Multiply Complex Numbers Calculator
Multiply complex numbers in rectangular or polar form with step-by-step solutions
Complex Number Multiplication
First Complex Number (z₁)
Second Complex Number (z₂)
Quick Examples:
Multiplication Result
Input Numbers
z₁ = 0
z₂ = 0
Rectangular Form
Imaginary: 0.000000
Polar Form
Phase: 0.000000 radians
Step-by-Step Solution
Method 1: Rectangular Form Multiplication
(0.000 + 0.000i) × (0.000 + 0.000i)
= 0.000 × 0.000 + 0.000 × 0.000i + 0.000i × 0.000 + 0.000i × 0.000i
= 0.000 + 0.000i + 0.000i + 0.000i²
= 0.000 + 0.000i + 0.000i + 0.000(-1)
= 0.000 - 0.000 + (0.000 + 0.000)i
= 0.000 + 0.000i
Formula: (a + bi) × (c + di) = (ac - bd) + (ad + bc)i
Remember: i² = -1
Method 2: Polar Form Multiplication
0.000 × e^(i×0.000) × 0.000 × e^(i×0.000)
= (0.000 × 0.000) × e^(i×(0.000 + 0.000))
= 0.000 × e^(i×0.000)
Formula: |z₁| × e^(iφ₁) × |z₂| × e^(iφ₂) = |z₁| × |z₂| × e^(i(φ₁ + φ₂))
Multiply magnitudes, add phases
Complex Number Forms
Rectangular Form
z = a + bi
a: real part
b: imaginary part
i: imaginary unit (i² = -1)
Polar Form
z = r × e^(iφ)
r: magnitude (|z|)
φ: phase (arg(z))
e^(iφ) = cos(φ) + i×sin(φ)
Multiplication Rules
Rectangular Method
(a + bi) × (c + di) = (ac - bd) + (ad + bc)i
Polar Method
r₁e^(iφ₁) × r₂e^(iφ₂) = r₁r₂e^(i(φ₁+φ₂))
Key Property
Polar form makes multiplication easier: just multiply magnitudes and add phases!
Special Cases
i × i = -1
Fundamental imaginary unit property
z × z* = |z|²
Complex number times its conjugate
z × 1 = z
Multiplicative identity
z × 0 = 0
Zero property
Form Conversions
Rectangular → Polar
r = √(a² + b²)
φ = atan2(b, a)
Polar → Rectangular
a = r × cos(φ)
b = r × sin(φ)
Mathematical Theory
Why Two Methods Work
Complex number multiplication can be performed in two equivalent ways because both rectangular and polar forms represent the same mathematical objects in the complex plane.
Geometric Interpretation
In the complex plane, multiplication corresponds to:
- • Scaling by the product of magnitudes
- • Rotating by the sum of angles
- • This explains why polar form is often easier
Applications
Engineering
AC circuit analysis, signal processing, control systems
Physics
Quantum mechanics, wave functions, oscillations
Mathematics
Complex analysis, Fourier transforms, fractals