Complex Number Calculator

How to use the complex number calculator

1. Type a number in Complex number z. You can write it in rectangular form (3+4i, 4i+3, -2i, i, 5), in polar form (5∠53.13° or 2 cis 45) or in exponential form (2e^(iπ/2)). Engineers can write j instead of i.
2. Choose the Operation: Add: z + w, Subtract: z − w, Multiply: z × w, Divide: z ÷ w, Power: zⁿ, n-th roots of z, or Modulus, argument, conjugate and forms of z.
3. Fill in the second field. Its label follows the operation: Second number w for the four basic operations, Exponent n (real or complex) for powers and Root degree n (1 to 100) for roots. It is hidden when you only want the properties of z.
4. Pick the Angle unit, Degrees or Radians. It is used for polar input without a unit symbol and for every argument in the results.
5. The result updates as you type; Calculate does the same and shows a message if an entry cannot be read. Copy copies the result, or the whole list of roots.

Under the result, you get the rectangular, polar and exponential forms, the modulus, the argument and the conjugate, followed by a Step by step section that repeats the calculation with your numbers.

The rules behind each operation

A complex number z = a + bi has a real part a and an imaginary part b, where i² = −1.

• Addition and subtraction work part by part: (a + bi) ± (c + di) = (a ± c) + (b ± d)i.
• Multiplication expands the brackets and replaces i² by −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. For example (3 + 4i)(1 − 2i) = 11 − 2i.
• Division multiplies the top and the bottom by the conjugate of the divisor, which makes the denominator real: (a + bi) / (c + di) = (a + bi)(c − di) / (c² + d²). So (3 + 4i) / (1 − 2i) = (−5 + 10i) / 5 = −1 + 2i.
• Modulus and argument: |z| = √(a² + b²) is the distance from the origin, and the argument θ = atan2(b, a) is the angle with the positive real axis. The tool gives the principal argument, between −180° and 180° (−π and π). The argument of 0 is undefined.
• Conjugate: z̄ = a − bi, the mirror image across the real axis. z × z̄ = |z|².

Polar and exponential forms

Any non-zero complex number can be written z = r(cos θ + i sin θ) = r∠θ = r·e^(iθ), with r = |z|. Going back, a = r cos θ and b = r sin θ. The exponential form always uses radians, because Euler’s formula e^(iθ) = cos θ + i sin θ is only true with θ in radians. For polar input, you can force the unit with ° or rad, or use π: 2∠π/2 is 2i whatever the selected unit.

Powers and roots

For a whole exponent, De Moivre’s formula gives zⁿ = rⁿ∠(nθ): raise the modulus to the power and multiply the angle. The tool also computes the result by repeated multiplication, so (1 + i)⁸ comes out as exactly 16. For a fractional or complex exponent, it uses the principal branch z^w = e^(w·Log z), with Log z = ln r + iθ. That is how i^i ≈ 0.2079, a real number.

A non-zero number has exactly n distinct n-th roots. They share the modulus ⁿ√r and their angles are (θ + k × 360°) / n for k = 0 … n − 1, so they sit evenly on a circle. The cube roots of −8 are 1 + 1.732i, −2 and 1 − 1.732i. The first one (k = 0) is the principal root.