Roots of UnityJEE Advanced

Concept

The nth roots of unity are the $n$ complex solutions of $z^n = 1$. Plotted on the Argand plane they sit equally spaced on the unit circle, forming a regular $n$-gon with one vertex at $1$.

Key formula

Derivation

Writing $z = e^{i\theta}$, the equation $z^n=1$ becomes $e^{in\theta}=1$, so $n\theta$ must be a multiple of $2\pi$.

That gives $\theta = 2\pi k/n$ for $k=0,\dots,n-1$ — the $n$ distinct roots. They are powers of the primitive root $\omega = e^{2\pi i/n}$, since $z_k = \omega^k$.

Scenarios to explore

• Roots of Unity — The n solutions of zⁿ = 1 on the unit circle.

Real-world applications

• Factorizing $z^n-1$ into linear factors.
• The Fast Fourier Transform uses roots of unity.
• Solving cyclotomic and symmetry problems.

JEE exam tips

• The sum of all nth roots of unity is 0 (for n ≥ 2); their product is (−1)ⁿ⁺¹.
• They are 1, ω, ω², …, ωⁿ⁻¹ with ωⁿ = 1.
• 1 + ω + ω² = 0 for the cube roots of unity.

Common mistakes

• Listing only the real roots and missing the complex ones.
• Forgetting k runs from 0 to n−1 (n roots, not n+1).
• Thinking the roots are randomly placed rather than equally spaced.

Exam traps to avoid

• A 'primitive' root generates all others; not every root is primitive (only when gcd(k,n)=1).
• Roots of zⁿ = a (a ≠ 1) lie on a circle of radius |a|^{1/n}, not the unit circle.