the following equation in the complex number system

  15. The number of solutions to a polynomial equation… What do you notice about the degree of polynomial and the number of solutions to the polynomial? In the complex plane, the number of solutions to a polynomial equals the degree!

  16. 7. a) Determine the character of solutions to a quadratic equation The number of solutions is determined by the discriminant: if then there are two unequal real solutions if then there is a repeated real solution if then there are two complex solutions that (not real) Complex solutions are conjugates of each other.

  17. Examples: Determine the Character of Solutions for each quadratic Ex. Ex. Ex. the equation has two unequal real solutions the equation has a repeated real solution the equation has two complex solutions that are not real and are conjugates of each other

  18. Practice. Now you try these! Solve the following in the complex system. 1) 2) 3) 4)

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