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Math

Complex Number Calculator

Add, subtract, multiply or divide complex numbers and inspect the result in rectangular and polar form.

Exact method explained belowEditable inputs
Complex plane
Complex plane

Calculate in rectangular form, then inspect magnitude, phase and position.

See the method

Enter real and imaginary parts

Runs on your device

Complex result

Your result

Let’s calculate.

Use the Calculate button to see your result.

Review the method below for assumptions and conventions.

How to use this tool

  1. Enter a: real part, a: imaginary part, b: real part, b: imaginary part, operation.
  2. Select Calculate to view the result.
  3. Check the method and assumptions below before using the result.

Input reference and sample values

Use the units and choices shown for each field. Sample values are provided for practice, not as recommendations.

InputFormat and unitsSample
A: real partNumeric value.3
A: imaginary partNumeric value.4
B: real partNumeric value.1
B: imaginary partNumeric value.-2
OperationChoose A × B / A ÷ B / A + B / A − B.A × B

Result for these sample inputs

11 − 2i
Real part
11
Imaginary part
-2
Magnitude
11.1803399
Phase in degrees
-10.3048465

Select Try the example in the form to load these values and calculate them again. This replaces the current form inputs. Compare the result with the method and worked explanation below.

The method, explained

Use i² = −1. For division, multiply numerator and denominator by the conjugate of B. Magnitude = hypot(real, imaginary); phase = atan2(imaginary, real).

A WORKED EXAMPLE

Using a: real part = 3, a: imaginary part = 4, b: real part = 1, b: imaginary part = -2, operation = multiply, the result is 11 − 2i. Change these example inputs to match your task; use the method above to check each step.

What to keep in mind

Finite real input components only. Phase uses the principal angle between −180 and 180 degrees; the phase of zero is undefined. Finite-precision arithmetic is used. Check the domain and the stated convention before using an approximation.

Common questions

What is rectangular form?

A complex number is written as a + bi, with a its real part and b its imaginary coefficient. Enter b alone in an imaginary-part input, without the letter i.

Does a negative imaginary part change the magnitude?

Magnitude uses the squares of both components and is nonnegative. The signs affect the position and phase in the complex plane.

Methodology maintained by ClarityKit. How these tools are built and checked.