Why don't we have just one type of charge? Why do we need two? Why do we need to admit two types of charges (positive and negative)? Can't there be a third type?
 A: If you take a collection of materials (glass, amber, polythene, perspex, pvc, polystyrene....) and rub them with a soft insulating material, they acquire charges. You discover that all these rubbed materials can be put into just two categories. All those in one category repel each other and attract any of those in the other category. Two categories; two sorts of charge.
This is the gist of the original argument, dating back some three hundred years. Subsequent discoveries have given us no reason to challenge it. For example, electrons are repelled by suitably rubbed amber or polythene but attracted by glass or perspex and therefore fall into the negative charge category.
What is more, equal quantities of the two sorts of charge can be represented by $+R$ and $-R$ in which $R$ is a positive real number with unit. For example the force on charge $Q_2$ due to charge $Q_1$ is
$$\mathbf F=\frac{Q_1 Q_2}{4 \pi \epsilon_0 r^2} \mathbf {\hat r} $$
and if $Q_1$ is negative and $Q_2$ is positive, $\mathbf{F}$ is in the opposite direction to $\mathbf {\hat r}$, the unit vector in the $Q_1Q_2$ direction.
