Complex Number Calculator
Perform arithmetic operations with complex numbers in rectangular and polar forms
Complex Number Operations
Enter two complex numbers in the form a + bi to perform various operations:
First Complex Number (z₁)
z₁ = 3 + 4i
Polar: 5 ∠ 53.130102°
Second Complex Number (z₂)
z₂ = 1 + 2i
Polar: 2.236068 ∠ 63.434949°
Results
Addition (z₁ + z₂)
Subtraction (z₁ - z₂)
Multiplication (z₁ × z₂)
Division (z₁ ÷ z₂)
Power (z₁^z₂)
ln(z₁)
√z₁
Conjugate z₁̄
Conjugate z₂̄
Step-by-Step Solutions
Addition
Add corresponding real and imaginary parts:
Subtraction
Subtract corresponding real and imaginary parts:
Multiplication
Use FOIL method: (a+bi)(c+di) = ac + adi + bci + bdi²:
Division
Multiply by conjugate of denominator:
Complex Number Properties
Modulus Property
The modulus of a product equals the product of moduli
Verification: 11.180340 = 11.180340
Conjugate Product
A complex number times its conjugate gives a real number
Verification: 3 + 4i × 3 - 4i = 25.000000
Polar Multiplication
In polar form, multiply moduli and add arguments
Verification: 11.18 ∠ 116.57°
Example Problems
Example 1
Example 2
Example 3
Example 4
Quick Reference
Operation Formulas
Polar Form Operations
Multiplication: multiply moduli, add arguments
Division: divide moduli, subtract arguments
Powers: raise modulus to power, multiply argument
Roots: take nth root of modulus, divide argument by n
Applications
Understanding Complex Numbers
What are Complex Numbers?
Complex numbers extend the real number system by introducing the imaginary unit i, where i² = -1. A complex number z = a + bi consists of a real part (a) and an imaginary part (b), enabling solutions to equations like x² + 1 = 0.
Rectangular vs. Polar Form
Complex numbers can be represented in rectangular form (a + bi) or polar form (r∠θ). The polar form is particularly useful for multiplication, division, and exponentiation, as these operations become simpler when working with magnitudes and angles.
Complex Plane Visualization
Complex numbers can be visualized on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. This geometric interpretation helps understand operations like addition (vector addition) and multiplication (rotation and scaling).
Key Properties
Conjugate Properties
- • z · z̄ = |z|² (always real and non-negative)
- • (z₁ + z₂)̄ = z̄₁ + z̄₂
- • (z₁ · z₂)̄ = z̄₁ · z̄₂
Modulus Properties
- • |z₁ · z₂| = |z₁| · |z₂|
- • |z₁/z₂| = |z₁|/|z₂|
- • |z₁ + z₂| ≤ |z₁| + |z₂| (triangle inequality)
Euler's Formula
e^(iθ) = cos θ + i sin θ
This connects complex exponentials with trigonometric functions.
Real-World Applications
AC Electrical Circuits
Complex numbers represent AC voltages and currents, where the real part represents the resistive component and the imaginary part represents the reactive component.
Signal Processing
The Fourier transform uses complex numbers to analyze frequency components of signals, enabling applications in audio processing, image compression, and communications.
Quantum Mechanics
Wave functions in quantum mechanics are complex-valued, and the squared magnitude gives the probability density of finding particles in specific states.