Friday, June 15, 2012

The nth Roots of Unity

The equation xn = 1 has n roots which are called the nth roots of unity. 
xn = 1 = cos 0 + i sin 0      = cos 2kπ + i sin 2kπ    [ k is an integer]
x = (cos 2kπ + i sin 2kπ)1/n 
  = (cos (2kpi/n) + i sin (2kpi/n)) where k = 0 , 1, 2 , 3 , 4 , ……… , (n-1) 

So each root of unity is cos[ (2kπ)/n] + i sin[(2kπ)/n] where 0 ≤ k ≤ n-1.

We know that complex numbers of the form x + iy can be plotted on the complex plane (Argand Diagram). If we compare each root of unity with x + iy ,  we get x = cos[(2kπ)/n] , y=sin[(2kπ)/n].
Now , we see that the values of x and y satisfy the equation of unit circle with centre (0,0) i.e. x2 + y2 = 1.

From above it can be concluded that the roots are located on the circumference of the unit circle with center at (0,0). If we join the points we always get a regular polygon of n sides.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

13 comments:

  1. find the special root of x^15-1=0.

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  2. what is the sum of nth roots of unity?

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    Replies
    1. For any polynomial equation, the coefficient of x^1 term gives the sum of the roots. Since this is zero for every equation of the form x^n=1, the sum of a nth roots of unity is zero.

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    2. *sum of all nth roots of unity is zero.

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    3. The coefficient of X to the power n-1 divided by coefficient of X to the power n with a negative sign represents sum of roots which is zero.

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    4. The coefficient of X to the power n-1 divided by coefficient of X to the power n with a negative sign represents sum of roots which is zero.

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  3. ( w^10+w^23) = ?
    where w represents omega

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  4. Can u pls find the polar form of I^27

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  5. How can I calculate cos {2*pi()/17}, from x^17 -1 = 0?

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