Complex Number Calculator
Calculate with complex numbers including addition, subtraction, multiplication, division, and conversions between rectangular and polar forms. Perfect for engineering, physics, and advanced mathematics.
Complex Number Operations
Common Complex Number Examples
Click on these links to see instant calculations with common complex numbers:
Complex Numbers
Complex numbers extend real numbers by including imaginary numbers. They are essential in engineering, physics, and advanced mathematics for representing quantities with both magnitude and phase.
Complex Number Forms
Polar Form: z = r(cos θ + i sin θ) = r∠θ
Where: a = real part, b = imaginary part
r = |z| = √(a² + b²) (magnitude)
θ = arg(z) = arctan(b/a) (argument)
Complex Number Operations
| Operation | Example | Result | Form |
|---|---|---|---|
| Addition | (3+4i) + (1+2i) | 4+6i | Rectangular |
| Subtraction | (5+3i) - (2+1i) | 3+2i | Rectangular |
| Multiplication | (2+3i) × (1+4i) | -10+11i | Rectangular |
| Division | (6+8i) ÷ (3+4i) | 2 | Rectangular |
| Polar Conversion | 3+4i | 5∠53.13° | Polar |
- Engineering: AC circuit analysis, signal processing, and control systems
- Physics: Quantum mechanics, wave functions, and electromagnetic fields
- Mathematics: Advanced calculus, differential equations, and number theory
- Computer Graphics: 2D rotations, transformations, and fractal generation
- Signal Processing: Fourier transforms and frequency domain analysis
Frequently Asked Questions
What are complex numbers?
Complex numbers are numbers that include both real and imaginary parts, written as a + bi where 'a' is the real part, 'b' is the imaginary part, and 'i' is the imaginary unit (√-1).
How do you add complex numbers?
To add complex numbers, add the real parts together and add the imaginary parts together. For example: (3 + 4i) + (2 + 5i) = (3+2) + (4+5)i = 5 + 9i.
What is the polar form of a complex number?
The polar form represents a complex number as r(cos θ + i sin θ) or r∠θ, where r is the magnitude (modulus) and θ is the argument (angle). It's useful for multiplication and division of complex numbers.