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9.3 Complex Numbers; The Complex Plane; Polar Form of Complex Numbers

9.3 Complex Numbers; The Complex Plane; Polar Form of Complex Numbers. In a complex number. a is the real part and bi is the imaginary part. When b=0, the complex number is a real number. When a 0 , and b 0, as in 5+8i, the complex number is an imaginary number.

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9.3 Complex Numbers; The Complex Plane; Polar Form of Complex Numbers

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  1. 9.3Complex Numbers; The Complex Plane;Polar Form of Complex Numbers

  2. In a complex number a is the real part and bi is the imaginary part. When b=0, the complex number is a real number. When a0, and b0, as in 5+8i, the complex number is an imaginary number. When a=0, and b0, as in 5i, the complex number is a pure imaginary number.

  3. The Complex Plane Imaginary Axis Real Axis O

  4. Let be a complex number. The magnitude or modulus of z, denoted by is defined As the distance from the origin to the point (x, y).

  5. Imaginary Axis y |z| Real Axis O x

  6. is sometimes abbreviated as

  7. Imaginary Axis 4 z =-3 + 4i Real Axis -3

  8. z = -3 + 4i is in Quadrant II x = -3 and y = 4

  9. z =-3 + 4i 4 Find the reference angle () by solving -3

  10. z =-3 + 4i 4 -3

  11. Find r:

  12. Imaginary Axis 4 Real Axis -3

  13. Find the reference angle () by solving

  14. Find the cosine of 330 and substitute the value. Find the sine of 330 and substitute the value. Distribute the r

  15. Write in standard (rectangular) form.

  16. Lesson Overview 9-7A

  17. Product Theorem

  18. Lesson Overview 9-7B

  19. Quotient Theorem

  20. 5-Minute Check Lesson 9-8A

  21. 5-Minute Check Lesson 9-8B

  22. Powers and Roots of Complex Numbers

  23. DeMoivre’s Theorem

  24. What if you wanted to perform the operation below?

  25. Lesson Overview 9-8A

  26. Lesson Overview 9-8B

  27. Theorem Finding Complex Roots roots

  28. Find the complex fifth roots of The five complex roots are: for k = 0, 1, 2, 3,4 .

  29. The roots of a complex number a cyclical in nature. That is, when the points are plotted on a polar plane or a complex plane, the points are evenly spaced around the origin

  30. Complex Plane

  31. Polar plane

  32. Polar plane

  33. To find the principle root, use DeMoivre’s theorem using rational exponents. That is, to find the principle pth root of Raise it to the power

  34. Example You may assume it is the principle root you are seeking unless specifically stated otherwise. Find First express as a complex number in standard form. Then change to polar form

  35. Since we are looking for the cube root, use DeMoivre’s Theorem and raise it to the power

  36. Example: Find the 4th root of Change to polar form Apply DeMoivre’s Theorem

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