1/0 = 0

1/0 = 0
hillelwayne.com

The issue with special definitions is you can use them to say anything you want, turning regular, common operations into weirdness.

What does it mean to take a factorial on the real numbers, or to add only on the even integers? In both cases, we're twisting what are generally accepted mechanics and domains into something else to get a funny result, like 1/0 = 0.

He gets to that point here:

" We can say that division’s domain is all the real numbers except zero. This is what we do in our day-to-day lives, and the way that Agda and Idris handle division. We can choose some value that isn’t a real number, such as “undefined” or infinity, and say x/0 = . This is what some mathematicians do with the Riemann sphere. We could choose some real number, like 19, and say that x/0 = 19. This is what Isabelle, Lean and Coq do.

"

So, tiny change in definition --> big change in outcome.

*I'm an undergrad

It actually does break something, the symmetry between division and multiplication and the many pieces of code that assume that (x / y) * y equals x. Here is a naive and non practical example, but it is not impossible to find a real world example where this simplified code manifests itself accidentally or by design

  function do_something_with_x(x, y) {
    let ratio = x / y;
    if (do_something_with_ratio(ratio)) {
      return x * y;
    }
    else return null;
  }