Division of Complex Numbers

Complex numbers are numbers with two parts: real and imaginary. They are represented as z = a + bi, where Re(z) = a and Im(z) = b are the real and imaginary parts, respectively. We use complex numbers to solve equations that cannot be solved with real numbers alone.

Let us look at some examples of complex number:

• z = 3 + 4i \(\to\) Re(z) = 3 and Im(z) = 4
   
• z = 7 − 2i \(\to\) Re(z) = 7 and Im(z) = -2
   
• z = −5 + 6i \(\to\) Re(z) = -5 and Im(z) = 6
   
• z = 2 − 9i \(\to\) Re(z) = 2 and Im(z) = -9

Complex numbers can be divided by using a method that involves the conjugate of the denominator.

Two complex numbers, \(z_1 = a_1 + iy_1\) and \(z_2 = a_2 + iy_2\), can be expressed as:

• \(\frac{4 + 6i}{2} = \frac{4 + 6i}{2 + 0i} = 2 + 3i\)
   
• \(\frac{3 + 3i}{1 + i} = 3\)
   
• \(\frac{10}{i} = \frac{10 + 0i}{0 + i} = -10i\)
   
• \(\frac{2 + 4i}{1 - i} = -1 + 3i\)
   
• \(\frac{4 - 2i}{3 + i} = 1 - i\)

Given two complex numbers \(z_1 = a + bi\) and \(z_2 = c + di\), the division is:

\(\frac{z_1}{z_2} = \left[ \frac{ac + bd}{c^2 + d^2} \right] + i \left[ \frac{bc - ad}{c^2 + d^2} \right]\)

Derivation:
The goal is to eliminate the imaginary unit i from the denominator. To do this, we use the property that a complex number multiplied by its conjugate yields a real number.

1. Set up the Fraction
Write the division as a fraction:

\(\frac{a + bi}{c + di}\)

2. Rationalize the Denominator
Multiply both the numerator and the denominator by the conjugate of the denominator.

• The denominator is c + di.
• The conjugate is c - di.

\(\frac{a + bi}{c + di} \times \frac{c - di}{c - di}\)

3. Expand the Numerator (Top)
Apply the FOIL method (First, Outer, Inner, Last):

\((a + bi)(c - di) = ac - adi + bci - bdi^2\)

Recall that \(i^2 = -1\). Substitute this into the last term:

\(ac - adi + bci - bd(-1)\)
 

\(= ac - adi + bci + bd\)

Group the Real parts together and the Imaginary parts together:

\(\text{Real: } (ac + bd)\)

\(\text{Imaginary: } (bc - ad)i\)

Numerator Result: (ac + bd) + i(bc - ad)

4. Expand the Denominator (Bottom)
Multiply the denominator by its conjugate (Difference of Squares):

\((c + di)(c - di) = c^2 - (di)^2\)
 

\(= c^2 - d^2i^2\)

Substitute \(i^2 = -1\):
 

\(= c^2 - d^2(-1)\)
 

\(= c^2 + d^2\)

Denominator Result: \(c^2 + d^2\)

5. Assemble the Final Formula
Place the new numerator over the new real denominator:

\(\frac{(ac + bd) + i(bc - ad)}{c^2 + d^2}\)

6. Separate into Standard Form (A + Bi)
Split the fraction into real and imaginary components:

\(\frac{ac + bd}{c^2 + d^2} + \frac{i(bc - ad)}{c^2 + d^2}\)

Which gives us the final derived formula:

\(\frac{z_1}{z_2} = \left[ \frac{ac + bd}{c^2 + d^2} \right] + i \left[ \frac{bc - ad}{c^2 + d^2} \right]\)
 

To divide two complex numbers \(z_1 = a + ib\) and \(z_2 = c + id\), we multiply both the numerator and denominator by the conjugate of the denominator. This removes the imaginary part from the denominator and gives the final formula:
 

\(\frac{z_1}{z_2} = \frac{(ac + bd)}{c^2 + d^2} + i\frac{(bc - ad)}{c^2 + d^2}\)
 

Let us see the steps to divide two complex numbers:

• Step 1: Ensure both the numerator and denominator are complex numbers written correctly in standard form.
   
• Step 2: Determine the denominator’s conjugate. For example, if the denominator is c + id, then the conjugate is c – id.
   
• Step 3: The conjugate should be multiplied by both numerator and denominator.
   
• Step 4: The denominator must be solved using the difference of squares formula.
   
• Step 5: Separate the result into its real and imaginary parts, expressed as a + bi.

Example:

• Divide \(\frac{3 + 4i}{2 + i}\)
  • Step 1 (Standard Form): Both are already in a + bi form.
     
  • Step 2 (Conjugate): The denominator is 2 + i, so the conjugate is 2 - i.
     
  • Step 3 (Setup Multiplication):
    
    \(\frac{(3 + 4i)(2 - i)}{(2 + i)(2 - i)}\)
     
  • Step 4 (Expand & Solve):
    • Numerator:
      \((3)(2) - 3i + 8i - 4i^2\\=6 + 5i - 4(-1) \rightarrow 6 + 5i + 4 = \mathbf{10 + 5i}\)
       
    • Denominator:
      \(2^2 + 1^2 \rightarrow 4 + 1 = \mathbf{5}\)
       
  • Step 5 (Final Result):
    Divide each term by the denominator:
    
    \(\frac{10 + 5i}{5} = \frac{10}{5} + \frac{5i}{5} = \mathbf{2 + i}\)

• Divide \(\frac{6 + 8i}{2i}\)
  • Step 1 (Rewrite in Standard Form): The denominator 2i lacks a real part.
    Rewrite it as 0 + 2i to see the standard form clearly.
     
  • Step 2 (Conjugate): The denominator is 0 + 2i.
    The conjugate is 0 - 2i (or simply -2i).
     
  • Step 3 (Setup Multiplication): Multiply numerator and denominator by the conjugate -2i.
    
    \(​​​​​​​\frac{(6 + 8i)(-2i)}{(2i)(-2i)}\)
     
  • Step 4 (Expand & Solve):
    • Numerator:
      \(6(-2i) + 8i(-2i)\\=-12i - 16i^2\\ = -12i - 16(-1) \to -12i + 16 = \mathbf{16 - 12i}\)
       
    • Denominator:
      \((2i)(-2i) \to -4i^2 \to -4(-1) = \mathbf{4}\)
       
  • Step 5 (Final Result):
    Divide each term by the denominator:
    
    \(\frac{16 - 12i}{4} = \frac{16}{4} - \frac{12i}{4} = \mathbf{4 - 3i}\)

Here, we will learn how to divide complex numbers in polar form.

If you have two complex numbers in polar form:

\(z_1 = r_1 (\cos \theta_1 + i \sin \theta_1)\)

\(z_2 = r_2 (\cos \theta_2 + i \sin \theta_2)\)

The division is calculated as:

\(\frac{z_1}{z_2} = \frac{r_1}{r_2} [\cos (\theta_1 - \theta_2) + i \sin (\theta_1 - \theta_2)]\)

Derivation:

1. Set up the Division

Write the ratio of the two numbers:

\(\frac{z_1}{z_2} = \frac{r_1 (\cos \theta_1 + i \sin \theta_1)}{r_2 (\cos \theta_2 + i \sin \theta_2)}\)

2. Isolate Moduli and Arguments

Separate the magnitudes (r) from the angular parts:

\(\frac{z_1}{z_2} = \left( \frac{r_1}{r_2} \right) \cdot \left[ \frac{\cos \theta_1 + i \sin \theta_1}{\cos \theta_2 + i \sin \theta_2} \right]\)

3. Multiply by the Conjugate

Multiply the numerator and denominator by the conjugate of the denominator, which is (\(\cos \theta_2 - i \sin \theta_2\)):

\(= \left( \frac{r_1}{r_2} \right) \cdot \left[ \frac{(\cos \theta_1 + i \sin \theta_1)(\cos \theta_2 - i \sin \theta_2)}{(\cos \theta_2 + i \sin \theta_2)(\cos \theta_2 - i \sin \theta_2)} \right]\)

4. Simplify the Denominator

Using the difference of squares and the identity \(\cos^2 x + \sin^2 x = 1\):

\(\text{Denominator} = \cos^2 \theta_2 - i^2 \sin^2 \theta_2\)

\(= \cos^2 \theta_2 - (-1)\sin^2 \theta_2\)

\(= \cos^2 \theta_2 + \sin^2 \theta_2 = 1\)

5. Expand the Numerator

Multiply the terms using FOIL:

\(\text{Numerator} = (\cos \theta_1 \cos \theta_2 - i \cos \theta_1 \sin \theta_2 + i \sin \theta_1 \cos \theta_2 - i^2 \sin \theta_1 \sin \theta_2)\)

\(= (\cos \theta_1 \cos \theta_2 + \sin \theta_1 \sin \theta_2) + i(\sin \theta_1 \cos \theta_2 - \cos \theta_1 \sin \theta_2)\)

6. Apply Trigonometric Identities

We use the angle difference identities:
 

• \(\cos(A - B) = \cos A \cos B + \sin A \sin B\)
   
• \(\sin(A - B) = \sin A \cos B - \cos A \sin B\)

Substituting these into the numerator:

\(\text{Numerator} = \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)\)

7. Final Result

Combining the simplified numerator with the ratio of the moduli:

\(\frac{z_1}{z_2} = \frac{r_1}{r_2} [\cos (\theta_1 - \theta_2) + i \sin (\theta_1 - \theta_2)\)

Dividing complex numbers relies on the concept of rationalizing the denominator to eliminate the imaginary unit and express the result in standard form (a + bi). While this process involves several algebraic steps—such as using conjugates and the difference of squares—mastering the underlying logic makes the calculations much more intuitive. Here are a few tips and tricks to help simplify the process and solve these problems with confidence.

• Link to "Rationalizing the Denominator": Remind students that i is just a square root (\(\sqrt{-1}\)). In algebra, we never leave a square root in the denominator (like\( \frac{1}{\sqrt{2}}\)). Explain how to solve division of complex numbers is the same process as rationalizing a denominator: you are simply rewriting the fraction so the "root" (i) isn't on the bottom.
   
• The "Multiply by 1" Magic: Students often feel they are changing the problem when they multiply by the conjugate. Show them that \(\frac{\text{Conjugate}}{\text{Conjugate}}\) is equal to 1. Emphasize that they are not changing the value of the number, only its appearance, effectively "dressing it up" in standard form.
   
• Master Multiplication First: Division is essentially a multi-step multiplication problem. Before starting division, ensure the student is 100% confident with FOIL (First, Outer, Inner, Last) and powers of i. If they struggle with multiplication and division of complex numbers, pause and drill multiplication of binomials first, as that is the engine that drives division.
   
• Color-Code the Conjugate: Visual aids reduce cognitive load. When showing examples, write the original complex numbers in one color and the conjugate in a distinct, contrasting color. This visually separates the original problem from the "tool" used to solve it, making the steps more straightforward.
   
• Process Over Formula: While the formula \(\frac{ac+bd}{c^2+d^2}\dots\) exists, it is mathematically "heavy" and hard to memorize. Advise students to ignore the formula and instead learn the algorithm: "Find Conjugate \(\rightarrow\) Multiply Top and Bottom \(\rightarrow\) Simplify." This procedural memory is more durable than rote memorization of variables.
   
• The "Difference of Squares" Safety Net: Teach students to predict the outcome of the denominator. When multiplying a number by its conjugate (c+di)(c-di), the result must be a real number (\(c^2 + d^2\)). If they end up with an i in the denominator after this step, they know immediately that they made a calculation error and can self-correct.
   
• Treat i Like a Variable (Until the End): To prevent confusion, tell students to treat i exactly like x during the expansion phase (3x + 4x = 7x). Only in the final step, when they see \(i^2\), should they swap it for -1. This prevents them from trying to do too much arithmetic in their heads simultaneously.

Complex numbers are used daily in fields like engineering and quantum mechanics. Here are a few examples where the division of complex numbers is applied:

• In electrical engineering, complex numbers are divided to analyze the alternating current (AC) circuits. For example, in electrical engineering, division of complex numbers is used in Ohm’s Law (V = IZ) to compute voltage or current when impedance Z is complex.

• In quantum mechanics, the division of complex numbers is used to understand wave function transformations and probability calculations.

• It is used to calculate ratios in the design of antennas, microwave circuits, and fiber optic communication systems. Here, engineers often work with signal reflection, impedance matching, and wave propagation, all of which involve complex numbers.