Thanks, this definition does seem problematic. In any case, it is not the only possible definition and in a/0=c, c does not have to be defined as a real number. We can define it as similarly to complex number with new rules that do not collide with existing reals.
There's a couple of mentions in other comments about the Riemann Sphere (https://en.wikipedia.org/wiki/Riemann_sphere) which does define division by zero, but sacrifices the numbers forming a field under addition and multiplication.
