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Complex Numbers1

Complex Numbers1. Who, what, where, why and how. Quadratic Equations. Girolamo Cardano, a contemporary of Leonardo DaVinci. ‘divide 10 into two parts such that the product is 40.’ x and 10 – x x (10 – x ) = 40. x 2 - 10 x + 40=0. ‘truly sophisticated’

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Complex Numbers1

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  1. Complex Numbers1 Who, what, where, why and how.

  2. Quadratic Equations • Girolamo Cardano, a contemporary of Leonardo DaVinci. • ‘divide 10 into two parts such that the product is 40.’ • x and 10 –x • x(10 – x) = 40

  3. x2 - 10x + 40=0 ‘truly sophisticated’ ‘as subtle as it would be useless’

  4. Roots of equations • What is meant by the roots of an equation? • An equation with roots -2 and 3 • (x - 3)(x + 2) = 0 • x2 - x - 6 = 0 • The roots of x2 + 3x + 5 = 0

  5. Imaginary Numbers

  6. Alternating systems

  7. Side view

  8. Alternating systems

  9. Powers of j

  10. Negative powers

  11. Complex Numbers • Have a real and imaginary part • For 3 + 4j • Real part Re(3 + 4j) = 3 • Imaginary part Im(3 + 4j) = 4 • If z = 5 - 2j • Re(z) = 5, Im(z) = -2

  12. The conjugate of a complex number • If z = -2 + 5j • z* = -2 - 5j • Only the imaginary part is in the opposite direction

  13. Addition and subtraction

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