Complex conjugate — A/L Combined Mathematics

How do I practise Complex conjugate?

1 question with a worked solution

Check before you answer: Proof of Distributive Properties: Proving algebraically that (z₁ z₂)⁻ = z̄₁ z̄₂ by expanding both left-hand and right-hand sides in terms of coordinates.

1. For any complex number z = x + iy with complex conjugate z̄ = x - iy, what is the value of the product z · z̄?
   1. A) 2iy
   2. B) 2x
   3. C) x² + y² (which equals |z|²)
   4. D) x² - y²
   Check my answer

   Correct answer: C) x² + y² (which equals |z|²)

   Computing: z · z̄ = (x + iy)(x - iy) = x² - (iy)² = x² - i² y² = x² - (-1)y² = x² + y². Since |z| = √(x² + y²), this is identical to |z|² (a purely real, non-negative quantity).

   Show worked solution

   Step 1: (x + iy)(x - iy) = x² - i² y² = x² + y². Step 2: Since |z| = √(x² + y²), z·z̄ = |z|². Answer: x² + y² (which equals |z|²).
   Modelled on: the A/L Combined Mathematics examination pattern

Get this auto-marked

Next in the syllabus

Where does Complex conjugate fit in the syllabus?

Year 2 · Term 1