Complex number calculator

How to use it

1. Type the first number as a+bi. Spaces are fine; i, -i and 2i are valid, and so is a plain real number.
2. Choose the operation. For a power or a root, type the whole number n in the second box instead of a complex number.
3. The result appears at once with its modulus, argument in radians and degrees, polar form, exponential form and conjugate.

How it works

• Add and subtract: real and imaginary parts separately.
• Multiply: (a+bi)(c+di) = (ac − bd) + (ad + bc)i.
• Divide: multiply by the conjugate of the divisor and divide by c² + d².
• Integer powers: repeated squaring, so results such as (1+i)⁸ = 16 come out exactly instead of with trigonometric rounding.
• Roots: r^(1/n)·e^(i(θ + 2πk)/n) for k = 0 … n−1, where r and θ are the modulus and argument of z₁.

Numbers are ordinary floating-point values, shown with up to 10 significant digits; parts that are only rounding dust are shown as 0.

Worked example

z₁ = 3+4i, z₂ = 1−2i:

• z₁ + z₂ = 4 + 2i
• z₁ × z₂ = 11 − 2i
• z₁ ÷ z₂ = −1 + 2i
• |z₁| = 5 and arg z₁ = atan2(4, 3) = 0.9273 rad = 53.13°, so z₁ = 5(cos 0.9273 + i sin 0.9273) = 5·e^(0.9273i).

Tips

• To solve a polynomial equation with complex roots, see the polynomial roots calculator or the cubic equation solver.
• The conjugate of a + bi is a − bi; multiplying a number by its conjugate gives |z|².