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Complex Numbers

Complex Numbers. Stephanie Golmon Principia College Laser Teaching Center, Stony Brook University June 27, 2006. Vectors. Vectors have both a magnitude and a direction. Magnitude = 20 mi Direction = 60˚ (angle of rotation from the east).

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Complex Numbers

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  1. Complex Numbers Stephanie Golmon Principia College Laser Teaching Center, Stony Brook University June 27, 2006

  2. Vectors • Vectors have both a magnitude and a direction. • Magnitude = 20 mi • Direction = 60˚ (angle of rotation from the east) *http://www.glenbrook.k12.il.us/GBSSCI/PHYS/Class/vectors/u3l1a.html

  3. Trigonometry • The black vector is the sum of the two red vectors D = 20 mi 20 Sin (60˚) 17.32 mi 60˚ 20 Cos (60˚) 10 mi

  4. 1 0.5 -1 -0.5 0.5 1 -0.5 -1 The Unit Circle R=1 60˚

  5. Radians vs. Degrees • radian: an angle of one radian on the unit circle produces an arc with arc length 1. • 2π radians = 360˚ • Example: 60˚= π/3 radians *http://mathworld.wolfram.com/Radian.html

  6. *http://www.math2.org/math/graphs/unitcircle.gif

  7. *http://members.aol.com/williamgunther/math/ref/unitcircle.gif*http://members.aol.com/williamgunther/math/ref/unitcircle.gif

  8. 1 0.5 -1 -0.5 0.5 1 -0.5 -1 Cosine, Sine, and the Unit Circle Cos(t) Sin(t)

  9. Imaginary Numbers

  10. Complex Numbers • Have both a real and imaginary part • General form: z = x +iy Z = 5 + 3i Real part Imaginary part

  11. Complex Plane Imaginary axis Real axis *http://www.uncwil.edu/courses/mat111hb/Izs/complex/cplane.gif

  12. Vectors in the complex plane z=x+iy • A point z=x+iy can be seen as the sum of two vectors • x=Cos(θ) • y=Sin(θ) • Z=Cos(θ) + i Sin(θ) R=1 i Sin(θ) θ Cos(θ)

  13. Euler’s Formula • Describes any point on the unit circle • θis measured counterclockwise from the positive x, axis

  14. Proof of Euler’s Formula

  15. Proof of Euler’s Formula

  16. Polar Coordinates • Of the form • r is the distance to the point from the origin, called the modulus • θis the angle, called the argument r= 20 θ= π/3

  17. Polar vs. Cartesian Coordinates • Any point in the complex plane can be written in polar coordinates ( ) or in Cartesian coordinates (x+iy) • how to convert between them:

  18. Multiplying Complex Numbers • (5+5i)(-3+3i)= ? -30 • ( )( )= ?

  19. Dividing Complex Numbers • (5+5i)/(-3+3i)= ? • ( )/( )= ?

  20. Roots of Complex Numbers

  21. The Most Beautiful Equation:

  22. Describing Waves is the amplitude is the phase is the phase constant *derivation from: Introduction to Electrodynamics, Third Edition. David J. Griffiths. Upper Saddle River, New Jersey: Prentice Hall, 1999.

  23. Continued… one period frequency angular frequency in complex notation:

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