What’s stopping mathematicians from defining a number for 1 ÷ 0, like what they did with √-1?
In: Mathematics
tldr: it’s not useful, because it leads to logical contradictions that force you to abandon extremely basic principles of what it means to be a number.
Declaring *i* to be a value such that *i^2 = -1* turns out to not break anything when you go to do algebra/arithmetic, and it turns out to have a bunch of very useful properties. Here by “break anything” I mean does it lead to logical contradictions when you apply familiar rules for how equations behave and such, and it’s somewhat surprising that everything works out so nicely when you do this.
If we try to do the same thing with 1/0, things break pretty much immediately. Let’s see how. Call that value f, so we have the definition `f=1/0`. The definition of multiplication/division then implies that `f*0=1`, and then from there the definition of zero implies that `0=1`. Uh-oh.
So what does that mean in terms of implications for mathematics? Nothing, really. What actually happened in the previous paragraph is I defined a *new* number system that works like the real numbers but has an extra element whose multiplicative inverse is zero. And then I ended up showing that oopsies, that number system actually collapses in on itself; the only value in it is zero and nothing meaningful can be done with it. Once you have *one* logical contradiction in a system, all bets are off; nothing is true and everything is permitted.
If you do the same thing with defining a *new* number system that works like the real numbers but has an extra element whose square plus one is zero, you can follow a similar process of applying known rules to see what happens and figure out how such a number system might look. And this time rather than the whole thing blowing up in your face, you get *extra* stuff that wasn’t there before and new useful properties that weren’t accessible before.
All that is to say “why does one work and the other doesn’t” really just boils down to checking what happens when you take an existing set of axioms (statements that are assumed as foundational truths) and add a new statement you declare to be true. If the new statement contradicts the existing ones in some way, the result is useless. If the new statement is a logical consequence of the existing ones in some way, the result is unchanged. If neither the new statement nor its negation contradicts the existing ones, congrats, you’ve found a new, larger logical structure with more stuff in it. Do some math and poke around to see how it works; maybe it’s useful for something.
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