Complex Number Calculator

This complex number calculator adds, subtracts, multiplies, and divides numbers of the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit (√-1). It is built for students, teachers, and engineers who need an accurate answer in seconds.

Every result is shown in three notations at once — rectangular (a + bi), polar (r∠θ), and exponential (re^iθ) — alongside the modulus, argument, and conjugate, plus an Argand diagram that plots z₁, z₂, and the result as vectors.

Private by design: every calculation runs entirely in your browser. The numbers you enter and the results you get are never uploaded to any server.

How to Use the Complex Number Calculator

Choose an operation

Pick a tab at the top: Add, Subtract, Multiply, or Divide. The operation symbol between z₁ and z₂ updates to match.

Enter the two numbers

For each number, type the real part in the first field and the imaginary part in the second. To enter 3 + 4i, type 3 then 4; for 3 − 4i, type 3 then −4.

Adjust the settings

Optionally set Decimals to 2, 4, 6, or 8 places and switch the angle between Deg and Rad. Changing a setting re-runs the current calculation instantly.

Calculate and read the result

Click Calculate or press Enter in any input. You get the rectangular answer up top, the polar and exponential forms, the modulus, argument, and conjugate, and the Argand diagram. Use the copy button to grab the result.

Features

Four Basic Operations

Add, subtract, multiply, and divide any two complex numbers. Division uses the conjugate to rationalize the denominator automatically.

Multiple Output Formats

See each answer in rectangular (a + bi), polar (r∠θ), and exponential (re^iθ) form at the same time, with no manual conversion.

Complex Number Properties

Instantly read the modulus |z|, the argument arg(z), and the conjugate z̄ of the result without extra steps.

Argand Diagram Visualization

Plot z₁ (blue), z₂ (green), and the result (purple) as vectors on the complex plane, each with coordinate labels. Collapse the diagram any time.

Customizable Settings

Choose 2, 4, 6, or 8 decimal places and switch the angle display between degrees and radians to match your work.

Quick Examples

Load a ready-made example with one click to fill in values, pick the operation, and see the calculation worked out.

Frequently Asked Questions

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 defined as √-1. Complex numbers extend the real number system and appear throughout mathematics, physics, and engineering.

How do you add complex numbers?

Add the real parts together and the imaginary parts together: (a + bi) + (c + di) = (a + c) + (b + d)i. Subtraction works the same way, subtracting each part separately. In the calculator, enter the two numbers and pick the Add or Subtract tab.

How do you divide complex numbers?

Multiply the numerator and denominator by the conjugate of the denominator, which turns the denominator into a real number: (a + bi) / (c + di) = [(ac + bd) + (bc − ad)i] / (c² + d²). The calculator does this automatically when you choose the Divide tab.

How do you convert a complex number to polar form?

Polar form is r∠θ, where r is the modulus √(a² + b²) and θ is the argument atan2(b, a). The calculator shows the polar form of every result automatically; use the Deg/Rad toggle to display the angle in degrees or radians.

What are the modulus and argument of a complex number?

The modulus |z| is the distance from the origin, |z| = √(a² + b²), and represents the magnitude of the number. The argument arg(z) is the angle measured counterclockwise from the positive real axis. Both appear in the Properties row after you calculate.

What is the conjugate of a complex number?

The conjugate of a + bi is a − bi — the reflection of the number across the real axis. Conjugates are used to divide complex numbers and to find the modulus, since |z|² = z × z̄. The calculator displays the conjugate of the result automatically.

What is the Argand diagram?

The Argand diagram (the complex plane) plots complex numbers graphically: the horizontal axis is the real part and the vertical axis is the imaginary part. Each number is drawn as a vector from the origin, so you can see how z₁, z₂, and the result relate geometrically.

Why can't I divide by zero?

Division by 0 + 0i is undefined, just like dividing by zero with real numbers. If you try it, the calculator shows an error message instead of a result.