Complex Number Calculator
Perform complex number operations with detailed calculations and visualizations
Complex Operations
Complex Number Results
z₁
3.000 + 4.000i
z₂
1.000 + 2.000i
Understanding Complex Numbers
What are Complex Numbers?
Complex numbers extend the real number system to include solutions to equations that have no real solutions. A complex number is expressed as a + bi, where a is the real part and b is the imaginary part, and i is the imaginary unit defined as i² = -1.
Complex Number Notation:
z = a + bi where a, b ∈ ℝ and i² = -1
Real part: Re(z) = a
Imaginary part: Im(z) = b
Example: z = 3 + 4i has Re(z) = 3, Im(z) = 4
The Imaginary Unit
The imaginary unit i is defined as the square root of -1. This definition allows us to solve equations like x² + 1 = 0, which has no real solutions but two complex solutions: x = i and x = -i.
- •Definition: i² = -1
- •Powers: i³ = -i, i⁴ = 1, i⁵ = i, ...
- •Geometric interpretation: 90° rotation in complex plane
- •Applications: Electrical engineering, quantum mechanics
Complex Number Operations
Addition: (a + bi) + (c + di) = (a+c) + (b+d)i
Add real parts and imaginary parts separately
Multiplication: (a + bi)(c + di) = (ac-bd) + (ad+bc)i
Use distributive property and i² = -1
Division: (a + bi)/(c + di) = [(ac+bd) + (bc-ad)i]/(c²+d²)
Multiply by conjugate of denominator
Modulus: |a + bi| = √(a² + b²)
Distance from origin in complex plane
Real-World Applications
Electrical Engineering
AC circuit analysis
- • Impedance calculations
- • Phasor analysis
- • Power systems
- • Signal processing
Quantum Mechanics
Wave functions
- • Schrödinger equation
- • Quantum states
- • Probability amplitudes
- • Entanglement
Signal Processing
Fourier analysis
- • Frequency domain
- • Digital filters
- • Modulation
- • Spectral analysis
Control Systems
System stability
- • Transfer functions
- • Pole-zero analysis
- • Frequency response
- • Feedback systems
Fluid Dynamics
Flow analysis
- • Potential flow
- • Vortex dynamics
- • Complex potentials
- • Conformal mapping
Mathematics
Advanced concepts
- • Complex analysis
- • Residue theorem
- • Contour integration
- • Riemann surfaces
The Complex Plane
Geometric Representation
Complex numbers can be visualized as points in a 2D plane called the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.
Rectangular Form: z = a + bi
Point (a, b) in the plane
Polar Form: z = r(cos θ + i sin θ)
Distance r from origin, angle θ from positive real axis
Exponential Form: z = re^(iθ)
Using Euler's formula: e^(iθ) = cos θ + i sin θ
Special Points and Regions
Origin (0, 0)
Zero complex number, additive identity
Real Axis (y = 0)
Pure real numbers, imaginary part = 0
Imaginary Axis (x = 0)
Pure imaginary numbers, real part = 0
Unit Circle |z| = 1
All complex numbers with modulus 1
Euler's Formula
e^(iθ) = cos(θ) + i·sin(θ)
This beautiful formula connects exponential functions with trigonometry through complex numbers. It's considered one of the most important formulas in mathematics.
When θ = 0:
e^(i·0) = 1 + 0i = 1
When θ = π:
e^(iπ) = -1 + 0i = -1
When θ = π/2:
e^(iπ/2) = 0 + 1i = i