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.
 A: In my experience, Mathematicians aren't the best at answering questions like, what is a number? Nonetheless, I'll try provide some justification why we have such properties with negative numbers. Please note this isn't a riguous approach, just an educational approach.
Negative "$a$" is the additive inverse of "$a$", that is$$a+(-a)=0$$
now, why is a negative number times a positive number negative? Let $-a$ be negative and $b$ be positive. I will show that $(-a)b=-(ab)$. 
$$ab+(-a)b=(a+(-a))b\:\:\:\text{because}\:\:\:yx+zx=(y+z)x$$
$$ab+(-a)b=0b=0\:\:\:\text{because}\:\:\:a+(-a)=0$$
$$\text{hence}\:\:\:ab+(-a)b=0$$
We know that negative $ab$ is $-(ab)$ and we know that $ab+(-(ab))=0$
$$\text{so}\:\:\:ab+(-a)b=0\:\:\:\text{and}\:\:\:ab+(-(ab))=0$$
$$\text{so we conclude that}\:\:\:(-a)b=-(ab)$$
Hence a negative number times by a positive number is negative
A: There is a joke:

Three retired professors sit in front of a building. They see two people entering it. Ten minutes later three people go out.
"They have multiplied", says the professor of biology.
"No, there is a distortion in time-space continuum", says the professor of physics.
"Never mind what has happened", says the professor of mathematics. "However, when one more person enters the building, we can consider it empty."

The answer of Jared is - in my opinion - perfect, however I would like to say my imagination about numbers. First of all, the mathematics itself does not exist. Physics does. There is no 4, no 8, no -10, no pi, no infinity. There is always "a number of something".
What can it be?
I see three cats. This weights 10 kilograms. It is 10 kilometers far away. The perimeter (eg. a tyre length) has pi centimeters when the diameter is 1 centimeter. I drive a car 100 km/h. And so on.
Understanding subtraction (leading to negative values) is clear - it is this what is too few, too small. In physics it is something that is measured from end to start. For example, if we measure distance from city A to city B, the negative values of speed may by understood that I am travelling in the opposite direction (from B to A). In finance negative values are loan, something that one owes, something that one paid too few and needs to pay back, but he/she does not have it at the moment.
The negative temperature is temperature calculated under freezing point of water (0 deg C). It is used to calculate energy that is required to heat up something from one temperature to another; however only temperature difference (in most cases) is required (please note: there is no negative temperature in Kelvins, as it would have no physical sense).
Understanding multiplication of negative values is a bit harder. We can multiply a scalar and a physical unit. We can calculate (-5) * (4 km) but we can also calculate (5) * (-4 km). The latter case is that we take 5 sections that have length 4 km each, but we count them from the city B.
The first case is similar, but you have a section directed from A to B and you want 5 of them, but also in opposite direction.
Division is similar, if you consider (20 km) / (-5) being (-20 km) / (5).
There is some problem eg. when you multiply two physical units eg. when calculate areas (why is (-5 km) * (-5 km) = 25 km^2 and (-5 km) * (5 km) = -25 km^2?). I've always liked the idea that multiplying something by -1 is some kind of rotation (sine for 180 deg is -1). This might require some theory of complex numbers and Euler notation from you.
A: 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 $0 - x$)
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.).
A: To say what a mathematical object is, definitively, ultimately requires its reduction to a set-theoretic formulation. For negative integers, this is best done by considering them as elements of $\Bbb Z$, which includes positive and negative integers constructed in a uniform way. In turn, $\Bbb Z$ is constructed from the set $\Bbb N$ of natural numbers, which was formulated elegantly by John Von Neumann: Starting with nothing, each natural number is the set of prior natural numbers. Thus $0=\varnothing,\, 1=\{0\}=\{\varnothing\},\, 2=\{0,1\}=\{\varnothing,\{\varnothing\}\}$, and so on. With addition defined on $\Bbb N$ in the usual way, we can, for each natural number $n$, define two equivalence classes, say $n_+$ and $n_-$, of ordered pairs of natural numbers by $$n_+=\{(a,b)\in\Bbb N^2:a=b+n\},\qquad n_-=\{(a,b)\in\Bbb N^2:b=a+n\},$$where $(a,b)=\{\{a\},\{a,b\}\}$ as usual denotes the ordered pairing of $a$ and $b$, and $\Bbb N^2$ is the set of ordered pairs of natural numbers. Note that $0_+=0_-$; let's call it $0_+$. We identify the set $\Bbb Z_{\geqslant0}=\{n_+:n\in \Bbb N\}$ of non-negative integers with the set $\Bbb N$ of natural numbers, and $\Bbb Z_{<0}=\{n_-:n\in\Bbb N\}\setminus\{0_+\}$ represents the negative integers. Finally, $\Bbb Z=\Bbb Z_+\cup\Bbb Z_-$.
The algebraic or arithmetic properties of $\Bbb Z$ are well addressed in the other answers.
A: The negative of a number $a$ is inverse in the additive group of numbers, represented as $-a$, such that $-a +a =0$.
A: A somewhat related question (but for rational numbers): How to make sense of fractions?.
Given the existence of "numbers" (the positive integers) and how to add and multiply them, one can use the same approach as in my answer there and define "new numbers" (signed integers) as equivalence classes of pairs $(a,b)$, where $a$ and $b$ are positive integers, and the equivalence relation is
$$
(a_1,b_1) \sim (a_2,b_2) \iff a_1 + b_2  = a_2 + b_1
.
$$
(We think of the pair $(a,b)$ as representing the signed integer $a-b$,
and the condition above is just a way of saying that
$$
(a_1,b_1) \sim (a_2,b_2) \iff a_1 - b_1  = a_2 - b_2
$$
without mentioning subtraction, which logically speaking isn't defined yet if $b_1 \ge a_1$ or $b_2 \ge a_2$.)
Then one can go on to define the operations of addition, subtraction and multiplication for these "new numbers" in a way which extends the operations for the "old numbers" (which constitute a subset of the new numbers, if we identify $n$ with the equivalence class containing $(n,0)$).
A: 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.


*

*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.

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

*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.

*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.
A: 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 ${\mathbb Z}$.
A: My favorite way to picture negative numbers is to think of them as directions (common in Physics). One easy way to visualise it is potentials. 
For instance, we assume gravitational potential of an object is zero at infinity and it gets more and more negative as it gets closer to the object. Also, a stretched spring has a negative potential to go back to its initial state. So the opposite kinds of forces are easy to see here. 
Many things in Physics, however, cannot be negative, like mass or time (in general, but there might be some other theories that treat these as negative). Negative values could show direction in Chemistry too. Exothermic and endothermic reactions comes to mind. Here's a link that explains this more in detail.
A: A negative number is a number $x$ such that $x+y=0$ for some positive number $y$.
In other words, a negative number is an object whose sum with some positive number equals the additive identity. A similar approach can be used to define other mathematical objects:

*

*$x=0$ is an object such that $a+x=a$

*$x=\frac{1}{a}$ is an object such that $xa=1$

*$x=i$ is an object such that $x^2=-1$

*$x=a^{-b}$ is an object such that $xa^b=1$

*$x=a^\frac{1}{b}$ is an object such that $x^b=a$
The pattern here is that you can define novel mathematical objects through their relationship with familiar ones. This approach also lends itself more easily to generalization.
Choosing a particular implementation of negative numbers in terms of sets and equivalence classes is not necessary to capture what we mean by a negative number, though it may be useful.
In the preface to The Road to Reality, Roger Penrose writes:

According to the mathematician's "equivalence class" notion, the fraction $\frac{3}{8}$, for example, simply is the infinite collection of all pairs
$(3,8),(-3,-8),(6,16),(-6,-16),(9,24),(-9,-24),(12,32)\ldots$
Although this ubiquitous equivalence class procedure, of which the above illustration is just a particular instance, is very powerful as a pure-mathematical tool for establishing consistency and mathematical existence... it hardly conveys to us the intuitive notion of what $\frac{3}{8}$ is.
I shall be more concerned with conveying the idea... inherent in many important mathematical notions. The idea of a fraction such as $\frac{3}{8}$ is simply that it is some kind of an entity which has the property that, when added to iself 8 times in all, gives 3.
One way to see that fractions do make consistent sense is, indeed, to use the 'definition' in terms of infinite collections of pairs of integers (whole numbers), as indicated above. But that does not mean that $\frac{3}{8}$ actually is such a collection.

This is more or less the way I tend to conceptualize mathematical objects.
A: Here is how it was thought to me. First you construct natural numbers. This can be done using Peanos axioms. A infinite set of symbols $\mathbb{S}$ is the set of natural numbers if there is a number $a$ (called the 1) and a bijective function
$$f : \mathbb{S} \rightarrow \mathbb{S}/\{a\}$$
This function is called the next function. $f(x)$ is basally the next symbol after x. As you know natural numbers have the structure $\{1, 2, 3 \ldots \}$. Once you have done that you define addition in this abstract set of symbols. After that you try to extend this set to a group. To do this you add new symbols $-\omega$ for every $\omega$ in $\mathbb{S}$. Also you add the additive identity 0. 
So basically the negative numbers come from natural numbers in an attempt to extend it to a group under addition. Check this article it has the full details.  
