Complex Number Calculator
Free complex number calculator. Add, subtract, multiply, and divide complex numbers (a + bi). Compute modulus, argument, conjugate, reciprocal, and square.
Loading tool...
What is Complex Number Calculator?
The Complex Number Calculator performs arithmetic on complex numbers of the form a + bi. Enter two complex numbers and instantly get their sum, difference, product, quotient, square, and reciprocal. The tool also computes the modulus (absolute value), argument (phase angle in degrees and radians), and complex conjugate for each number. Every calculation happens locally in your browser — no data is ever uploaded.
Key Benefits
- Perform all four complex arithmetic operations — add, subtract, multiply, and divide — in one place
- Get the modulus, argument, and conjugate automatically to understand the geometry of each number
- Handles edge cases like division by zero and the reciprocal of zero gracefully
- Clean output formatting for pure real, pure imaginary, and general a + bi forms
- Ideal for checking algebra, physics, and engineering homework involving complex numbers
Common Use Cases
- •Students verifying complex arithmetic in algebra and precalculus courses
- •Engineers computing impedance, phasors, and AC circuit values
- •Physicists working with quantum mechanics wave functions
- •Programmers implementing signal processing and control systems
- •Anyone learning about the complex plane and polar representation
How to Use the Complex Number Calculator
- Enter z₁: Type the real part (a) and imaginary part (b) of the first complex number z₁ = a + bi.
- Enter z₂: Type the real part (c) and imaginary part (d) of the second complex number z₂ = c + di.
- View arithmetic: The tool instantly shows z₁ + z₂, z₁ − z₂, z₁ × z₂, z₁ ÷ z₂, z₁², and 1/z₁.
- Check properties: Review the modulus, argument (phase angle), conjugate, and radians for each number, then copy all results.
Key Features
- Full complex arithmetic: add, subtract, multiply, and divide any two complex numbers (a + bi).
- Computes modulus |z|, argument (phase angle in degrees and radians), and complex conjugate.
- Supports reciprocal (1/z) and square (z²) with safe handling of the zero case.
- Clean, readable output formatting — handles pure real, pure imaginary, and general cases.
- One-click copy of all results for homework, spreadsheets, and notes.
- 100% client-side — no sign-up, no server calls, works offline.
Frequently Asked Questions
What is a complex number?
A complex number has the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = −1. The number a is called the real part and b is the imaginary part. Complex numbers extend the real numbers to solve equations like x² + 1 = 0 that have no real solution.
How do I multiply two complex numbers?
Use the distributive rule and i² = −1. For (a + bi)(c + di), expand to get ac + adi + bci + bdi², then substitute i² = −1: the real part is ac − bd and the imaginary part is ad + bc. So (a + bi)(c + di) = (ac − bd) + (ad + bc)i.
What are the modulus and argument of a complex number?
The modulus |z| = √(a² + b²) is the distance from the origin to the point (a, b) in the complex plane. The argument arg(z) = atan2(b, a) is the angle (in radians or degrees) from the positive real axis to the line from the origin to (a, b). Together, the modulus and argument give the polar form z = |z|(cos θ + i sin θ).
Why does division by zero give an error?
Dividing by a complex number z₂ means multiplying by 1/z₂ = conj(z₂) / |z₂|². When z₂ = 0, the denominator |z₂|² is zero, so division is undefined — just like with real numbers. The calculator shows "undefined" and flags the result so you know the operation is not valid.