Complex conjugate — A/L Combined Mathematics

What must I remember about Complex conjugate?

• Definition of Complex Conjugate: For complex number z = x + iy, its complex conjugate is z̄ (or z*) = x - iy; reflection of z across the real axis (x-axis) in Argand diagram.
• Conjugate of sum and product: (z1 ± z2)* = z1* ± z2*; (z1 * z2)* = z1* * z2*; (z1 / z2)* = z1* / z2*.
• Product of number and its conjugate: z * z* = (x + iy)(x - iy) = x² + y² = |z|^2, strictly real and non-negative.
• Real and imaginary parts via conjugate: Re(z) = (z + z*) / 2; Im(z) = (z - z*) / (2i).
• Conjugate root theorem: If a polynomial P(x) with real coefficients has a non-real complex root z = α + iβ, then its conjugate z* = α - iβ MUST also be a root of P(x) = 0.

Test yourself (1 minute)

1. Purely real condition via conjugate

   z is purely real if and only if z = z*.

2. Purely imaginary condition via conjugate

   z is purely imaginary if and only if z = -z* (with z ≠ 0).

3. Power of conjugate

   (zⁿ)* = (z*)ⁿ for any positive integer n.

Practise Complex conjugate questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 21

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Where does Complex conjugate fit in the syllabus?

Year 2 · Term 1