A complex number is a pair $z=x+iy$ where $x$ and $y$ are real numbers;

$x=\Re z$ is the real part of $z$, whereas $y=\Im z$ is the imaginary part of $z$.

The conjugate of $z$ is $\bar{z}=x-iy$ and the modulus of $z$ is $|z|=\sqrt{x^{2}+y^{2}}$.

The Argand diagram or complex plane is the usual coordinate system where we represent $x+iy$ as the point $(x,y)$ in the plane. The horizontal axis is called the real axis $\{x:x\in{\mathbb{R}}\}$; the vertical axis is called the imaginary axis $\{iy:y\in{\mathbb{R}}\}.$