Complex Number Calculator

Add, subtract, multiply and divide complex numbers and find modulus, argument, conjugate, polar and exponential forms.

Your result will appear here

Fill in the fields and press Calculate, or pick one of the examples.

About the Complex Number Calculator

Performs arithmetic on complex numbers written as a + bi and describes the result in every common form: rectangular, polar and exponential, with its modulus, argument and conjugate.

Inputs can be expressions, so (1 + i)^2, e^(i*pi) or sqrt(-9) are accepted.

Limitations

The argument is the principal value in (−π, π]. The argument of 0 is undefined.

How to use it

Choose an operation, or “Properties of z₁” to analyse a single number.
Enter the numbers using i for the imaginary unit, for example 3 + 4i.
Press Calculate.

Example

(3 + 4i) × (1 − 2i)

→ 11 − 2i

Formulas used

(a + bi)(c + di) = (ac - bd) + (ad + bc)i

Multiplication, using $i^2 = -1$ .

\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^{2} + d^{2}}

Division multiplies by the conjugate of the denominator.

z = r(\cos\theta + i\sin\theta) = r e^{i\theta},\quad r = |z|,\ \theta = \arg z

Polar and exponential forms.

Root CalculatorSquare roots, cube roots and nth roots with simplified radical form, plus the complex roots of negative numbers.Quadratic Equation SolverSolve quadratic equations with the discriminant and quadratic formula shown step by step, including exact radical and complex roots.Polar GrapherPlot polar curves r = f(θ) on a polar grid with an adjustable angular range — roses, cardioids, spirals and limaçons.

About the Complex Number Calculator

Limitations

How to use it

Formulas used

Related tools

About the Complex Number Calculator

Limitations

How to use it

Formulas used