I'm trying to get to an abstract definition of a negative number that would fit in with the basic concept of addition/subtraction.
There are questions here about multiplying and dividing negative numbers that really point to this basic question.
The common analogy is of monetary debt. This analogy is useful, and I would like to abstract the fundamental concept of 'negative number' from it. Other analogies which may provide a different perspective would also be helpful. But the aim is to distill the fundamental concept from these analogies.
In answering the question one might talk about what a number is. It seems that 'negative' is an adjective describing number. I want to assume that the concept of 'number' is generally understood though, so we don't have to go too deeply in that.
What is a negative number? Please give your thoughts.