Complex Number Calculator — Add, Multiply, Modulus & Polar

Use this free complex number calculator to compute with numbers of the form a + bi. Enter A and B, choose add, subtract, multiply, divide, modulus, conjugate or polar form, and you’ll get the result in standard form with the worked algebra and an Argand-plane diagram.

How to use the calculator

1. Enter the real and imaginary parts of A (and B when needed).
2. Choose an operation.
3. Read the result, the modulus and argument, and see the numbers plotted.

Key rules

Multiplication uses (a + bi)(c + di) = (ac − bd) + (ad + bc)i. Division multiplies top and bottom by the conjugate of the denominator. The modulus is |a + bi| = √(a² + b²), and the polar form is r(cosθ + i sinθ).

The Ultimate Algebra Bundle: From Pre-Algebra to Algebra II

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

Frequently asked questions

How do you multiply complex numbers?

Use FOIL and replace i² with −1: (3+4i)(1−2i) = 3 − 6i + 4i − 8i² = 11 − 2i.

How do you divide complex numbers?

Multiply the numerator and denominator by the conjugate of the denominator, which makes the denominator a real number.

The Ultimate Middle School Math Bundle: Grades 6-8

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

What is the modulus of a complex number?

Its distance from the origin in the complex plane: |a + bi| = √(a² + b²). For 3 + 4i it’s 5.

The Ultimate Elementary School Math Bundle: Grade 3-5

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

Read the full lesson: learn the method step by step.

Pre-Algebra for Beginners 2026: The Ultimate Step by Step Guide to Preparing for the Pre-Algebra Test

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.