How do you divide complex numbers?

Multiply numerator and denominator by the conjugate of the denominator: (a+bi)/(c+di) = (a+bi)(c−di)/(c²+d²), which makes the denominator real. The simplified a+bi result is shown.

What are the modulus and argument?

|z| = √(a²+b²) is the distance from the origin in the complex plane; the argument θ = atan2(b, a) is the angle from the positive real axis, given in both degrees and radians.

How do I find the nth roots of a complex number?

By De Moivre’s theorem: take the nth root of the modulus, divide the argument by n and add 360°/n for each further root — all n of them are listed.

What is the square root of i?

Enter z₁ = 0 + 1i and choose 2 roots: ±(0.7071 + 0.7071i), that is ±(√2/2)(1+i).

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🌀Complex Number Calculator - Add, Multiply, Divide, Powers and Roots

Enter complex numbers as a+bi and add, subtract, multiply, divide, raise to a power or take nth roots. Every answer comes with |z|, the argument in degrees and radians, the conjugate, the polar form r(cosθ+isinθ) and the exponential form re^(iθ); roots list all n values.

Free · No sign-up · Runs in your browserLast updated: 2026-10-26

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How to use Complex Numbers

1Choose the operation: +, −, ×, ÷, power or nth roots
2Enter the real and imaginary parts (and n for powers and roots)
3Read the result with its modulus, argument and polar form

FAQ

How do you divide complex numbers?: Multiply numerator and denominator by the conjugate of the denominator: (a+bi)/(c+di) = (a+bi)(c−di)/(c²+d²), which makes the denominator real. The simplified a+bi result is shown.
What are the modulus and argument?: |z| = √(a²+b²) is the distance from the origin in the complex plane; the argument θ = atan2(b, a) is the angle from the positive real axis, given in both degrees and radians.
How do I find the nth roots of a complex number?: By De Moivre’s theorem: take the nth root of the modulus, divide the argument by n and add 360°/n for each further root — all n of them are listed.
What is the square root of i?: Enter z₁ = 0 + 1i and choose 2 roots: ±(0.7071 + 0.7071i), that is ±(√2/2)(1+i).