>>13757631What is minus and what is infinity?
You might think these are absurd questions, but they are not - you can answer them in different ways.
1. In analysis your answer would be that inf - inf represents lim(a_n-b_n) as n goes to infinity for two sequences of real numbers which both go to infinity. This is a limit of a sequence and there is a lot you can do with it - you can have it grow to +inf, -inf, have a unique point it converges to or even have it have all of its limit points be all positive real numbers, all integers or even all real numbers, if you like very pathological things.
2. Another is to think of inf as a set, then you can interpret inf - inf as the set theoretic difference of A and B where A and B are two infinite sets. This is very general and can evaluate to any subset of A, where A can be the integers.
3. Algebra. Now we are dealing with an abelian group and a symbol in it we will call inf, satisfying this identity
a+inf=a.
But because this is an abelian group, this implies that inf=0, meaning that the only abelian group with such a symbol is the group consisting of one element - inf. So you lose all elements.
Alternatively, lets forget we have a group at all, we just have a commutative monoid with two symbols -inf and inf, satisfying these identities for any a
a+inf=inf
a+(-inf)=-inf
But if we do this, something funny happens, these identities have to hold for inf and -inf too, so we get inf+(-inf)=-inf and -inf + inf=inf
This means we either don't have commutativity, so we don't have addition or don't have -inf and inf be distinct. You want addition, so lets do addition - we now have the identity inf=-inf, so inf-inf=-inf.
But there is actually a nice structure, relevant to real mathematics that is even useful to the real world, that uses this.
https://en.wikipedia.org/wiki/Tropical_semiringSo as you can see, this can be answered in many ways in different parts of mathematics, each way to answer it being fun somehow.
