Roots of Unity - Maple Help
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Complex Roots of Unity

Main Concept

A root of unity, also known as a de Moivre number, is a complex number z which satisfies , for some positive integer n.

Solving for the  roots of unity

Note that Maple uses the uppercase letter I, rather than the lowercase letter i, to denote the imaginary unit: .

 

Since  is a polynomial with complex coefficients and a degree of n, it must have exactly n complex roots according to the Fundamental Theorem of Algebra.

To solve for all the  roots of unity, we will use de Moivre's Theorem: , where x  is any complex number and n is any integer (in this particular case x will be any real number and n will be any positive integer).

 

First, convert the complex number z to its polar form: , where  is the modulus of z and q is the angle between the positive real axis (Re) and the line segment joining the point z to the origin on the complex plane. Since , it must be true that , and so the previous equation simply becomes  .

 

Also, converting the real number  to polar form, we get  for any integer k.

 

Now,  and so using de Moivre's Theorem, this equation becomes . From this form of the equation, we can see that  , or equivalently, .

Therefore, the  roots of unity can be expressed using the formula , for .   

Using Euler's formula: , we can write this formula for the  roots of unity in its most common form: , for .

 

When the roots of unity are plotted on the complex plane (with the real part [Re] on the horizontal axis and the imaginary part [Im] on the vertical axis), we can see that they all lie on the unit circle and form the vertices of a regular polygon with n sides and a circumradius of 1.

Degree of Polynomial, n

 

 

 

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