Complex Number Calculator
Perform arithmetic operations on complex numbers, convert between rectangular and polar forms, calculate powers using De Moivre's theorem, and find all nth roots.
Complex Arithmetic
Add, subtract, multiply, divide complex numbers or find conjugate
How to Use the Complex Number Calculator
Enter complex numbers in rectangular form (a + bi) by providing the real and imaginary parts separately. Select the operation type and click Calculate to see results in multiple formats.
Calculator Modes
Arithmetic Operations
Perform basic operations on two complex numbers:
- Addition: (a + bi) + (c + di) = (a+c) + (b+d)i
- Subtraction: (a + bi) - (c + di) = (a-c) + (b-d)i
- Multiplication: (a + bi)(c + di) = (ac-bd) + (ad+bc)i
- Division: Uses the conjugate to rationalize the denominator
- Conjugate: The conjugate of (a + bi) is (a - bi)
Form Conversion
Convert between three representations of complex numbers:
- Rectangular: a + bi (Cartesian form)
- Polar: r ∠ θ where r is magnitude and θ is angle
- Exponential: re^(iθ) using Euler's formula
Powers (De Moivre's Theorem)
Raise a complex number to any integer power. De Moivre's theorem states: (r∠θ)^n = r^n ∠ nθ. This makes power calculations much simpler in polar form.
Roots
Find all n distinct nth roots of a complex number. A complex number has exactly n nth roots, evenly spaced around a circle in the complex plane.
What is a Complex Number?
A complex number has the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit (i² = -1). Complex numbers extend the real numbers and are essential in many areas of mathematics, physics, and engineering.
Common Applications
- Electrical engineering (AC circuit analysis)
- Signal processing and Fourier transforms
- Quantum mechanics
- Control systems
- Fractal geometry (Mandelbrot set)
Key Formulas
- Magnitude: |z| = √(a² + b²)
- Angle: θ = arctan(b/a)
- Euler's formula: e^(iθ) = cos(θ) + i·sin(θ)