Friday, April 8, 2016

DeMoivre's Theorem for Powers and Roots of Complex Numbers

DeMoivre’s Theorem  (pronounced də ‘mwavʁ)
is a handy way to find powers and roots of complex numbers.  Just think of the Draconian feat required to raise (3 + 4i ) to the 7th power this way…

   (3 + 4i)(3 + 4i)(3 + 4i)(3 + 4i)(3 + 4i)(3 + 4i)(3 + 4i) = ?  

I doubt that even a 100% grade on a homework assignment could inspire me to work this out longhand.  Thanks to Abraham deMoivre (1667 - 1754), we can significantly cut down the required work.

 Raising a complex number to a higher power is a pretty straightforward process with few steps.
        (a + bi)^n

1)  Write the complex number 
     (a + bi) in polar form  

      z = r(cos theta + i sin theta)

      by calculating r (modulus, 
             r =  √(a^2 + b^2)
      theta (argument)   
             theta = tan^(-1) b/a 

 (If the degree comes up negative, add 180˚ or 360˚ to get it into the range [0,360].  You’ll want to remember some inverse trig concepts here to ensure you’re maintaining the positive and negative natures of functions. I highly recommend graphing the complex number to provide a suitable location map.)

 2) from r(cos theta + i sin theta)
    a)  raise r to the desired power n
    b)  multiply theta by n
        r^n [cos n(theta) + i sin n(theta)]

EXAMPLE:  (7 - 24i)^5

    r = 25
    theta =>    Tan^(-1) (-24/5) = -78.2˚  (180 - 78.2 = 101.8˚)

    POLAR:    z = 25 (cos 101.8 + sin 101.8)

            z^5 = 25^5 [cos (5•101.8) + i sin (5•101.8)]
                  = 9,765,625 (-.857 + i .515)
                  = -8,369,140.63 + i 5,029,296.88  


WOW!  Those numbers got very big, very fast!  I don't even want to check the arithmetic.  The classroom is more likely to use a less intimidating example like ------->


Finding the n roots is much more arithmetically intensive and is well guided by a consistent structure that emphasizes the repetitious nature of the calculations.

 1)  Write the complex number (a + bi) in polar form....             z = r(cos theta + i sin theta).

      r (modulus, magnitude) = 
            √(a^2 + b^2)
      theta (argument) =   
             tan^(-1) b/a

Look at the number of roots you need and list Polar forms that represent that many coterminal angles by adding 360˚ or 2π, depending on whether you're working in degrees or radians.


2)  Raise the modulus to the fractional exponent and multiply the argument by the same fraction. 

z^(1/n) = r^(1/n)[cos theta/n + i sin theta/n]

NOTICE that to raise a complex number to a HIGHER POWER, the strategy is to RAISE the modulus to a power, N, and MULTIPLY theta by n.  To find the ROOTS, the exponent on the modulus is a FRACTION, 1/n, and multiplying theta by a fraction with a numerator of 1 is the same as dividing by the denominator.  The exponent tells everything you need to know about the steps to take.  Keep in mind the rules of exponents. 

3)  Calculate.  Now the process turns into simple arithmetic (combined with a healthy knowledge of trig values of angles).

NOTICE that once theta is reduced, the distance between the angles is directly proportional to the number of roots
                                     360     OR    
                                       n                  n
   Number of Roots
                           2        180˚………… π
                           3        120˚…………
                           4          90˚………… π
                           5          72˚…………        Use this factoid to avoid
                                                            5         mistakes on the arithmetic.



NOTICE also that the final roots are evenly distributed around the circle.  So a visual display may be another tip for avoiding errors.

When the classroom curriculum turns to finding powers and roots of complex numbers, the process is well defined.  It requires an established system to organize the process and strict attention to arithmetic and trig calculations. 

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