Complex Number Conjugate and Division
Learn about complex conjugates and how to divide complex numbers
Introduction
Before we can properly divide complex numbers, we need to understand the concept of a complex conjugate. This fundamental operation is essential for division and appears throughout complex analysis.
The conjugate of a complex number has important algebraic properties that make division possible and allow us to simplify complex expressions.
Complex Number Conjugate
The complex conjugate (or just "conjugate") of a complex number is obtained by changing the sign of the imaginary part only.
For a complex number \(z = a + bi\), the complex conjugate is:
\(\displaystyle \overline{z} = a - bi\)
The conjugate is denoted with a horizontal line (overline) above the variable: \(\overline{z}\) or sometimes as \(z^*\).
Conjugate Examples
Conjugate Examples
- \(z = 5 + 3i \quad \Rightarrow \quad \overline{z} = 5 - 3i\)
- \(z = 2 - 7i \quad \Rightarrow \quad \overline{z} = 2 + 7i\)
- \(z = 4 \quad \Rightarrow \quad \overline{z} = 4\) (purely real)
- \(z = 6i \quad \Rightarrow \quad \overline{z} = -6i\) (purely imaginary)
- \(z = -3 - 2i \quad \Rightarrow \quad \overline{z} = -3 + 2i\)
Key Property: Product with Conjugate
A crucial property is that the product of a complex number and its conjugate is always a real number.
For any complex number \(z = a + bi\):
\(\displaystyle z \cdot \overline{z} = (a + bi)(a - bi) = a^2 + b^2\)
Example: Product with Conjugate
For \(z = 5 + 3i\):
\(\displaystyle = 25 + 9 = 34\)
Result: The product is the real number 34.
This property is the key to eliminating imaginary numbers from denominators during division. By multiplying by the conjugate, we convert the denominator into a real number.
Properties of Complex Conjugates
Double Conjugate
\(\displaystyle \overline{\overline{z}} = z\)Taking the conjugate twice returns the original number
Sum Property
\(\displaystyle \overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}\)Conjugate of sum equals sum of conjugates
Product Property
\(\displaystyle \overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}\)Conjugate of product equals product of conjugates
Quotient Property
\(\displaystyle \overline{\frac{z_1}{z_2}} = \frac{\overline{z_1}}{\overline{z_2}}\)Conjugate of quotient equals quotient of conjugates
Division of Complex Numbers
To divide one complex number by another, we use the conjugate of the denominator. By multiplying both numerator and denominator by the conjugate, we transform the denominator into a real number, making the division straightforward.
For complex numbers \(z_1 = a + bi\) and \(z_2 = c + di\) (with \(z_2 \neq 0\)):
\(\displaystyle \frac{z_1}{z_2} = \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(a + bi)(c - di)}{c^2 + d^2}\)
Step-by-Step Division Process
1Identify the denominator
Write the division problem and identify \(z_2 = c + di\)2Find the conjugate
The conjugate of \(c + di\) is \(c - di\)3Multiply by conjugate
Multiply both numerator and denominator by the conjugate: \(\displaystyle \frac{\overline{z_2}}{\overline{z_2}}\)4Expand and simplify
Use FOIL on the numerator; use the conjugate product formula for the denominator5Separate real and imaginary parts
Divide numerator terms by the real denominatorDivision Examples
Example 1: Basic Division
Divide \(\displaystyle \frac{3 + i}{1 - 2i}\)
Step 1-2: The denominator is \(1 - 2i\), so the conjugate is \(1 + 2i\)
Step 3: Multiply by the conjugate:
Step 4: Expand numerator using FOIL:
Step 4: Denominator using conjugate formula:
Step 5: Separate real and imaginary parts:
Result: \(\displaystyle \frac{3 + i}{1 - 2i} = \frac{1}{5} + \frac{7}{5}i\)
Example 2: Division with Negative Terms
Divide \(\displaystyle \frac{4 - 2i}{2 + i}\)
Conjugate: The conjugate of \(2 + i\) is \(2 - i\)
Multiply:
Numerator:
Denominator:
Result:
Example 3: Division by Pure Imaginary
Divide \(\displaystyle \frac{6}{2i}\)
Conjugate of \(2i\): \(-2i\)
Conjugate and Division Summary
| Concept | Formula/Notation | Property/Example |
|---|---|---|
| Complex Number | \(z = a + bi\) | Example: \(5 + 3i\) |
| Conjugate | \(\overline{z} = a - bi\) | Example: \(5 - 3i\) |
| Conjugate Product | \(z \cdot \overline{z} = a^2 + b^2\) | Always real: \(25 + 9 = 34\) |
| Division Formula | \(\displaystyle \frac{z_1}{z_2} = \frac{z_1 \cdot \overline{z_2}}{z_2 \cdot \overline{z_2}}\) | Multiply by conjugate to rationalize |
Common Mistakes to Avoid
WRONG: Conjugate of \(5 + 3i\) is \(-5 - 3i\) ✗
RIGHT: Conjugate of \(5 + 3i\) is \(5 - 3i\) (only change imaginary sign) ✓
WRONG: \(\displaystyle \frac{3 + i}{1 - 2i} = \frac{3 + i}{1 + 2i}\) (only changed denominator) ✗
RIGHT: \(\displaystyle = \frac{(3+i)(1+2i)}{(1-2i)(1+2i)}\) (multiply both) ✓
WRONG: \((3+i)(1+2i) = 3 + 6i + 1i + 2i^2 = 4 + 7i - 2\) (wrong grouping) ✗
RIGHT: \(= 3 + 6i + i + 2i^2 = 3 + 7i - 2 = 1 + 7i\) ✓
Practice Problems
Conjugate Identification
Find the Conjugate
- \(z = 3 + 4i \quad \Rightarrow \quad \overline{z} = 3 - 4i\)
- \(z = -2 + 5i \quad \Rightarrow \quad \overline{z} = -2 - 5i\)
- \(z = 7 - i \quad \Rightarrow \quad \overline{z} = 7 + i\)
- \(z = -3i \quad \Rightarrow \quad \overline{z} = 3i\)
Division Problems
Divide These Complex Numbers
- \(\displaystyle \frac{2 + i}{1 + i} = ?\) → Answer: \(\frac{3}{2} - \frac{1}{2}i\)
- \(\displaystyle \frac{5 - 2i}{2 + i} = ?\) → Answer: \(\frac{8}{5} - \frac{9}{5}i\)
- \(\displaystyle \frac{1 + 2i}{3 - i} = ?\) → Answer: \(\frac{1}{10} + \frac{7}{10}i\)
- \(\displaystyle \frac{6}{1 + i} = ?\) → Answer: \(3 - 3i\)
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