Roots of Unity - Maple Help

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 ${\mathbit{n}}^{\mathbit{th}}$ 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 ${\mathbit{n}}^{\mathbf{th}}$ roots of unity, we will use de Moivre's Theorem: ${\left(\mathrm{cos}\left(x\right)+\mathrm{I}\mathbf{}\mathrm{sin}\left(x\right)\right)}^{n}=\mathrm{cos}\left(n\mathbf{}x\right)+\mathrm{I}\mathbf{}\mathrm{sin}\left(n\mathbf{}x\right)$, 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: $z=\left|z\right|\mathbf{}\left(\mathrm{cos}\left(\mathrm{\theta }\right)+I\mathbf{}\mathrm{sin}\left(\mathrm{\theta }\right)\right)$, where $\left|z\right|$ 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  $z=\mathrm{cos}\left(\mathrm{\theta }\right)+I\mathbf{}\mathrm{sin}\left(\mathrm{\theta }\right)$.   Also, converting the real number  to polar form, we get $1=\mathrm{cos}\left(2\mathbf{}\mathrm{\pi }\mathbf{}k\right)+I\mathbf{}\mathrm{sin}\left(2\mathbf{}\mathrm{\pi }\mathbf{}k\right)$ 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: ${ⅇ}^{\mathrm{I}\mathbf{}\mathrm{θ}}=\mathrm{cos}\left(\mathrm{\theta }\right)+\mathrm{I}\mathbf{}\mathrm{sin}\left(\mathrm{θ}\right)$, 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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