Modulus, conjugate and inverse

For z = a + ib:

• The conjugate is z̄ = a – ib: change the sign of the imaginary part only. The conjugate of 5 + 12i is 5 – 12i, and of –2 – 7i it is –2 + 7i.
• The modulus is |z| = √(a² + b²), never negative. For 5 + 12i, |z| = √(25 + 144) = 13.

Multiplying a number by its conjugate removes i: z z̄ = (a + ib)(a – ib) = a² + b² = |z|². For 5 + 12i, z z̄ = 169.

That gives the inverse of any z ≠ 0: z⁻¹ = z̄/|z|². So (5 + 12i)⁻¹ = (5 – 12i)/169. Every non-zero complex number has an inverse; only 0 + 0i has none.

Division: multiply top and bottom by the conjugate of the denominator, just like rationalising a surd.

• (3 + 2i)/(1 – 2i) = (3 + 2i)(1 + 2i)/((1 – 2i)(1 + 2i)) = (3 + 6i + 2i + 4i²)/(1 + 4).
• = (–1 + 8i)/5 = –1/5 + (8/5)i.

Properties of the conjugate:

• The conjugate of z̄ is z itself.
• The conjugate of a sum, difference, product or quotient is the sum, difference, product or quotient of the conjugates.
• z + z̄ = 2 Re z and z – z̄ = 2i Im z. So z = z̄ exactly when z is purely real, and z̄ = –z exactly when z is purely imaginary.

Properties of the modulus:

• |z| = |z̄| = |–z|, and |z| = 0 only for z = 0.
• |z₁z₂| = |z₁||z₂| and |z₁/z₂| = |z₁|/|z₂|: you may take moduli piece by piece in products and quotients.
• But not in sums: z₁ = 3 and z₂ = –3 give |z₁ + z₂| = 0, while |z₁| + |z₂| = 6.

This saves a lot of work. To find the modulus of (3 + 4i)(5 – 12i)/(8 + 6i), do not multiply anything out: it is 5 × 13/10 = 13/2.