Roots of Unity

The roots of z=1 are called roots of unity. The nth roots are evenly spaced around the circle of radius 1, starting with the first root at 1+0i. Putting this together with what we learned above, we see that the n nth roots of unity are given by
Exp[i 2kPi/n)], k=0, 1, 2, 3, . . ., n-1

In[49]:=

  fifthroots=Table[N[Exp[I 2 k Pi/5]],{k,0,4}]

Out[49]=

  {1., 0.309017 + 0.951057 I, -0.809017 + 0.587785 I, 
   
    -0.809017 - 0.587785 I, 0.309017 - 0.951057 I}

In[50]:=

  points=Table[{Re[fifthroots[[i]]],Im[fifthroots[[i]]]},
      {i,1,5}]

Out[50]=

  {{1., 0}, {0.309017, 0.951057}, {-0.809017, 0.587785}, 
   
    {-0.809017, -0.587785}, {0.309017, -0.951057}}

In[51]:=

  p5=ListPlot[points,DisplayFunction->Identity, 
    PlotStyle->{RGBColor[1,0,0],PointSize[.02]}];
  p6=PolarPlot[1,{t,0, 2 Pi},DisplayFunction->Identity];
  Show[p5,p6,DisplayFunction->$DisplayFunction, 
  AspectRatio->Automatic]

Out[52]=

  -Graphics-

Another important aspect of the roots of unity is illustrated in the following:

In[53]:=

  MatrixForm[fifthroots]

Out[53]=

  1.
  
  0.309017 + 0.951057 I
  
  -0.809017 + 0.587785 I
  
  -0.809017 - 0.587785 I
  
  0.309017 - 0.951057 I

In[54]:=

  MatrixForm[Table[fifthroots[[2]]^k,{k,1,4}]]

Out[54]=

  0.309017 + 0.951057 I
  
  -0.809017 + 0.587785 I
  
  -0.809017 - 0.587785 I
  
  0.309017 - 0.951057 I

That is, if w is an nth root of unity, then so are w^2, w^3, . . . , w^(n-1).
This idea will be useful in working with the Fast Fourier Transform.

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