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Manipulate real and complex numbers and solve equations

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Manipulate real and complex numbers and solve equations

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  1. Manipulate real and complex numbers and solve equations AS 91577

  2. Worksheet 1

  3. Quadratics General formula: General solution:

  4. Example 1 Equation cannot be factorised.

  5. Using quadratic formula We use the substitution A complex number

  6. The equation has 2 complex solutions Imaginary Real

  7. Equation has 2 complex solutions.

  8. Example 2

  9. Example 2

  10. Example 2

  11. Adding complex numbers Subtracting complex numbers

  12. Example

  13. Example

  14. (x + yi)(u + vi) = (xu – yv) + (xv + yu)i. Multiplying Complex Numbers

  15. Example

  16. Example

  17. Example 2

  18. Conjugate If The conjugate of z is If The conjugate of z is

  19. Dividing Complex Numbers

  20. Example

  21. Example

  22. Example

  23. Solving by matching terms Match real and imaginary Real Imaginary

  24. Solving polynomials Quadratics: 2 solutions 2 real roots 2 complex roots

  25. If coefficients are all real, imaginary roots are in conjugate pairs

  26. If coefficients are all real, imaginary roots are in conjugate pairs

  27. Cubic Cubics: 3 solutions 3 real roots 1 real and 2 complex roots

  28. Quartic Quartic: 4 solutions 2 real and 2 imaginary roots 4 real roots 4 imaginary roots

  29. Solving a cubic This cubic must have at least 1 real solutions Form the quadratic. Solve the quadratic for the other solutions x = 1, -1 - i, 1 + i

  30. Finding other solutions when you are given one solution. Because coefficients are real, roots come in conjugate pairs so Form the quadratic i.e. Form the cubic:

  31. Argand Diagram

  32. Just mark the spot with a cross

  33. Plot z = 3 + i z

  34. z = i z = -1 z =1 z = -i

  35. Multiplying a complex number by a real number.(x + yi) u = xu + yu i.

  36. Multiplying a complex number by i.zi = (x + yi) i = –y + xi.

  37. Reciprocal of z Conjugate

  38. Rectangular to polar form Using Pythagoras Modulus is the length Argument is the angle Check the quadrant of the complex number

  39. Modulus is the length

  40. Example 1 Rectangular form Polar form

  41. Example 2

  42. Example 3

  43. Converting from polar to rectangular

  44. Multiplying numbers in polar form Example 1

  45. Multiplying numbers in polar form Example 2 Take out multiples of