Complex Number Calculator
Add, subtract, multiply, and divide two complex numbers instantly.
How it works
- 1Enter the real and imaginary parts of complex numbers a and b using the four input fields.
- 2All four operations (addition, subtraction, multiplication, division) are computed at once using standard complex arithmetic.
- 3Division multiplies numerator and denominator by the conjugate of b; if b is zero the quotient is undefined (—).
Use cases
- Electrical engineering: impedance in AC circuits combines resistance and reactance as a complex number.
- Signal processing: Fourier transforms and phasor analysis rely on complex multiplication and division.
- Physics: quantum mechanics and wave equations express amplitudes as complex numbers.
Frequently asked questions
Why does multiplication look different from regular numbers?
Complex multiplication applies the distributive law and substitutes i² = −1, so (a + bi)(c + di) = (ac − bd) + (ad + bc)i.
How does complex division work?
To divide a by b, multiply both by the conjugate of b (re − im·i). This makes the denominator real (|b|² = re² + im²), and the result follows directly.
What happens when b is zero?
Division by zero is undefined for complex numbers just as for reals — the quotient shows — when both parts of b are 0.
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