Complex Number Calculator
Arithmetic on complex numbers, plus instant conversion between rectangular (a + bi) and polar (r ∠ θ) form. This free online math calculator runs entirely in your browser — no signup, no data sent anywhere.
Inputs
Results
How It Works (Formula & Method)
Addition is component-wise: (a + bi) + (c + di) = (a + c) + (b + d)i. Multiplication uses the FOIL rule with i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. Division multiplies numerator and denominator by the conjugate of the denominator. Polar form represents a complex number by its magnitude r = √(a² + b²) and argument θ = atan2(b, a).
Worked Example
Below is a worked example using the calculator's default values. The same numbers are pre-filled in the form above so you can press Calculate and see the result without typing anything.
Inputs used:
- A: Real Part: 3
- A: Imaginary Part: 4
- B: Real Part: 1
- B: Imaginary Part: 2
With these inputs, the calculator computes the metrics shown in the Results panel. Change any value and press Calculate again to see how the result responds — the live widget and the chart both update instantly.
About the Complex Number Calculator
Complex numbers extend the real number line into a two-dimensional plane by adding the imaginary unit i, defined by i² = −1. They are essential in electrical engineering (AC circuit analysis), signal processing (Fourier transforms), quantum mechanics, and control theory.
How to Use This Calculator
Enter the real and imaginary parts of each complex number A and B in rectangular form: A = a_re + a_im · i. The calculator returns the sum, product, quotient (when defined), and the polar-form representation of A.
Tips & Considerations
- Polar form is convenient for multiplication and powers (de Moivre's formula): magnitudes multiply, angles add.
- The complex conjugate of a + bi is a − bi. Multiplying a complex number by its conjugate gives a real number: a² + b².
- Every nonzero complex number has exactly n distinct n-th roots, evenly spaced on a circle in the complex plane.
Frequently Asked Questions
What is i?
The imaginary unit, defined by i² = −1. It does not exist on the real number line but is a perfectly valid mathematical object with consistent algebraic rules.
Why are complex numbers useful?
They unify rotation and scaling into one operation (multiplication), which makes them indispensable in physics, electrical engineering, and signal processing.
Can I divide by zero in complex arithmetic?
No — division by 0 + 0i is still undefined. Our calculator handles this gracefully.
What does the Complex Number Calculator compute?
The Complex Number Calculator takes 4 input values and returns 4 results. Add, subtract, multiply, and divide complex numbers. Converts between rectangular (a + bi) and polar (r ∠ θ) form.