To deepen the concept.
Most people are forgetting the main aspect of mathematics: Number is always a representation of something. The sign of a number represents the direction of something(for example a direction of rotation). The absolute value is the length/size of that something.
Example: "You have 5 apples and you give 2 apples to your friend. How many apples do you have?"
mathematical solution: 5-2=3
the 5 and 3 represents the amount of apples (technically it has no sign - you simply have certain amount of apples). -2 represents the change (change can be positive or negative). You solve the problem by adding the change to the amount (so what is really happening is 5+(-2) )
The absolute value of the change is 2 = the size of the change, disregarding the direction. Sign is a property, that the number technically doesn't need to have, depending on what that number represents. The absolute value of amount of 5 apples is also 5 apples (because it was an amount to start with).
Math was invented to let us (logically) operate with things, without thinking of the things themselves - all we think about is their properties of amount, length, size, direction, orientation, probability, complexity,... Numbers, with all their properties are just general representations of the properties of real things, letting us apply general concepts to them (for example, in real life adding apples is quite different from adding areas, but it's driven by identical concepts).
For me, this is the real beauty of mathematics, that for some reason most people overlook... Probably because of how mathematics is being thought at schools these days. Teachers take so much effort to teach children how to separate the math from the reality (so that they can solve similar problems in general abstract way), that they miss the link between numbers, operations and relationships and the real things they represent.
For example, the prime number. Most people know that it is "a number divisible only by 1 and by itself". What people don't know is the consequences of it in real life! Many species of insects have reproductive cycle of prime number of years, so that there is smaller likelyhood of being in sync with reproductive cycles of your predators. Thanks to that, they do not serve as a main source of food for them (predator can't rely on them to be around when he is born / at certain age).
wow
I've gone quite philosophical today...