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) *

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

60˚ The Unit Circle R=1

Radians vs. Degrees radian: an angle of one radian on the unit circle produces an arc with arc length 1. 2 π radians = 360˚ Example: 60˚= π /3 radians *

*

*

Cosine, Sine, and the Unit Circle Cos(t) Sin(t)

Imaginary Numbers

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

Complex Plane Real axis Imaginary axis *

Vectors in the complex plane 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 ) θ R=1 i Sin( θ) Cos( θ) z=x+iy

Euler’s Formula Describes any point on the unit circle θ is measured counterclockwise from the positive x, axis

Proof of Euler’s Formula

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

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:

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

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

Roots of Complex Numbers

The Most Beautiful Equation:

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.

Continued… one period frequency angular frequency in complex notation: