Condition for z to Be Pure Real or Pure Imaginary

The conjugate of a complex number $z=a+bi$ is $a-bi$.

We will denote it as $Z$ in this topic.

Why is the conjugate of a complex number significant?

Notice that $z+Z=2a, z \cdot Z = a^2+b^2$. $2a$ and $a^2+b^2$ are both rea, and in fact Z is the only complex number satisfying that $z+Z, zZ$ are both real.

By the logic above, it's not hard to see that a complex number z is real if and only if $z=Z$.

It's not hard to see that complex number z is pure imaginary if and only if $z=-Z$.

As you may know, in $z = a + bi$, $a$ is the real part and $bi$ is the imaginary part. In other words, $a$ is pure real and $bi$ is pure imaginary. Remember, the title of this post is Condition for $z$ to be pure real or pure imaginary. You may have just realized that I have given you the answers.