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Complex Numbers

Complex Numbers. Lesson 3.3. It's any number you can imagine. The Imaginary Number i. By definition Consider powers if i. Using i. Now we can handle quantities that occasionally show up in mathematical solutions What about. Imaginary part. Real part. Complex Numbers.

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Complex Numbers

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  1. Complex Numbers Lesson 3.3

  2. It's any number you can imagine The Imaginary Number i • By definition • Consider powers if i

  3. Using i • Now we can handle quantities that occasionally show up in mathematical solutions • What about

  4. Imaginary part Real part Complex Numbers • Combine real numbers with imaginary numbers • a + bi • Examples

  5. Try It Out • Write these complex numbers in standard form a + bi

  6. Operations on Complex Numbers • Complex numbers can be combined with • addition • subtraction • multiplication • division • Consider

  7. Operations on Complex Numbers • Division technique • Multiply numerator and denominator by the conjugate of the denominator

  8. Complex Numbers on the Calculator • Possible result • Reset modeComplex formatto Rectangular • Now calculator does desired result

  9. Complex Numbers on the Calculator • Operations with complex on calculator Make sure to use the correct character for i. Use 2nd-i

  10. Warning • Consider • It is tempting to combine them • The multiplicative property of radicals only works for positive values under the radical sign • Instead use imaginary numbers

  11. Try It Out • Use the correct principles to simplify the following:

  12. Complex roots The Discriminant Return of the • Consider the expression under the radical in the quadratic formula • This is known as the discriminant • What happens when it is • Positive and a perfect square? • Positive and not a perfect square? • Zero • Negative ?

  13. Example • Consider the solution to • Note the graph • No intersectionswith x-axis • Using the solve andcsolvefunctions

  14. Fundamental Theorem of Algebra • A polynomial f(x) of degree n ≥ 1 has at least one complex zero • Remember that complex includes reals • Number of Zeros theorem • A polynomial of degree n has at most n distinct zeros • Explain how theorems apply to these graphs

  15. Conjugate Zeroes Theorem • Given a polynomial with real coefficients • If a + biis a zero, then a – bi will also be a zero

  16. Assignment • Lesson 3.3 • Page 211 • Exercises 1 – 78 EOO

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