1-3 The conjugate and the division of complex numbers

The conjugate

• The conjugate of z = a + bi is z̄ = a − bi.
• z + z̄ = 2a — always real.
• z − z̄ = 2bi — purely imaginary.
• z · z̄ = a² + b² — real and non-negative.
• The conjugate of (z + w) is z̄ + w̄; the conjugate of (z · w) is z̄ · w̄.

Division

• Multiply both numerator and denominator by the conjugate of the denominator, so the denominator becomes a real number.
• Example: (3 + 2i)/(1 − i) = (3 + 2i)(1 + i) / (1 + 1) = (3 + 3i + 2i + 2i²)/2 = (1 + 5i)/2 = 1/2 + (5/2)i.