Complex Numbers

Complex Conjugate

z̄ = reflection over real axis

  • High school
  • about 3 min
  • 5 of 18 in the family
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z = 2+2iz̄ = 2-2i
a = 2
b = 2
z=2+2i,zˉ=2−2iz = 2+2i, \quad \bar{z} = 2-2i
z⋅zˉ=22+22=8z \cdot \bar{z} = 2^2 + 2^2 = 8

The conjugate z̄ is z reflected across the real axis. Their product z·z̄ = |z|² is always a positive real – that's why division works (multiply numerator and denominator by the conjugate).