What is the meaning of the negative numbers?

Question

I am having troubles understanding what is the "meaning" or best way to think of the negative numbers. I am not really sure where is my confusion, so I would set some examples that make me trouble. Sorry for my informal language.

1. When thinking of "5 - 3" should I think about it as "taking away three elements from five elements" or is it "five positive elements and three negative elements together"? I mean, are negative numbers really numbers (or some quantity) or are they just a natural number under a subtraction operation.

2. From there come I think the rest of my questions. What does "10/-5" really mean? Am I dividing "10 elements" into "five negative groups"? How is that possible? How can there be negative groups?. How can the result be negative, since the elements I am dividing are positive?

I know how to get the results of these cases. But I do not really think I understand what a negative number really mean.

Can anyone help me see where is my confusion and if possible recomend some content to read?

Answer 1

Numbers don't just correspond with physical objects. We can think of directions as well. Moving one unit forwards +1, or one unit backwards -1. Directing a backwards direction backwards is then tantamount to forwards.

Answer 2

I think of numbers geometrically, equipped with some intrinsic quality of direction. In this view "-" is a unary function that sends a number to one of equal magnitude but opposite direction.

Answer 3

Well, the natural numbers are given by the Peano axioms. They form a commutative monoid with addition and a commutative monoid with multiplication, where multiplication is distribute over addition.

The next step is then to find the smallest commutative ring contained the set of natural numbers extending their operations. And here you land at the ring of integers.