Conjugate of a Complex Number

Conjugate of a complex number z = (a + ib) is defined as \[\overline{z}=\mathrm{a-ib}\].

• Double Conjugate
  z̄̄ = z
• Sum with Conjugate
  z + z̄ = 2 Re(z)
• Difference with Conjugate
  z − z̄ = 2i Im(z)
• Purely Real Condition
  z = z̄ ⇔ z is purely real
• Purely Imaginary Condition
  z + z̄ = 0 ⇔ z is purely imaginary
• Addition
  \[\overline{z_{1}+z_{2}}=\overline{z}_{1}+\overline{z}_{2}\]
• Subtraction
  \[\overline{z_1-z_2}=\overline{z}_1-\overline{z}_2\]
• Multiplication
  \[\overline{z_1z_2}=\overline{z}_1\overline{z}_2\]
• Division
  \[\overline{\left(\frac{z_1}{z_2}\right)}=\frac{\overline{z}_1}{\overline{z}_2},z_2\neq0\]
• z · z̄ = [Re(z)]² + [Im(z)]²
• \[\overline{z^{n}}=\left(\overline{z}\right)^{n}\]
• z₁z̄₂ + z̄₁z₂ = 2 Re(z₁z̄₂)