Conjugate of a Complex Number

Conjugate of a Complex Number

For $z=a+ib$, the conjugate of $z$, written $\bar z$, is obtained by reversing the sign of the imaginary part only:

Geometrically (once the Argand plane is introduced in §3) $\bar z$ is the mirror image of $z$ in the real axis. Taking the conjugate twice returns the original number:

The key identity. Multiplying $z$ by its own conjugate always produces a non-negative real number:

since $-i^2=-(-1)=1$. This quantity $a^2+b^2$ is exactly the squared modulus of $z$ (§4), and this identity — turning a product into a real number by multiplying by the conjugate — is the single trick behind computing reciprocals (§2.4) and simplifying any fraction of complex numbers.

Further standing properties, each proved directly from $\bar z=a-ib$:

• $z+\bar z=(a+ib)+(a-ib)=2a=2\,\mathrm{Re}(z)$ — always real.
• $z-\bar z=(a+ib)-(a-ib)=2ib=2i\,\mathrm{Im}(z)$ — always purely imaginary.
• $z=\bar z \iff b=-b \iff b=0 \iff z$ is real.
• $z=-\bar z \iff a=-a \iff a=0 \iff z$ is purely imaginary (or zero).
• $\overline{z_1+z_2}=\bar z_1+\bar z_2$ and $\overline{z_1z_2}=\bar z_1\,\bar z_2$ — the conjugate of a sum (product) is the sum (product) of the conjugates; both follow by expanding both sides with $z_1=a+ib,\ z_2=c+id$ and comparing real and imaginary parts. …