eli5: Why haven’t mathematicians invented a symbol for x/0 like they have pi and imaginary numbers?
In: 277
Because x/0 is undefined. Pi and imaginary numbers are defined. The squareroot of -1 could be undefined but mathematicians found a way to make sense of it. Treating i as a different kind of number works. Its consistent with already existing maths most importantly. Treating x/0 valid isnt consistent with maths. To see why: if a/b=c then c×b=a if b=0 this is incorrect becaus c×0=0 and not a. What if a=0? Then 0/0=c so c×0=0. The second one is true but with the first one the issue is that c could be any number. There is no way to make sense of this so we just say that this x/0 is undefined. Its not useful unlike i or pi.
Why would you create a variable or define a nonexistent quality? It would be absolutely pointless because it wouldn’t be used for anything.
In computing there is [NaN](https://en.wikipedia.org/wiki/NaN) (Not a Number) for results that are undefined or unrepresentable. It is there really to signal and propagate error state. There are number of ways to [get NaN](https://en.wikipedia.org/wiki/NaN#Operations_generating_NaN) including x/0, but once you have an NaN all further operations take that NaN will keep returning NaN.
While useful for computing as far I know it’s not particularly useful for mathematicians?
Because π and I can participate in mathematical expressions in a way that x/0 can’t. You can’t use the infinity symbol in expressions, because It’s just a tendency, not a value.
This is the premise of the Riemann sphere — created by one of the most famous historical mathematicians. There are many other analogues, but that’s a great place to first get into the topic.
Pi and imaginary numbers have a mathematical value, and coresponding symbols and variables to represent those values. X/0 does not have a value to define. It cannot be anything other than undefined.
X/0 being undefined is different from i being imaginary. because the square root of a negative number is not undefined, its just impossible to represent with real numbers. So we defined a symbol to represent this non-real number. i very much is defined.
X/0 has more to do with philosophy than mathematics. We can describe why x/0 is undefined with mathematics, but beyond that it cannot have useful implication. You could say that x/0 is a symbol to represent an undefined value.
Because it doesn’t make sense. How can you split 5 apples into zero groups?
Because x/0 doesn’t work consistently with the mathematics of finite numbers, basically – and there’s no point in giving a symbol to something that you’re not going to use. Dividing by zero breaks arithmetic and leads to invalid answers. Things like negative numbers and i (and, indeed, zero) may have seemed ludicrous when they were first proposed – but they’re well behaved and don’t mess the place up, so we give them house room. (They also turn out to be very useful, to boot.)
Pi is different, in that it’s a very specific number. It crops up all over the place – but it’s irrational, so the only practical way to write it down is to give it a name/symbol.
As an addition of the already mentioned points if you construct a modulo ring with a non prime, you can have the product of two numbers which gives 0. E.g. For Z_6 you can have 2*3=0 since 2*3 =6 mod 6 =0.
Math breaks down when you divide by zero. There are wonderful “proofs” you can do where sneakily dividing by zero leads to things like 1=2.
Replacing 1/0 with a symbol like i doesn’t help- the same breakdowns in logic happen regardless.
Once logic is broken, you can “prove” anything at all, even contradicting statements. That makes math not useful anymore.
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