Complex Number Calculator

Complex Number Calculator

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

An online Complex Number Calculator allows you to perform basic mathematical operations to simplify complex expressions. This free imaginary number calculator can quickly handle addition, subtraction, multiplication, division, and more, providing step-by-step solutions. It also supports operations like finding the polar form, conjugate, modulus, and inverse.

What is a Complex Number?

A complex number is a combination of a real number and an imaginary number. It is expressed as z = x + yi, where:

x is the real part
y is the imaginary part
i (also called iota) is the imaginary unit, defined as i² = -1

Complex numbers can be added, subtracted, multiplied, or divided using a complex number calculator.

Basic Operations with Complex Numbers

Think of complex numbers as vectors when performing operations. Let:

A = x + yi and B = m + ni

Addition:

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

Where 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 only valid if B ≠ 0.

Operations Using Polar Form

Complex numbers can also be expressed 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 power: (|z₁| * exp(iφ₁))ᶜ = |z₁|ᶜ * exp(iφ₁c)
General formula using polar form: AB = |z₁||z₂| * exp(i(φ₁ + φ₂))
Complex logarithm: ln(z) = ln(|z|) + iφ

For function composition, you may also use an online Composite Function Calculator.

How the Complex Number Calculator Works

Input:

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

Output:

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

FAQ

Is every number complex?

Yes. Every real number is also a complex number (with imaginary part = 0). However, not every complex number is real. The set of complex numbers is represented by Z ∈ C. The set of purely imaginary numbers is Z ∈ C − R.

Where are complex numbers used in real life?

Complex numbers are used in electrical engineering to calculate voltage, current, and resistance in AC circuits (alternating current).

How does a complex number work?

A complex number is a + bi, where the real part a is 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 perform addition, subtraction, multiplication, and division of complex numbers. It allows you to input purely real or purely imaginary numbers (by setting the other part to 0) and also provides the polar form of the expression.

References

Wikipedia: Cartesian and Polar Complex Plane
Brilliant: Complex Numbers, Imaginary Unit, and Arithmetic
Varsity Tutors: Complex Plane and Imaginary Numbers