Roots of Unity
The n nth-roots of unity on the unit circle
About this calculator
The Roots of Unity Calculator finds all n complex solutions of zⁿ = 1. It plots the roots as equally spaced points on the unit circle, lists each root in rectangular form e^(2πik/n) = cos(2πk/n) + i·sin(2πk/n) with its angle, and counts the primitive roots φ(n). Use it for complex analysis, abstract algebra, and signal-processing study.
How to use the Roots of Unity calculator
- Enter the values for your problem into the input fields.
- Read the result — it updates instantly as you type.
- Check the formula and the visual explanation to follow how the answer was found.
- Copy the page URL to share the exact calculation.
Common examples
- n = 1 → z = 1
- n = 2 → z = 1, −1
- n = 3 → z = 1, −0.5 ± 0.8660i (equilateral triangle)
- n = 4 → z = 1, i, −1, −i (square); primitive roots i and −i
- n = 6 → six roots forming a regular hexagon; φ(6) = 2 primitive roots
Frequently asked questions
How does the Roots of Unity calculator work?
Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.
When would I use the Roots of Unity calculator?
It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.
How accurate are the results?
Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.
Can I share a specific calculation with someone else?
Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.