De Moivre'S Theorem And Nth Roots Worksheet Page 4
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6Example 7
Find the sixth roots of
3 + i
The modulus of
3 + i is 2 and the argument is
The sixth roots are
,
,
,
,
and
It is easy and important to find the nth roots of 1
n
i.e. complex numbers such that z
= 1
Such numbers are often referred to as the nth roots of unity.
Roots of unity
n
The nth roots of unity are those numbers that satisfy the equation z
= 1
Since 1 = cos 2 + i sin 2 , it follows that
is an nth root of unity.
But 1 can be written using different arguments as follows:
Hence dividing the argument in each case by n gives the following nth roots of unity.
Note that arguments increase by
each time. The roots of unity are regularly spaced in an Argand diagram.
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