Complex Conjugates - Algebra-2

1. Fundamental Concepts

Complex conjugates refer to pairs of complex numbers where, for a complex number \(a + bi\), its conjugate is \(a - bi\) (i.e., the real part remains unchanged, while the sign of the imaginary part is reversed).
Complex conjugates play a crucial role in complex number operations, particularly in simplifying complex division or eliminating the imaginary unit from denominators.

Definition: For a complex number \(z = a + bi\) (where a and b are real numbers), its complex conjugate is denoted as \(\overline{z} = a - bi\).
Key Properties:
- The sum of a complex number and its conjugate is a real number: \(z + \overline{z} = (a + bi) + (a - bi) = 2a\) (containing only the real part).
- The product of a complex number and its conjugate is a non-negative real number: \(z \cdot \overline{z} = (a + bi)(a - bi) = a^2 + b^2\) (using the difference of squares formula to eliminate the imaginary unit).
Application Scenarios: Primarily used to rationalize the denominator in complex division (convert the denominator to a real number) for simplifying calculations.

Easy difficulty: Find the complex conjugate of \(3 + 5i\).
Solution: Keep the real part 3 unchanged and reverse the sign of the imaginary part.
The conjugate is \(3 - 5i\).
Medium difficulty: Calculate the product of \(2 - 7i\) and its complex conjugate.
Solution: Its conjugate is \(2 + 7i\).
The product is:\((2 - 7i)(2 + 7i) = 2^2 - (7i)^2 = 4 - 49i^2 = 4 - 49(-1) = 4 + 49 = 53\).
Hard difficulty: Simplify \(\frac{1 + 3i}{4 - 2i}\) using complex conjugates.
Solution: The conjugate of the denominator is \(4 + 2i\).
Multiply the numerator and denominator by this conjugate:\(\frac{(1 + 3i)(4 + 2i)}{(4 - 2i)(4 + 2i)} = \frac{4 + 2i + 12i + 6i^2}{4^2 - (2i)^2}\)
Simplify the numerator: \(4 + 14i + 6(-1) = 4 + 14i - 6 = -2 + 14i\)
Simplify the denominator: \(16 - 4i^2 = 16 - 4(-1) = 20\)
Final result: \(\frac{-2 + 14i}{20} = -\frac{1}{10} + \frac{7}{10}i\).

Step 1: Identify the complex conjugate: For a complex number \(a + bi\), directly write its conjugate as \(a - bi\) (reverse the sign of the imaginary part while keeping the real part unchanged).
Step 2: Simplify using the product of conjugates: When calculating the product of a complex number and its conjugate, apply the difference of squares formula \((x - y)(x + y) = x^2 - y^2\) and replace \(i^2\) with \(-1\) to quickly obtain a real result.
Step 3: Rationalize complex division: When the denominator is a complex number \(c + di\), multiply both the numerator and denominator by its conjugate \(c - di\) to convert the denominator to \(c^2 + d^2\) (a real number). Then expand and simplify the numerator into the standard form \(a + bi\).