Roots of unity

Roots of Unity

Concept of Roots of Unity

  • Roots of unity refer to the solutions of the equation z^n = 1, where n is a positive integer.
  • These roots are complex numbers which, when raised to the power of n, equal 1.
  • Understand that roots of unity are regularly spaced around the unit circle in Argand Diagram.
  • There are exactly n different roots of unity for any given n.

Expression of Roots of Unity

  • Recognize that roots of unity can be expressed in the form cis(2kπ/n), where k = 0, 1, 2,…, (n - 1).
  • Understand that the cis function stands for cosine plus i sine and is a convenient way to write complex numbers in polar form.

Geometric Interpretations

  • Appreciate that the roots of unity can be visualised as points on the unit circle in the complex plane that are equally spaced around the origin.
  • Know that the principal nth root of unity is denoted ωn = cis(2π/n), representing the root with the smallest positive argument.

Properties of Roots of Unity

  • Note that the sum of the roots of unity for a given n is zero.
  • When multiplied together, the roots of unity give 1 if n is even and -1 if n is odd.
  • Understand that the roots of unity form the vertices of a regular n-gon on the unit circle in the complex plane.

Applications of Roots of Unity

  • Comprehend that roots of unity are used in various areas of mathematics, including number theory, field theory, Fourier analysis, and many areas of computer science, such as Fast Fourier Transforms.
  • Understand that the roots of unity provide a systematic way to deal with complex numbers and their powers.

Revise these points thoroughly; understanding roots of unity is essential to exploit the power of complex numbers across various mathematical problems.