You know.. dividing is like breaking something up.. If you divide by 2, you break it up into 2 pieces.. if you divide by 0, from some standpoints, you do have nothing, because you broke it up into 0 pieces, and 0 pieces is 0.
And dividing by 0.5 is breaking into how many pieces exactly?
*disclamer: I have no math background whatso ever
The reason it's a useful function is it has some properties:
y=1/x is continuous except where x==0. That is, as you approach 0 from either side, it grows to either +- infinity. The developers of this language want to make this function ==0 when x is 0.
You can do that, sure, but it doesn't change the fact that when you go from +0.0000001 to -0.0000001, you now have 2 non-continuous jumps instead of just one.
So all that to say: We can't treat 0 like a number.
*disclaimer: I have no math background whatsoever, this is just a fun topic.
Multiplying by the square root of 2 is also a fun one since you need limits to show people that real numbers only make sense with the least upper bound theorem and limits and things they expect to work "intuitively" (like division or multiplication) only have approximate meaning when they use the common operations.
Reals are fundamentally different and are not found in things like computers and in general are of no practical use. For work with values (as opposed to counts) we use something called floating point.
This explains exactly why we shouldn't divide by 0. The pieces after breakage should, if one is so inclined, be able to be re-assembled into the whole; but, once you break something into 0 pieces, it is gone. Thus at best `0/0` could be defined … but the problem there is not that there's no answer but that every answer is equally good.
You know.. dividing is like breaking something up [..]