Difference between imaginary and complex numbers

Every complex number can be written as $z=a+bi$, where $a,b\in \mathbb{R}$ (real numbers). The number $a$ is called real part of $z$ and the number $b$ is the imaginary part of $z$.

If the real part is zero then we call $z=bi$ as pure imaginary complex number.

Here is a diagram to show the inclusions:

Imaginary numbers are numbers than can be written as a real number multiplied by the imaginary unit $i$, and complex numbers are imaginary numbers, plus numbers that has both real and imaginary parts. $i$ is both imaginary and complex. The imaginaries are a subset of the complex numbers, as the naturals are a subset of the integers.