Mathematics
Complex number calculator
Write numbers as 3+4i, 2∠60° or 2e^(iπ/3) and choose the operation: get the result in rectangular, polar and exponential form, with modulus, argument and a drawing on the Argand plane.
The three forms of a complex number
A complex number a + bi is a point on the Argand plane: a on the real axis, b on the imaginary one. The same point can be described by its distance from the origin, the modulus r = √(a² + b²), and its angle to the real axis, the argument θ. That gives the polar (trigonometric) form r(cos θ + i sin θ) and the exponential form r·e^(iθ).
Each form suits something. Sums and differences are done in rectangular form, component by component. Products and quotients are simpler in exponential form: moduli multiply and arguments add. For example (1 + i)⁸: the modulus √2 to the eighth is 16, the argument π/4 times eight is 2π, and the result is 16.
The n-th roots are n points on a circle: all with modulus ⁿ√r and arguments (θ + 2kπ)/n, spaced 2π/n apart. The three cube roots of 1 are 1 and −1/2 ± (√3/2)i, the corners of an equilateral triangle.
Common mistakes
- Taking the argument as arctan(b/a) without checking the quadrant: −1 − i has argument −3π/4, not π/4.
- Forgetting that i² = −1 in products: (3 + 4i)(1 − 2i) = 3 − 6i + 4i + 8 = 11 − 2i.
- Finding only one n-th root: in the complex numbers there are always exactly n.
Frequently asked questions
How do you divide by a complex number?
Multiply numerator and denominator by the conjugate of the denominator: the denominator becomes the real number a² + b². Or, in exponential form, divide the moduli and subtract the arguments.
What is Euler's formula?
It is e^(iθ) = cos θ + i sin θ, the bridge between exponential and polar form. With θ = π it gives e^(iπ) + 1 = 0.
What are complex numbers used for?
In electrical engineering they represent alternating voltages and currents as phasors, in signal processing they describe frequencies and phases, and in mathematics they guarantee that every polynomial of degree n has n roots.
How this calculation works
z = a + bi; modulus r = √(a² + b²); argument θ = atan2(b, a), in (−π, π]. Polar form r(cos θ + i sin θ), exponential r·e^(iθ). Product: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. Quotient: multiply by the conjugate of c + di. Power (De Moivre): zⁿ = rⁿ(cos nθ + i sin nθ). Roots: wₖ = ⁿ√r·e^(i(θ + 2kπ)/n), k = 0 … n − 1.
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