MIT18_03S10_c06

# MIT18_03S10_c06 - 18.03 Class 6 Complex exponential[1...

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18.03 Class 6 , Feb 16, 2010 Complex exponential [1] Complex roots [2] Complex exponential, Euler's formula [3] Euler's formula continued [1] More practice with complex numbers: Magnitudes Multiply: |zw| = |z||w| Arguments Add: Arg(zw) = Arg(z) + Arg(w) so: |z^n| = |z|^n and Arg(z^n) = n Arg(z) For example: Let's take some powers of z = 1+i . |z| = sqrt 2 Arg(z) = pi/4 |z^2| = 2 Arg(z^2) = pi/2 |z^3| = 2 sqrt 2 Arg(z^3) = 3pi/4 |z^4| = 4 Arg(z^4) = pi Notice that these numbers march out along a spiral. This continues for all powers of 1+i , even negative ones. How about fractional powers? i.e. roots: of unity, first. What are the cube roots of 1? Well 1 is one. If z is one, then |z|^3 = |z^3| = 1 , so it lies on the unit circle. The argument has to be so that 3 times it is zero - or 2pi, or 4pi, or . ... Thus the argument is 0 , giving 1 , or 2pi/3 , giving (-1+sqrt(3)i)/2 or 4pi/3 , giving (-1-sqrt(3)i)/2 . The nth roots of unity divide the unit circle into n equal pieces. How about cube roots of -8 . If z^3 = -8 , then |z|^3 = |z^3| = 8 . |z| is a positive real number so |z| = 2 : all cube roots of -8 lie on the circle with center 0 and radius 2. The argument of -8

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MIT18_03S10_c06 - 18.03 Class 6 Complex exponential[1...

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