What is a negative number?

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.

To limit my liability, let's only consider integers and then once we start talking about division, rational numbers. First off we intuitively understand addition:

If I have two apples and you give me three more apples, then I now 
have 2 + 3 = 5 apples.

There are two routes we can go down: 1) we can accept negative numbers exist and deal with it or 2) we can accept subtraction as a valid operation that we definitely understand. I think the second approach is best for your question (since we do not want to accept negative numbers a priori).

So just as addition is intuitive to us, subtraction also is:

If I have 5 apples and you take 3 from me, then I am left with 2 apples: 5 - 3 = 2.

The value zero now becomes very important because I know that if I have $x$ apples and you take away $x$ apples then I am left with none, $0$:

$$ x - x = 0 $$

So now what happens when I have $5$ apples and you take away $6$? How many apples am I left with? Obviously, intuition now breaks down because you do not have $6$ apples to give up, but the math can remain:

$$ 5 - 6 = 5 - (5 + 1) = 5 - 5 - 1 = 0 - 1 $$

We are happy in every step up until the last when I get to $0 - 1$ which we have no value for! I don't understand what $0 - 1$ represents in exactly the same way that I do not understand what $\sqrt{-1} = i$ represents--it's a definition! I am now defining that $0 - 1 = -1$--$-1$ is now a symbol for that value (which I do not fully comprehend). (and ultimately when I say $-x$, I really mean $-1*x$ just as when I say $ai$, I mean $a*i$)

So now that we have this new symbol, what can we do with it? Well we can try and add it to values: $5 + -1 = 5 + (0 - 1) = 5 + 0 - 1 = 5 - 1 = 4$--we see that $5 + -1$ is the same as $5 - 1$! What about $5 - -1$? This is a little trickier. Now obviously we can write $5 - (0 - 1)$, but this doesn't help us because we don't know how to subtract a negative (in fact the above expression just devolves into $5 - -1$--the original question)! What we really need to show now is the following:

$$ 0 - (0 - 1) = 0 - -1 = +1 $$

So we can do this through some algebraic manipulation:

$$ 0 - (0 - 1) = x \\ 0 = x + (0 - 1) \\ 0 = x + 0 - 1 = x - 1 \\ 0 + 1 = x + 1 - 1 = x + 0 = x\\ x = 1 $$

So notice that I used only addition to get to this result! This proves that $0 - -1 = +1$ therefore we can rewrite:

$$ 5 - -1 = 5 + 0 - -1 = 5 + (0 - -1) = 5 + 1 = 6 $$

At this point, I hope that we both accept negative numbers as they are. The next question is for multiplication and division. If I have $5*-2$, then what should the result be? Well that one is easy:

$$ 5*-2 = (-2) + (-2) + (-2) + (-2) + (-2) = -10 $$

What's not so easy is $-2*5$! There are two ways to approach this: 1) we accept that multiplication is commutative and thus $-2*5 = 5*-2 = -10$ (as we already showed) or 2) a negative multiplier means something "different" from a positive multiplier. A positive multiplier means to add the thing being multiplied whereas a negative multiplier means to subtract the thing being multiplied. The latter definition will help us define also a negative times a negative.

So what exactly is multiplication? Multiplication means taking a value and adding it to zero $x$ times (whatever the multiplier is). If the multiplier is negative, then it means subtracting from zero. For instance:

$$ 5*-2 = 0 + (-2) + (-2) + (-2) + (-2) + (-2) = -10 \\ -2*5 = 0 - (5) - (5) = -10 \\ -2*-5 = 0 - (-5) - (-5) = 5 + 5 = +10 \\ -5*-2 = 0 - (-2) - (-2) - (-2) - (-2) - (-2) = +10 $$

From the above definition we see that a negative times a positive results in a negative value, a positive times a positive results in a positive value, and a negative times a negative results in a positive value. I don't want to go much further--division can be considered somewhat elementary (just as subtraction to addition) but, at this point, I think it's easier to accept division as the inverse of multiplication and prove the same laws apply (i.e. a division by a positive and negative gives a negative, etc.).

The negative of a number $a$ is inverse in the additive group of numbers, represented as $-a$, such that $-a +a =0$.

You seem to be asking the ontological question. In other words, what do mathematical negative numbers mean really. You correctly note that it depends on the meaning of a number. But there is more than one meaning for a number.

1. Geometric magnitudes. These were the "floating point numbers" in mathematics from classical Greece until only a couple of hundred years ago. These magnitudes don't really have a negative. You can talk about opposite displacements, and these did appear in Euclid's geometry, but they were not really thought of as negative numbers as such. Even in Euler's writings in the 18th century, he did his calculus with "lines", not numbers. Even at that time, what we call "real numbers" could not be abstracted from geometric magnitudes.

2. Counting numbers can be either ordinal numbers or cardinal numbers. Clearly the negative of an ordinal or cardinal number is fairly meaningless.

3. The negative "real numbers" of Renaissance polynomial algebra were initially thought of as fictional, and were rejected as meaningless solutions. But as we all know, they were accepted within about a hundred years or so, especially when it was found that inclusion of the complex numbers gave $n$ solutions for every $n$th degree polynomial equation. These negative real numbers were initially a convenience to make arithmetic work more smoothly.

4. Complex numbers. Within the complex numbers, we know that the negative is either a rotated positive number, or once again a solution of algebraic equations.

There are various other kinds of number contexts, where the meaning of "negative" in each case is a bit different. Just in the last 24 hours, I received a book in the mail where the author axiomatically defines positive integers, then positive rationals, and then positive real numbers. He makes the interesting comment that all of the positive numbers were defined historically before the negative numbers. And in terms of axiomatic development, it's actually much easier to do everything for positive numbers first. In out modern education system, we learn negative integers before negative rationals/reals. But actually negative numbers are very abstract compared positive rationals and reals.

In the realm of addition/subtraction it is not possible to distinguish positive from negative numbers since $x\mapsto -x$ is an isomorphism of the additive group ${\mathbb Z}$.