Finding roots of unity using Euler and De Moivreś

Nasser M. Abbasi

June 14,2006 compiled on — Thursday September 14, 2017 at 12:23 AM
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To find the roots of

f(x) = xn − 1

Solving for x  from

pict

Now   1
1 n  is evaluated. Since

     i(2π)
1 = e

Substituting (2) in the RHS of (1) gives

pict

Using De Moivre’s formula

                     (         )       (         )
(cos α+ isinα )1n = cos  α-+ k2π-  + isin   α-+ k2π-      k = 0,1,⋅⋅⋅n − 1
                       n     n           n     n

Therefore (3) is rewritten as

       ( 2π    2π )       (2π     2π)
x = cos  n--+ k-n-  + isin  -n-+  kn--      k = 0,1,⋅⋅⋅n − 1

The above gives the roots of f(x) = xn − 1  . The following examples illustrate the use of the above.

  1. Solve f(x) = x2 − 1  . Here n = 2  , therefore k = 0,1  . For k = 0
    pict

    And for k = 1

    pict

    Hence the two roots are {1,− 1}

  2. Solve f(x) = x3 − 1  . Here n = 3  , hence for k = 0
    pict

    And for k = 1

    pict

    And for k = 2

    pict

    Therefore the roots are {1,         √ -       √-
−  1+ i--3,− 1− i-3}
   2    2    2    2