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

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

This complex number calculator handles all four basic arithmetic operations on complex numbers entered in rectangular form (a + bi). Results are displayed in both rectangular form and polar form with magnitude and argument, making it ideal for electrical engineering, signal processing, and mathematics coursework.

How it works

Enter the real and imaginary parts of the first complex number (z1).
Enter the real and imaginary parts of the second complex number (z2).
Select the operation: addition, subtraction, multiplication, or division.
The result appears instantly in rectangular form (a + bi).
The magnitude (r) and argument (θ) in both degrees and radians are shown in polar form.

Use cases

Solve AC circuit analysis problems involving impedance and phasors.
Check complex arithmetic homework answers in seconds.
Convert between rectangular and polar representations.
Work through signal processing calculations involving complex exponentials.
Verify results from manual complex number multiplication or division.
Explore how operations affect magnitude and phase angle.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu