What is a complex number?

A complex number has the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit (i² = -1). For example, 3 + 4i has real part 3 and imaginary part 4.

Polar form expresses a complex number as r∠θ, where r is the magnitude (distance from origin) and θ is the argument (angle from positive real axis). Magnitude r = √(a² + b²) and argument θ = atan2(b, a).

Why can I not divide by zero?

Division by a complex number whose real and imaginary parts are both zero is undefined, just as division by zero is undefined for real numbers.

How is complex number multiplication done?

(a + bi)(c + di) = (ac − bd) + (ad + bc)i. Equivalently in polar form, magnitudes multiply and angles add.

What angle range does the argument use?

The argument θ is reported in the range (−180°, 180°] (principal argument), which corresponds to (−π, π] radians.

Can I enter negative imaginary parts?

Yes. Simply enter a negative number in the imaginary part field, such as −5. The calculator handles negative real and imaginary parts correctly.

Complex Number Calculator

Name: Complex Number Calculator
Availability: InStock
Author: Nham Vu

Add, subtract, multiply, or divide two complex numbers and instantly see the result in both rectangular (a+bi) and polar (r∠θ) forms.

Enter Complex Numbers

First number (z1)

Real part (a)

Imaginary part (b)

Operation

Second number (z2)

Real part (c)

Imaginary part (d)

Result

Enter values and press Calculate

Rectangular Form

—

Real part—

Imaginary part—

Polar Form

—

Magnitude |z|—

Argument (deg)—

Argument (rad)—

Operation Formulas

Add: (a+c) + (b+d)i

Sub: (a−c) + (b−d)i

Mul: (ac−bd) + (ad+bc)i

Div: (ac+bd)/(c²+d²) + (bc−ad)/(c²+d²)i

Summary