Conjugate of a Complex Number

Conjugate of a Complex Number

Definition. The conjugate of a complex number $z=a+ib$ is defined as $a-ib$, and is denoted by $\bar z$ (read "$z$-bar"). In words: keep the real part exactly the same, and flip the sign of the imaginary part.

Worked examples of conjugation (see the accompanying table): the conjugate of $3+4i$ is $3-4i$; the conjugate of $7i-2$ (i.e. $-2+7i$) is $-2-7i$; the conjugate of the real number $\sqrt3$ is $\sqrt3$ itself (a real number is unaffected by conjugation, since its imaginary part is already $0$); the conjugate of the purely imaginary $5i$ is $-5i$; and the conjugate of $7+\sqrt5\,i$ is $7-\sqrt5\,i$.

Three standing properties of the conjugate, used constantly in this chapter's proofs:

1. $\overline{\overline z}=z$ — conjugating twice returns the original number (flipping the sign of the imaginary part twice cancels out).
2. If $z=\bar z$, then $z$ is purely real — this can only happen if the imaginary part is its own negative, i.e. zero.
3. If $z=-\bar z$, then $z$ is purely imaginary — this can only happen if the real part is its own negative, i.e. zero. …