Complex Numbers Calculator
Perform addition, subtraction, multiplication and division of complex numbers with this free online calculator. Enter complex numbers in standard form (a+bi) and get step-by-step solutions.
Results
| Standard Form | - |
|---|---|
| Polar Form | - |
| Magnitude (r) | - |
| Phase (φ) | - |
Polar Form Explanation
The polar form represents a complex number in terms of its magnitude (r) and phase angle (φ).
r = √(a² + b²)
φ = atan2(b, a)
Polar form: r(cosφ + i·sinφ)
About Complex Numbers Calculator
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 that satisfies the equation i² = -1. Complex numbers extend the concept of one-dimensional real numbers to the two-dimensional complex plane.
Complex Number Operations
This calculator performs the four basic operations on 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
(a + bi) ÷ (c + di) = [(ac + bd)/(c² + d²)] + [(bc - ad)/(c² + d²)]i
Complex Number Forms
Complex numbers can be represented in different forms:
Standard Form
a + bi (rectangular form)
Polar Form
r(cosφ + i·sinφ) where r is the magnitude and φ is the phase angle
Complex Number Properties
| Property | Description |
|---|---|
| Real Part | The 'a' in a + bi |
| Imaginary Part | The 'b' in a + bi (coefficient of i) |
| Complex Conjugate | a - bi (changes sign of imaginary part) |
| Magnitude (Modulus) | √(a² + b²) |
| Phase (Argument) | atan2(b, a) in radians |
Applications of Complex Numbers
Complex numbers are fundamental in many areas of mathematics, physics, and engineering:
- Electrical engineering (AC circuit analysis)
- Signal processing (Fourier transforms)
- Control theory (system stability analysis)
- Quantum mechanics (wave functions)
- Fluid dynamics (potential flow)