Complex Root Calculator
Find all n-th roots of a complex number a+bi using De Moivre's theorem. Enter real and imaginary parts and the root degree to compute every root in polar form.
About This Calculator
The Complex Root Calculator computes all n distinct n-th roots of a complex number a+bi using De Moivre's theorem. Enter the real part (a), imaginary part (b), and the root degree (n) to get every root with its magnitude and argument.
For a degree of n, there are always exactly n roots equally spaced around a circle in the complex plane. This calculator works for any positive integer n, including square roots (n=2), cube roots (n=3), fourth roots (n=4), and beyond.
Frequently Asked Questions
What are complex roots?
Complex roots are the n-th roots of a complex number a+bi. For a degree n, there are exactly n distinct roots evenly spaced around the complex plane, all with the same magnitude r^(1/n) but different arguments.
How do you calculate complex roots?
Using De Moivre's theorem: convert a+bi to polar form r·e^(itheta), take the n-th root of the magnitude r, and divide the angle theta by n. Each of the n roots is given by r^(1/n)·e^(i(theta+2pik)/n) for k=0,...,n-1.
Can the real part be zero?
Yes, the real part (a) can be zero. For example, the cube roots of i (0+1i) are three roots equally spaced on the unit circle.
Is this tool free?
Yes, all calculators on Calculy are completely free to use with no registration or limits.
What is the magnitude of a complex number?
The magnitude (or modulus) of a+bi is r = sqrt(a^2+b^2). It represents the distance from the origin to the point (a,b) in the complex plane.
What does the argument of a complex number mean?
The argument theta = atan2(b,a) is the angle the complex number makes with the positive real axis, measured in radians. It determines the direction of the complex number in the plane.