What is the Complex Root Calculator?
Every nonzero complex number z = a + bi has exactly n distinct n-th roots. This calculator finds all of them. It converts z to polar form (modulus r and argument θ), then applies De Moivre's theorem to list each root in rectangular a + bi form, together with its angle. The roots are evenly distributed around a circle of radius r^{1/n} in the complex plane, separated by 360^\circ/n.
How to use it
Enter the real part (a) and imaginary part (b) of your complex number, then choose the root degree n (for example, 2 for square roots, 3 for cube roots). The tool returns the modulus |z|, the argument θ in degrees, the root modulus r^{1/n}, and a full table of all n roots. The principal root (k = 0) is highlighted at the top.
The formula explained
First write z in polar form: r = √(a^2 + b^2) and θ = atan2(b, a). The n-th roots are then:
Each root shares the same modulus r^{1/n}; only the angle changes, increasing by 2π/n with each step.
Worked example
Find the square roots of z = -1 (a = -1, b = 0, n = 2). Here r = 1 and θ = 180^\circ. The root modulus is 1^{1/2} = 1. The angles are 180^\circ/2 = 90^\circ and (180^\circ + 360^\circ)/2 = 270^\circ. So the roots are \cos 90^\circ + i·\sin 90^\circ = i and \cos 270^\circ + i·\sin 270^\circ = −i. These are exactly the two square roots of -1.
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