Divide Complex Numbers Calculator
Calculate the division of two complex numbers in rectangular or polar form
Complex Number Division
Dividend (Z₁)
Divisor (Z₂)
Division Results: Z₁ ÷ Z₂
Step-by-Step Solution (Rectangular Form)
Given: Z₁ = 4 + 3i, Z₂ = 2 + 1i
Formula: (a + bi) ÷ (c + di) = [(ac + bd) + i(bc - ad)] ÷ (c² + d²)
1. Calculate ac = 4 × 2 = 8
2. Calculate bd = 3 × 1 = 3
3. Calculate bc = 3 × 2 = 6
4. Calculate ad = 4 × 1 = 4
5. Numerator real part: ac + bd = 8 + 3 = 11
6. Numerator imaginary part: bc - ad = 6 - 4 = 2
7. Denominator: c² + d² = 2² + 1² = 5
8. Real part: 11 ÷ 5 = 2.200000
9. Imaginary part: 2 ÷ 5 = 0.400000
Result: 2.2+0.4i
Polar Form Division (Much Simpler!)
Formula: |Z₁|e^(iφ₁) ÷ |Z₂|e^(iφ₂) = (|Z₁|/|Z₂|)e^(i(φ₁-φ₂))
1. Divide magnitudes: |Z₁|/|Z₂| = 5.000000/2.236068 = 2.236068
2. Subtract phases: φ₁ - φ₂ = 36.869898 - 26.565051 = 10.304846°
Result: 2.236068×e^(i10.304846°)
Example Problems
Example 1: Basic Complex Division
Problem: Divide (4 + 3i) ÷ (2 + i)
Solution:
• Using formula: (a + bi) ÷ (c + di) = [(ac + bd) + i(bc - ad)] ÷ (c² + d²)
• ac + bd = 4×2 + 3×1 = 8 + 3 = 11
• bc - ad = 3×2 - 4×1 = 6 - 4 = 2
• c² + d² = 2² + 1² = 4 + 1 = 5
• Answer: (11 + 2i) ÷ 5 = 2.2 + 0.4i
Example 2: Division by Pure Imaginary
Problem: Divide (3 + 4i) ÷ 2i
Solution:
• Multiply by conjugate: (3 + 4i) × (-2i) ÷ [2i × (-2i)]
• Numerator: (3 + 4i) × (-2i) = -6i - 8i² = -6i + 8 = 8 - 6i
• Denominator: 2i × (-2i) = -4i² = 4
• Answer: (8 - 6i) ÷ 4 = 2 - 1.5i
Example 3: Polar Form Division
Problem: Divide 5e^(i60°) ÷ 2e^(i30°)
Solution:
• Divide magnitudes: 5 ÷ 2 = 2.5
• Subtract phases: 60° - 30° = 30°
• Answer: 2.5e^(i30°) = 2.5(cos30° + i sin30°) ≈ 2.165 + 1.25i
Division Formulas
Rectangular Form
z₁/z₂ = [(ac+bd) + i(bc-ad)]/(c²+d²)
Multiply by conjugate of denominator
Polar Form
|z₁|e^(iφ₁) ÷ |z₂|e^(iφ₂) = (|z₁|/|z₂|)e^(i(φ₁-φ₂))
Divide magnitudes, subtract phases
Special Cases
• 1/i = -i
• z/z* = |z|²
• z/1 = z
Key Properties
Division by Zero
Cannot divide by 0 + 0i
Conjugate Method
Multiply by conjugate of denominator
Polar Advantage
Much simpler in polar form
Inverse Operation
Division is inverse of multiplication
Understanding Complex Number Division
What are Complex Numbers?
Complex numbers extend the real number system by including the imaginary unit i, where i² = -1. They can be represented in rectangular form (a + bi) or polar form (r×e^(iφ)), where r is the magnitude and φ is the phase angle.
Division Methods
There are two main approaches to dividing complex numbers: the conjugate method for rectangular form and the magnitude/phase method for polar form. The polar method is generally simpler and more intuitive.
Applications
- •Electrical engineering (AC circuit analysis)
- •Signal processing and communications
- •Quantum mechanics and physics
- •Control systems engineering
Mathematical Foundation
Complex Conjugate
The key to rectangular division is multiplying by the complex conjugate:
z̄ = a - bi (conjugate of a + bi)
This eliminates the imaginary part in the denominator.
Polar Form Advantage
In polar form, division becomes much simpler:
- • Divide the magnitudes
- • Subtract the phase angles
- • No complex algebra required!
Division Properties
- 1. |z₁/z₂| = |z₁|/|z₂|
- 2. arg(z₁/z₂) = arg(z₁) - arg(z₂)
- 3. (z₁/z₂)* = z₁*/z₂*
- 4. z/z̄ = |z|²