This complex number calculator performs arithmetic on complex numbers a + bi — addition, subtraction, multiplication, division — and reports each number’s modulus, argument (angle) and conjugate, the full working vocabulary of complex analysis and AC circuit math.
Arithmetic with i² = −1
Add and subtract componentwise; multiply by FOIL with the one rule that matters, i² = −1; divide by rationalizing with the conjugate. In polar form the geometry sings: moduli multiply/divide, arguments add/subtract — multiplication is a rotation-and-scaling, which is why complex numbers run signal processing, control theory and AC analysis. They also appear uninvited whenever the Quadratic Formula meets a negative discriminant. Part of our Math Calculators collection.
Worked Examples (Verified)
- z₁ = 3 + 4i, z₂ = 1 − 2i (defaults): sum 4 + 2i · difference 2 + 6i
- Product: (3+4i)(1−2i) = 3 − 6i + 4i − 8i² = 11 − 2i (i² = −1 rescues the last term into +8)
- Quotient: (3+4i)/(1−2i) = (3+4i)(1+2i)/5 = −1 + 2i — suspiciously clean; chosen so you can verify by hand
- |3 + 4i| = 5 — the 3-4-5 triangle lives in the complex plane too; its argument is 53.13°
The Ideas Behind the Buttons
- Conjugate (a − bi): reflection across the real axis; z × z̄ = |z|², always real — the engine inside division and modulus identities.
- Argument: measured with atan2, mindful of quadrant — plain arctan would confuse 1+i with −1−i.
- Powers get easy in polar form (De Moivre): (1+i)¹⁰ = (√2)¹⁰ ∠(45°×10) = 32∠450° = 32i — try it via repeated multiplication here if you doubt it.
- Every polynomial factors completely over ℂ (Fundamental Theorem of Algebra) — complex numbers are where algebra stops saying “no solution.”
Related Tools
- Quadratic Formula Calculator — where complex roots first appear
- Trigonometry Calculator — the cos/sin inside polar form
- Math Calculators — the full collection
Frequently Asked Questions
How do you multiply complex numbers?
FOIL, then replace i² with −1: (a+bi)(c+di) = (ac − bd) + (ad + bc)i. Example: (3+4i)(1−2i) = (3 + 8) + (−6 + 4)i = 11 − 2i. In polar form: multiply moduli, add angles.
How do you divide complex numbers?
Multiply numerator and denominator by the denominator’s conjugate; the denominator becomes the real modulus-squared. (3+4i)/(1−2i) = (3+4i)(1+2i)/(1+4) = (−5 + 10i)/5 = −1 + 2i.
What are modulus and argument?
The polar coordinates of the number as a point: modulus = distance from origin = √(a² + b²) (5 for 3+4i — Pythagorean), argument = angle from the positive real axis (53.13° here). Together they rewrite z as r(cos θ + i sin θ) = re^(iθ).
What is the complex conjugate for?
ā = a − bi reflects across the real axis. Products z·z̄ = |z|² power division, impedance calculations in AC circuits, and probability amplitudes in quantum mechanics (|ψ|² is a conjugate product) — humble notation, heavy lifting.
Why do complex numbers matter outside math class?
AC impedance is complex (resistance + reactance), Fourier transforms decompose signals into complex exponentials, control systems place poles in the complex plane, and fractals are iterated complex maps. The “imaginary” label is history’s worst branding — they’re the working numbers of electrical engineering.
Is this complex number calculator free?
Yes — free, instant, no sign-up.