Math Problem Statement

what are cube roots of unity?

Solution

The cube roots of unity are the three complex numbers that satisfy the equation:

x3=1x^3 = 1

These roots are solutions to the equation and can be expressed as:

  1. Unity (1): This is the real and positive root of the equation.

  2. Non-real Complex Roots: The other two roots are complex numbers and can be expressed as:

    ω=e2πi/3=12+i32\omega = e^{2\pi i / 3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2} ω2=e4πi/3=12i32\omega^2 = e^{4\pi i / 3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2}

Properties of Cube Roots of Unity:

  1. ω3=1\omega^3 = 1 and (ω2)3=1(\omega^2)^3 = 1.
  2. ω2+ω+1=0\omega^2 + \omega + 1 = 0.
  3. The three roots satisfy the relationship: 1,ω,ω21, \omega, \omega^2, where ω2\omega^2 is the square of ω\omega.
  4. Multiplying all the cube roots gives 1ωω2=11 \cdot \omega \cdot \omega^2 = 1.

Visual Representation:

The cube roots of unity can be represented geometrically in the Argand plane as the vertices of an equilateral triangle inscribed in the unit circle.


Would you like further details, such as derivation or applications? Here are five related questions:

  1. What are the general nn-th roots of unity?
  2. How are cube roots of unity used in solving cubic equations?
  3. Can you explain the geometric representation of cube roots of unity in detail?
  4. How do cube roots of unity simplify trigonometric expressions?
  5. What is the significance of roots of unity in Fourier transforms?

Tip: Roots of unity are often useful for simplifying cyclic sums and products in algebra.

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Math Problem Analysis

Mathematical Concepts

Algebra
Complex Numbers
Roots of Unity

Formulas

x^3 = 1
ω = e^(2πi / 3) = -1/2 + i√3/2
ω^2 = e^(4πi / 3) = -1/2 - i√3/2
ω^3 = 1
ω^2 + ω + 1 = 0

Theorems

Euler's Formula
Roots of Unity Theorem

Suitable Grade Level

Grades 11-12