Why are electrons written with a proton number? Why are electrons written as $\require{mhchem}\ce{^{0}_{-1}e}$ when they have no protons, meaning the proton number should be $0$ and not $-1$?
Is it also acceptable to write $\beta$ particles in this manner, such as $\ce{^0_{-1}\beta}$, considering that a beta particles is basically an electron?
 A: Proton number is really applicable in Chemistry mostly but in Nuclear equations in physics the proton number and charge is essentially the same in representation. Thus as charge is conserved in a nuclear reaction (e.g. during $\beta$ decay) so thus the sum of all the so called 'proton number' that is the charge must be same as both sides of the equation so as electrons and protons have equal magnitude of charge for protons it corresponds to +1 while it is -1 for electrons. Hence, this notation is sometimes used. 
Edit: And, yes beta particles can be written in that manner.
A: These numbers are the atomic mass number, or nucleon number (top) and the electric charge (bottom), not the number of protons. Nuclear physicists use these particular numbers - rather than the number of a specific particle like protons - because in nuclear reactions, the atomic mass number and the electric charge are conserved, whereas particle counts are not. You can add up the values on top and on the bottom across a reaction and they should be the same on both sides, which provides a useful check for mistakes.
Fun fact: mass is not conserved during nuclear reactions, but the atomic mass number is not quite the mass, it's more like the mass rounded to the nearest integer number of atomic mass units. And the changes in mass during a nuclear reaction are small enough not to change the rounding. (Unless large quantities of antimatter are involved, but then you're way beyond the scope of nuclear physics.)
