Complex Number Calculator
Add, subtract, multiply, divide complex numbers.
Overview
The Complex Number Calculator performs the four basic operations on numbers of the form a + bi: addition, subtraction, multiplication and division. It accepts inputs in rectangular form and returns answers in rectangular form, along with the polar representation r ∠ θ for both operands and the result.
This is the daily tool for electrical engineers manipulating phasors, signal-processing students working with Fourier coefficients and anyone reviewing a complex-analysis problem set without firing up MATLAB or Mathematica.
How it works
For complex numbers z1 = a + bi and z2 = c + di, addition is (a + c) + (b + d)i and subtraction is (a - c) + (b - d)i. Multiplication uses the FOIL expansion (a + bi)(c + di) = (ac - bd) + (ad + bc)i.
Division multiplies numerator and denominator by the conjugate of the denominator: (a + bi) / (c + di) = ((ac + bd) + (bc - ad)i) / (c^2 + d^2). Polar form uses r = sqrt(a^2 + b^2) and θ = atan2(b, a).
Examples
(2 + 3i) + (1 - i) → 3 + 2i
(1 + i) * (1 - i) → 2 + 0i
(3 + 4i) / (1 + 2i) → 2.2 - 0.4i
polar form of 1 + i → √2 ∠ 45°
FAQ
What is i?
The imaginary unit defined by i^2 = -1. Complex numbers extend the reals with a perpendicular axis and behave as ordered pairs.
Why divide using the conjugate?
Multiplying top and bottom by the conjugate of the denominator turns the denominator into a real number, which gives a clean a + bi form on the result.
Can I input numbers in polar form?
This tool takes rectangular a + bi but shows polar r ∠ θ on every result, so you can convert by entering r * cos θ + r * sin θ * i.
Does it support more than two operands?
Apply operations stepwise. Each pair gives a complex result you can feed into the next calculation.
What about complex powers and roots?
For powers and nth roots, see the Quaternion Calculator (for 4D) or compute manually with polar form: (r ∠ θ)^n = r^n ∠ (n * θ).