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Complex Number Calculator

Write down any arithmetic combination of complex numbers, and the calculator will simplify them, with the steps displayed.

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Our online Complex Number Calculator lets you quickly perform various operations on complex numbers. This free calculator can handle addition, subtraction, multiplication, division, and more, providing detailed step-by-step solutions. It also supports computing the conjugate, modulus, inverse, and polar form.

What is a Complex Number?

A complex number combines a real part with an imaginary part and is written as z = x + yi, where:

x is the real component
y is the imaginary component
i is the imaginary unit, defined as i² = -1

You can add, subtract, multiply, or divide complex numbers using a complex number calculator.

Basic Operations with Complex Numbers

Think of complex numbers like vectors when performing arithmetic. Suppose:

A = x + yi and B = m + ni

Addition:

A + B = (x + m) + (y + n)i

Here, Re(A + B) = x + m (real part) and Im(A + B) = y + n (imaginary part).

Subtraction:

A - B = (x - m) + (y - n)i

Multiplication and Division

Multiplication:

(x + yi)(m + ni) = (xm - yn) + (xn + ym)i

Division:

A / B = (x + yi) / (m + ni) = [(x + yi)(m - ni)] / [(m + ni)(m - ni)] = [(xm + yn) + (ym - xn)i] / (m² + n²)

Division is valid only if B ≠ 0.

Operations Using Polar Form

Complex numbers can also be represented in polar form: z = |z| * exp(iφ)

Multiplication: |z₁||z₂| * exp(i(φ₁ + φ₂))
Division: |z₁ / z₂| * exp(i(φ₁ - φ₂))

Complex Powers and Logarithms

Complex powers: (|z₁| * exp(iφ₁))ᶜ = |z₁|ᶜ * exp(iφ₁c)
Multiplication using polar form: AB = |z₁||z₂| * exp(i(φ₁ + φ₂))
Complex logarithm: ln(z) = ln(|z|) + iφ

You can also use an online Composite Function Calculator for function composition.

How the Complex Number Calculator Works

Input:

Enter the expression containing real and imaginary components.
Click the calculate button.

Output:

The calculator provides the inverse, conjugate, modulus, and polar form.
Step-by-step solutions for addition, multiplication, division, and other operations are displayed.

FAQ

Is every number complex?

Yes. All real numbers are also complex numbers (imaginary part = 0). Not all complex numbers are real. Complex numbers are denoted by Z ∈ C, while purely imaginary numbers are Z ∈ C - R.

Where are complex numbers used?

They are widely used in electrical engineering to calculate voltage, current, and impedance in AC circuits (alternating current).

How does a complex number work?

A complex number a + bi has the real part a plotted on the horizontal axis and the imaginary part b on the vertical axis in the complex plane.

Conclusion

Use this online Complex Number Calculator to easily add, subtract, multiply, and divide complex numbers. You can input purely real or purely imaginary numbers and get the polar form along with stepwise solutions.

References

Wikipedia: Complex Numbers in Cartesian and Polar Form
Brilliant: Complex Numbers and Arithmetic
Varsity Tutors: Imaginary Numbers and the Complex Plane

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