Complex Conjugates and Inequality

📌 CONCEPT: The complex conjugate of a complex number z = a + bi is given by θ = a - bi.

📐 RULE / FORMULA: We know that the modulus of the product of two complex numbers is the product of their moduli, i.e., |zθ| = |z||θ|.

💡 WORKED EXAMPLE: Let's consider a complex number z = 4 + 3i. Its conjugate is θ = 4 - 3i. Given |z| > 3, i.e., |4 + 3i| > 3, we need to show that |zθ| > 9. Using the rule, |zθ| = |z||θ| = |4 + 3i||4 - 3i| = √(4^2 + 3^2) * √(4^2 + (-3)^2) > 3√(16 + 9) = 3√25 = 15 > 9. Hence, |zθ| > 9.

⚠️ COMMON MISTAKE: Students often forget to consider the absolute value when multiplying complex numbers, leading to incorrect results.