Complex Number Calculator
Enter two complex numbers (a + bi) and an operation to compute the result, modulus, argument and see both numbers as vectors on the complex (Argand) plane.
Operation
- 1
Real part
3 + 1 = 4 - 2
Imaginary part
4 + -2 = 2 - 3
Result modulus
√(4² + 2²) = 4.4721
How does this calculator work?
Complex arithmetic: add/subtract component-wise; multiply with FOIL and i² = −1; divide by multiplying by the conjugate of the denominator. Modulus |z| = √(a² + b²), argument = atan2(b, a). The Argand diagram shows both numbers and the result as vectors from the origin.
Formula
How this is calculated
A complex number z = a + bi has a real part a and an imaginary part b, where i² = −1. Addition and subtraction are component-wise: real parts add to real, imaginary to imaginary. Multiplication uses FOIL and the identity i² = −1, giving (ac − bd) + (ad + bc)i.
Division requires rationalising the denominator: multiply numerator and denominator by the conjugate (c − di) of the denominator. The denominator becomes c² + d² (a real number), and the result has real part (ac + bd)/(c² + d²) and imaginary part (bc − ad)/(c² + d²). Division is undefined when c = d = 0.
The modulus |z| = √(a² + b²) is the distance from the origin on the Argand plane — equivalent to the magnitude of a 2-D vector. The argument arg(z) = atan2(b, a) is the angle from the positive real axis, shown in degrees. The vector diagram plots both input numbers and the result as arrows from the origin.
Frequently asked questions
An imaginary number is a multiple of i, where i is defined as √(−1). Since no real number squares to a negative, i is "imaginary" — but complex numbers are very real in engineering, physics and signal processing.
Set the imaginary part to 0 for a real number (e.g. 5 + 0i = 5), or the real part to 0 for a purely imaginary number (e.g. 0 + 3i). Both cases are handled correctly by all four operations.
On the Argand plane, a complex number is a point (or vector). The modulus is the distance from the origin — the length of that vector. For multiplication, |z₁ × z₂| = |z₁| × |z₂|; the moduli multiply and the arguments add, which is why complex multiplication is a rotation-plus-scaling in the plane.
Also known as
TG we-Calculate Editorial Team. (2026). Complex Number Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/complex-number-calculator
TG we-Calculate Editorial Team. "Complex Number Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/complex-number-calculator.
TG we-Calculate Editorial Team, "Complex Number Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/complex-number-calculator
@misc{wecalculate_complex_number_calculator, title = {Complex Number Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/complex-number-calculator}}, year = {2026}, note = {TG we-Calculate} }
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