Engineering Math

Complex Number Calculator

Perform arithmetic on complex values, then inspect magnitude and argument to connect algebraic output with geometric meaning.

Complex Input

Result

How to use

Enter two complex numbers in a+bi format.
Select the operation you want.
Click Calculate and review value, magnitude, and argument.

How this concept works

Complex numbers combine a real part and an imaginary part. Multiplication and division follow algebraic rules where i² = -1, allowing rotation and scaling behavior in the complex plane.

Quick examples

(3+4i) + (1-2i) → 4+2i
(3+4i) × (1-2i) → 11-2i

FAQ

Can I enter decimals?

Yes, decimals are supported, for example 2.5-3.1i.

Why do I see radians in argument?

Argument is provided in radians by default for mathematical consistency.

What happens when dividing by 0+0i?

The tool blocks that operation and returns an explicit division-by-zero complex error.

Concept overview

What complex numbers represent

A complex number has two components: a real axis value and an imaginary axis value. Instead of seeing a+bi as only symbolic math, you can interpret it as a point or vector in the complex plane.

This is why complex arithmetic is so useful in circuits, signal processing, control systems, and graphics. Multiplication can be interpreted as rotation and scaling, while magnitude and argument capture distance and angle.

Input format rules

Required format: a+bi or a-bi
Decimals are allowed for both parts
Include both real and imaginary components explicitly

Manual conversion

Worked multiplication example

Compute (2+3i)(4-5i):

Multiply outer terms: 2×4 = 8
Cross terms: 2×(-5i) + 3i×4 = -10i + 12i = 2i
Imaginary product: 3i×(-5i) = -15i² = +15
Total: 8 + 15 + 2i = 23 + 2i

The calculator should output this exact algebraic form. Then check magnitude/argument for geometric insight.

Common mistakes

Forgetting the i suffix in imaginary part
Using 3+i instead of 3+1i
Confusing degrees and radians for argument interpretation

Extended FAQ

Complex calculator questions

Can this tool show polar form directly?

It currently shows magnitude and argument, which are the core polar components. Polar string formatting can be derived directly from those values.

Why is argument sometimes negative?

Angle direction depends on the quadrant; negative values represent clockwise orientation from the positive axis.

Are very small decimals supported?

Yes, but extremely small values may display rounded due to floating-point precision limits.

Can I use this for AC circuit phasors?

Yes. It is suitable for quick rectangular-form arithmetic and angle/magnitude checks.

Where can I study the basics first?

Use the Learn page for conceptual background before advanced workflows.