Division of Complex Numbers

Online calculator for dividing complex numbers with step-by-step explanation

Division Calculator

Division of Complex Numbers

The division of complex numbers is performed by multiplying by the conjugate of the denominator, which makes the denominator real.

Dividend (Numerator) z₁ = a + bi
+
i
Divisor (Denominator) z₂ = c + di
+
i
Calculation Result
Quotient z₁/z₂ =

Division - Properties

Basic Principle

Division is performed by multiplying by the conjugate of the denominator. This makes the denominator real.

Formula
\[\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{(c+di)(c-di)}\]
Denominator becomes real
\[(c+di)(c-di) = c^2 + d^2\]
Result
\[\frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2}i\]
Important Notes
  • Divisor must not be 0 + 0i
  • Multiply by conjugate of denominator
  • Denominator becomes real (c² + d²)
  • In polar form: divide magnitudes, subtract angles
Calculation Rules
  • \(\frac{z_1}{z_2} \cdot z_2 = z_1\) (Inverse of multiplication)
  • \(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\) (Magnitude of quotient)
  • \(\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)\) (Angle)
  • \(\frac{1}{z} = \frac{\overline{z}}{|z|^2}\) (Reciprocal)


Description of Division of Complex Numbers

The division of complex numbers is performed by a clever trick: multiplying the fraction by the conjugate of the denominator.

Problem Statement

Given: \(\frac{3+i}{1-2i}\)
Problem: Complex number in the denominator

Solution

Multiply by \(\overline{1-2i} = 1+2i\)
Denominator becomes real: \((1-2i)(1+2i) = 5\)

Step-by-Step Example

Calculation: \(\frac{3+i}{1-2i}\)
Step 1: Determine the conjugate

Denominator: \(1 - 2i\)

Conjugate: \(\overline{1-2i} = 1 + 2i\)

Step 2: Multiply
\[\frac{3+i}{1-2i} = \frac{(3+i)(1+2i)}{(1-2i)(1+2i)}\]
Step 3: Compute denominator

\((1-2i)(1+2i) = 1^2 - (2i)^2\)

\(= 1 - 4i^2 = 1 - 4(-1)\)

\(= 1 + 4 = 5\)

Step 4: Compute numerator

\((3+i)(1+2i)\)

\(= 3 + 6i + i + 2i^2\)

\(= 3 + 7i + 2(-1)\)

\(= 1 + 7i\)

Step 5: Result

\[\frac{1+7i}{5} = \frac{1}{5} + \frac{7}{5}i\]

Decimal: \(0.2 + 1.4i\)

Verification

Check: \((0.2 + 1.4i) \cdot (1 - 2i)\)
\(= 0.2 - 0.4i + 1.4i - 2.8i^2\)
\(= 0.2 + 1.0i + 2.8 = 3 + i\) ✓

Alternative Calculation Methods

Method 1: Standard form (Cartesian)
Formula:
\[\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{c^2+d^2}\]
Advantage: Direct computation with real and imaginary parts
Method 2: Polar form (easier!)
Formula:
\[\frac{r_1 e^{i\phi_1}}{r_2 e^{i\phi_2}} = \frac{r_1}{r_2} e^{i(\phi_1-\phi_2)}\]
Advantage: Easy division of magnitudes, subtraction of angles
Comparison: Example \(\frac{3+i}{1-2i}\)
Standard form:
Multiply by (1+2i)
Denominator: 1² + 2² = 5
Expand numerator
Result: \(\frac{1}{5} + \frac{7}{5}i\)
Polar form:
\(|3+i| = \sqrt{10}\), \(\phi_1 = \arctan(1/3)\)
\(|1-2i| = \sqrt{5}\), \(\phi_2 = \arctan(-2)\)
\(\frac{\sqrt{10}}{\sqrt{5}} = \sqrt{2}\), \(\phi = \phi_1 - \phi_2\)
Back to standard form

Division of Complex Numbers - Detailed Description

The complex conjugate

The complex conjugate \(\overline{z}\) is obtained by changing the sign of the imaginary part.

Definition:
• If \(z = a + bi\), then \(\overline{z} = a - bi\)
• Geometrically: reflection across the real axis
• Important property: \(z \cdot \overline{z} = |z|^2\)
• For division: \((c+di)(c-di) = c^2 + d^2\) (real!)

Permanence principle

According to the permanence principle, the arithmetic rules of real numbers should also hold for complex numbers. Therefore we extend fractions.

Why does this work?

Multiplying by \(\frac{\overline{z_2}}{\overline{z_2}}\) is effectively multiplying by 1, so the value doesn't change. The denominator, however, becomes real!

Practical applications

Division of complex numbers is used in many technical fields:

Applications:
• Electrical engineering: Impedance calculations (Z = U/I)
• Control engineering: Transfer functions
• Signal processing: Filter design
• Physics: Wave and vibration theory

Common mistakes

Caution!
  • Wrong: Cancel numerator and denominator separately
  • Right: Multiply by the conjugate
  • Wrong: Forget or miscompute \(i^2\) as +1
  • Right: Always use \(i^2 = -1\)
  • Note: Division by 0 + 0i is not defined!

Special cases

  • Division by a real number: \(\frac{a+bi}{c} = \frac{a}{c} + \frac{b}{c}i\)
  • Division by an imaginary number: \(\frac{a+bi}{di} = \frac{b}{d} - \frac{a}{d}i\)
  • Reciprocal: \(\frac{1}{a+bi} = \frac{a-bi}{a^2+b^2}\)
  • Division by itself: \(\frac{z}{z} = 1\) for \(z \neq 0\)
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