Complex Number Arithmetic

(a + bi) ± (c + di), ×, ÷. Real part, imaginary part, modulus, argument.

Inputs

Result

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How to use this calculator

  • Pick operation.
  • Enter z₁ = a + bi and z₂ = c + di.
  • Read result + modulus + argument.

About this calculator

Complex numbers extend reals with i = √−1. Addition/subtraction: combine real and imaginary parts. Multiplication: (a+bi)(c+di) = (ac−bd) + (ad+bc)i. Division: multiply numerator and denominator by conjugate. Modulus |z| = √(a²+b²); argument θ = atan2(b, a). Polar form: r∠θ. Used in electrical engineering (impedance), signal processing (FFT), and quantum mechanics.

Frequently asked

What is i?+
Imaginary unit, defined by i² = −1. Allows solving equations like x² + 1 = 0 that have no real solution.
Polar vs. rectangular?+
Rectangular: a + bi. Polar: r∠θ (modulus + angle). Multiplication is easier in polar (multiply moduli, add arguments).
Why complex multiplication formula?+
(a+bi)(c+di) = ac + adi + bci + bdi² = (ac−bd) + (ad+bc)i since i² = −1.
Conjugate?+
a − bi for a + bi. Used in division: multiply by conjugate over conjugate to make denominator real.
Real-world use?+
AC circuit impedance (V = IZ, complex Z), signal processing (FFT, phase), quantum probability amplitudes.

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