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Mathematics · Ch 2 — Complex Numbers

Properties of Complex Conjugates

2.4.2

Properties of Complex Conjugates

Ten properties of complex conjugates (for complex numbers z,z1,z2z,z_1,z_2 and integer nn):

  1. z1+z2‾=z1‾+z2‾\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}
  2. z1−z2‾=z1‾−z2‾\overline{z_1-z_2}=\overline{z_1}-\overline{z_2}
  3. z1z2‾=z1‾ z2‾\overline{z_1z_2}=\overline{z_1}\,\overline{z_2}
  4. (z1z2)‾=z1‾z2‾, z2≠0\overline{\left(\dfrac{z_1}{z_2}\right)}=\dfrac{\overline{z_1}}{\overline{z_2}},\ z_2\ne0
  5. Re⁡(z)=z+z‾2\operatorname{Re}(z)=\dfrac{z+\overline z}{2}
  6. Im⁡(z)=z−z‾2i\operatorname{Im}(z)=\dfrac{z-\overline z}{2i}
  7. zn‾=(z‾)n\overline{z^n}=(\overline z)^n
  8. zz is real   ⟺  z=z‾\iff z=\overline z
  9. zz is purely imaginary   ⟺  z=−z‾\iff z=-\overline z
  10. z‾‾=z\overline{\overline z}=z

Proof (property 1). Let z1=x1+iy1z_1=x_1+iy_1, z2=x2+iy2z_2=x_2+iy_2. Then

z1+z2‾=(x1+x2)+i(y1+y2)‾=(x1+x2)−i(y1+y2)=(x1−iy1)+(x2−iy2)=z1‾+z2‾.\overline{z_1+z_2}=\overline{(x_1+x_2)+i(y_1+y_2)}=(x_1+x_2)-i(y_1+y_2)=(x_1-iy_1)+(x_2-iy_2)=\overline{z_1}+\overline{z_2}.

This generalises by induction to any finite number of terms: z1+z2+⋯+zn‾=z1‾+z2‾+⋯+zn‾\overline{z_1+z_2+\cdots+z_n}=\overline{z_1}+\overline{z_2}+\cdots+\overline{z_n}.

Proof (property 3). With z1,z2z_1,z_2 as above, direct expansion gives z1z2=(x1x2−y1y2)+i(x1y2+x2y1)z_1z_2=(x_1x_2-y_1y_2)+i(x_1y_2+x_2y_1), so

z1z2‾=(x1x2−y1y2)−i(x1y2+x2y1),\overline{z_1z_2}=(x_1x_2-y_1y_2)-i(x_1y_2+x_2y_1),

while z1‾ z2‾=(x1−iy1)(x2−iy2)=(x1x2−y1y2)−i(x1y2+x2y1)\overline{z_1}\,\overline{z_2}=(x_1-iy_1)(x_2-iy_2)=(x_1x_2-y_1y_2)-i(x_1y_2+x_2y_1) — the same expression, so z1z2‾=z1‾ z2‾\overline{z_1z_2}=\overline{z_1}\,\overline{z_2}.

Proof (property 9). Let z=x+iyz=x+iy; then z‾=x−iy\overline z=x-iy, so z=−z‾  ⟺  x+iy=−(x−iy)=−x+iy  ⟺  2x=0  ⟺  x=0z=-\overline z\iff x+iy=-(x-iy)=-x+iy\iff 2x=0\iff x=0, which is exactly the condition for zz to be purely imaginary. …

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