Roots Of Unity

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The sum of all the distinct fifth roots of unity in the complex plane, taken together as complex numbers, equals which value?

RankGuru · Accepted Answer

The n distinct nth roots of unity are the solutions of z^n = 1 and are equally spaced on the unit circle, a configuration central to JEE Advanced. They are the roots of the polynomial z^5 - 1 = 0. By Vieta's formulas, the sum of all roots of z^5 - 1 equals the negative of the coefficient of z^4 divided by the leading coefficient. Since z^5 - 1 has no z^4 term, that coefficient is 0, so the sum of the roots is 0. Geometrically, the five roots are vertices of a regular pentagon centred at the origin, and their position vectors cancel by symmetry. Option 1 counts only the root z = 1. Option 5 wrongly adds the count of roots. Option -1 has no symmetric basis. Hence the sum is 0. Plausibility check: for any n ≥ 2 the nth roots of unity sum to zero because they form a symmetric set about the origin, and the absent z^{n-1} term in z^n - 1 confirms this algebraically.

Roots Of Unity

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