Does the absolute value of +3 lose its positive direction yet have its positive value?

Question

We have no sigh with the absolute value of +3, yet its value is positive.(Wikipedia) Does this mean that the absolute value doesn’t have its positive direction (+3 is located on positive direction from origin; while -3 is on negative), yet its value is positive –– no positive in point of direction; positive in point of value?

Answer 1

This is a really good question, and I do not understand why it has attracted downvotes. It depends on how you think of the absolute value, and how you think about numbers having directions.

In the real numbers $\Bbb R$ we think of numbers as having one of two directions, either positive or negative, with 0 as usual being an exception; it has neither direction, or both, depending on how you want to look at it.

We can't say that the directions are east or north, because they are not spatial directions. They're just positive and negative, and we can't locate them in space, except to say that they are opposite to one another. We often draw the number line with the positive numbers on the right and the negative ones on the left, but this is just a convention, and it could just as easily have gone the opposite way, or we could have a convention of always imagining positive numbers on the upper left and negative ones on the lower right.

The only really interesting thing you can say about the positive and negative directions is that they are opposite. Their only properties are defined relative to one another.

Now consider $\Bbb R^+$, the set of just the non-negative real numbers. Do these numbers have a direction? If so, it seems much less meaningful, because there is only one direction, not two, so there is no contrast to be drawn.

Now consider the absolute value function, which I will call $\def\abs{\operatorname{abs}}\abs$. One way to think of it is as a function from $\Bbb R$ to $\Bbb R$, one which takes a real number and gives you back a real number, one which happens to be in the positive direction. Seen this way, the absolute value function takes a number and gives you back a number with the same size but positive direction.

But another way to view $\abs$ is that it takes a real number and gives you back an element of $\Bbb R^+$, which is a number with no meaningful direction. We could say that it tells you the size of a number without any associated direction.

I think your questions is, which of these views is correct? Both are correct. Which is most useful may depend on context. or it may be that both are equally useful, since the difference between them is very small.

Answer 2

The absolute value of $3$ is very much a positive number. Here's why.

There's at least two ways of formalize the absolute value function. One is that its a mapping $\mathbb{R} \rightarrow \mathbb{R},$ in which case the absolute value of a number certainly has a sign, namely it is always non-negative. On the other hand, we could (if we wanted to) formalize the absolute value function as a mapping $\mathbb{R} \rightarrow [0,\infty).$ It could perhaps be argued that under this second formalization, the absolute value of a number does not have a sign. However, I wouldn't buy this argument. There is only one canonical mapping $[0,\infty) \rightarrow \mathbb{R},$ namely the inclusion; if you try to "embed" $[0,\infty)$ into $\mathbb{R}$ via the function $f(x) = -x$, you'll quickly notice that its not a multiplicative homomorphism. So $[0,\infty)$ is very much a subset of $\mathbb{R}$, not just by definition but also for deeper, structural reasons. So yes, I think that the absolute value of $3$ is very much positive, even if we don't conceptualize the absolute value function a mapping $\mathbb{R} \rightarrow \mathbb{R}$.