Complex Number Calculator – Add, Subtract, Multiply, Divide
Calculate complex number operations with magnitude and phase
Table of Contents
How to Use
- Enter the real and imaginary parts of the first complex number
- Select the operation (add, subtract, multiply, or divide)
- Enter the real and imaginary parts of the second complex number
- Click calculate to see the result with magnitude and phase
What are Complex Numbers?
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit satisfying i² = -1. The real part is 'a' and the imaginary part is 'b'.
Complex numbers extend the concept of one-dimensional number lines to a two-dimensional complex plane using the horizontal axis for the real part and the vertical axis for the imaginary part.
Complex Number Operations
Operations on complex numbers follow these rules:
- 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: (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] / (c² + d²)
Key Properties
- Magnitude (modulus): |z| = √(a² + b²), the distance from the origin
- Argument (phase): arg(z) = atan2(b, a), the angle from the positive real axis
- Conjugate: z* = a − bi, reflection across the real axis
- Euler's formula: e^(iθ) = cos(θ) + i·sin(θ)
Applications
- Electrical engineering: AC circuit analysis and impedance
- Signal processing: Fourier transforms and frequency analysis
- Quantum mechanics: Wave functions and probability amplitudes
- Control theory: Transfer functions and system stability
- Fluid dynamics: Potential flow and conformal mapping
Frequently Asked Questions
- What is the imaginary unit i?
- The imaginary unit i is defined as the square root of -1. While no real number satisfies i² = -1, introducing i allows us to work with square roots of negative numbers and solve equations that have no real solutions.
- How do you divide complex numbers?
- To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator. This eliminates the imaginary part in the denominator, leaving a real number that you can then divide by.
- What is the difference between magnitude and argument?
- Magnitude (or modulus) is the distance of the complex number from the origin in the complex plane. Argument (or phase) is the angle formed with the positive real axis. Together they form the polar representation of a complex number.