When is a complex number $z$ real? When discussing the order relation on $\mathbb{C}$, it is said that such a statement as $z_1 < z_2$ where $z_1, z_2 \in \mathbb{C}$ is meaningless, unless $z_1$ and $z_2$ are real.
My question is, when will a complex number $z$ be real? I know that if $\bar{z}$ is the conjugate of $z$, then
$$z + \bar{z} = 2a$$
$$z\bar{z} = a^2+b^2$$
produce real numbers, but it is easy to add $0i$ to either equation to produce a complex number.

A complex number is strictly real iff $z=\overline{z}$
