What the complex number calculator does
Enter two complex numbers in the form a + bi, choose add, subtract, multiply or divide, and the calculator returns the result in rectangular form together with its magnitude and angle (polar form). Complex numbers are the standard tool in electrical engineering (AC circuits, impedance), signal processing, control theory, quantum mechanics and any problem involving rotation or oscillation — and the arithmetic is easy to get wrong by hand. Runs in your browser.
The arithmetic
i² = −1
Add: (a + bi) + (c + di) = (a + c) + (b + d)i
Subtract: (a + bi) − (c + di) = (a − c) + (b − d)i
Multiply: (a + bi)(c + di) = (ac − bd) + (ad + bc)i
Divide: (a + bi) / (c + di) = [(ac + bd) + (bc − ad)i] / (c² + d²)
(3 + 2i)(1 − 4i) = 3 − 12i + 2i − 8i² = 3 − 10i + 8 = 11 − 10i
(3 + 2i)/(1 − 4i) = (3 + 2i)(1 + 4i) / (1 + 16) = (3 + 12i + 2i − 8) / 17 = (−5 + 14i)/17Division multiplies top and bottom by the conjugate of the divisor (c − di), which makes the denominator real. The calculator does this for you and reports an error only when dividing by zero.
Rectangular and polar forms
z = a + bi (rectangular)
Magnitude |z| = √(a² + b²)
Angle θ = atan2(b, a) in degrees or radians
Polar z = |z| (cos θ + i sin θ) = |z| e^(iθ)
3 + 4i: |z| = 5, θ = 53.13°| Operation | Easier in |
|---|---|
| Addition, subtraction | Rectangular — add the parts |
| Multiplication | Polar — multiply magnitudes, add angles |
| Division | Polar — divide magnitudes, subtract angles |
| Powers and roots | Polar — De Moivre: (r e^(iθ))ⁿ = rⁿ e^(inθ) |
Where complex numbers are used
- AC circuits: impedance Z = R + jX combines resistance and reactance; engineers write j instead of i to avoid confusion with current. Voltage, current and impedance obey Ohm's law as complex numbers.
- Signal processing: the Fourier transform represents signals as sums of complex exponentials; filters are designed by placing poles and zeros in the complex plane.
- Control systems: stability is decided by where the roots of a polynomial lie relative to the imaginary axis.
- Geometry and graphics: multiplying by e^(iθ) rotates a point by θ; quaternions extend the idea to 3D.
- Quantum mechanics: wave functions and amplitudes are complex-valued by nature.
- Solving equations: every polynomial of degree n has exactly n complex roots.
Quick facts
- The conjugate of a + bi is a − bi; a number times its conjugate is |z|² — always real and non-negative.
- Powers of i cycle: i, −1, −i, 1, i, …
- √(−1) = i, so √(−16) = 4i; the quadratic formula gives complex roots when the discriminant is negative.
- Euler's identity, e^(iπ) + 1 = 0, links five fundamental constants through complex exponentials.
- Complex numbers cannot be ordered — there is no meaningful “greater than”, only comparison of magnitudes.
Frequently asked questions
How do I enter a purely real or purely imaginary number?
Set the other part to zero: 5 is 5 + 0i, and 3i is 0 + 3i.
What angle convention is used?
The angle is measured counter-clockwise from the positive real axis, from −180° to 180°, computed with atan2 so all four quadrants are correct.
Can I compute powers or roots?
Multiply repeatedly for small integer powers. For general powers and roots, convert to polar form (shown in the results) and apply De Moivre's formula by hand.
Why do engineers write j instead of i?
In electrical engineering i already denotes current, so the imaginary unit is written j. The mathematics is identical.
Is the magnitude the same as absolute value?
Yes — |z| generalises absolute value to the complex plane as the distance from the origin.
Are my values stored?
No. The calculation runs in your browser.