Roots of Unity

Trigonometry · Axiom Academy

Discover the beautiful symmetric patterns formed by the nth roots of 1 on the unit circle The nth roots of unity are complex numbers that satisfy the equation: For example, when n = 2, we're looking for numbers where z ² = 1. The solutions are z = 1 and z = −1. 2. The Formula Using Trigonometry Using Euler's formula and De Moivre's theorem, we can express the nth roots of unity in terms of trigonometric functions: This formula tells us that the roots are evenly spaced around the unit circle, separated by angles of 2π/ n radians (or 360°/ n ). 3. Beautiful Symmetric Patterns Let's visualize the nth roots of unity for different values of n . Notice how they always form regular polygons inscribed in the unit circle: 4. Key Properties and Insights The roots of unity have remarkable mathematical properties: Perfect Symmetry: The roots are evenly distributed with rotational symmetry of order n . Sum to Zero: For n > 1, all nth roots of unity sum to zero. Closure Under Multiplication: The product of any two nth roots of unity is also an nth root of unity. Powers Generate All Roots: All roots can be generated as powers of the primitive root: ω, ω², ω³, …, ω n −1 , ω n = 1

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