Lets say we have two complex numbers and the product of which is a real number.
The idea of multiplying two imaginary numbers to give a real number is something which I am unable to grasp.
Are the imaginary numbers not that imaginary? Do they have some sort of link to the real world? How can we interpret this answer?
Updated:
Each complex number $z = x + i y$ ($x, y \in \mathbb{R})$ can be interpreted as a point $(x, y)$ in the Gaussian plane.
Pure imaginary numbers $z = i y$ are lying on the $y$-axis of that plane.
Pure real numbers $z = x$ are lying on the $x$-axis of that plane.
Multiplication of complex numbers is convenient in polar coordinates: $$ z_1 z_2 = r_1 e^{i \phi_1} r_2 e^{i \phi_2} = r_1 r_2 e^{i(\phi_1 + \phi_2)} $$ so it can be interpreted as changing the radius and the angle.
For two pure imaginary numbers one gets on of these cases $$ z_1 z_2 = r_1 e^{i (\pm\pi/2)} r_2 e^{i (\pm \pi/2)} = \pm r_1 r_2 $$ which all happen to lie on the $x$-axis, which consist of the real numbers.