Complex Numbers Calculator

The Complex Numbers Calculator adds, subtracts, multiplies, or divides two complex numbers in the form a + bi. Enter the real and imaginary parts of z₁ and z₂, pick an operation, and it instantly returns the result along with its modulus, conjugate, and polar form — plus a step-by-step derivation showing exactly how the answer was reached.

It's built for algebra and engineering students checking homework, teachers preparing worked examples, and anyone working with electrical circuits, signal processing, or complex analysis who needs a quick, reliable computation.

How to use it

1. Choose the operation: add, subtract, multiply, or divide.
2. Enter the real part a and imaginary part b of z₁.
3. Enter the real part c and imaginary part d of z₂.
4. Read the Result, Modulus, Conjugate, and Polar form fields, plus Step-by-Step Solution for the full derivation.

Results update automatically as you type. For example, adding z₁ = 3 + 4i and z₂ = 1 + 2i returns 4 + 6i.

The four operations

Add:      (a + c) + (b + d)i
Subtract: (a − c) + (b − d)i
Multiply: (ac − bd) + (ad + bc)i
Divide:   ((ac + bd) / (c² + d²)) + ((bc − ad) / (c² + d²))i

Multiplication and division both rely on i² = −1: multiplying (a + bi)(c + di) expands to ac + adi + bci + bdi², and the bdi² term flips to −bd once i² is replaced with −1. Division multiplies both sides by z₂'s conjugate (c − di) to clear the imaginary part from the denominator, leaving c² + d² — the squared modulus of z₂.

What are the modulus, conjugate, and polar form?

• Modulus — the distance from the origin to the point (re, im) on the complex plane: |z| = √(re² + im²).
• Conjugate — the same real part with the imaginary part's sign flipped: re − im·i. Conjugates are what division uses to clear the denominator.
• Polar form — the same number expressed as a magnitude and angle instead of real/imaginary parts: r(cos θ + i sin θ), where r is the modulus and θ (in degrees, from −180° to 180°) is the angle from the positive real axis.

Why does division reject z₂ = 0 + 0i?

Dividing by zero is undefined for complex numbers the same way it is for real numbers — the denominator c² + d² would be zero, making the result infinite/undefined. The calculator flags this rather than silently returning Infinity or NaN.

Complex numbers also show up when solving equations that don't have real solutions — see the Quadratic Formula Solver, whose complex-conjugate-root branch uses this same a + bi notation.

Privacy

This calculator runs entirely in your browser. Your numbers are never uploaded, logged, or stored — the computation happens locally on your device.