Is absolute value a function or property?

Question

Not the most groundbreaking curiosity, but entertaining my brain.

In basic algebra, is absolute value a function, counting the distance between the value and zero?

Or is it a property, that every numbers unsigned representation is its absolute value?

Answer 1

Mathematics does not distinguish between "functions" and "properties" in the way you seem to imagine. The same thing can be called a "function" in one breath and treated like a "property" in the next without anyone batting an eye.

For example, the arctangent is undoubtedly a function. Yet we also happily say that the arctangent of $1$ is $\pi/4$, suggesting that $\pi/4$ is somehow a property, i.e. something that we can find attached to the number $1$ with the label "arctangent". But I don't think anyone would seriously say that "its arctangent is $\pi/4$" is inherently part of what the number one really is.

I suspect you may be thinking of a real number as somehow "made up of" an absolute value together with a sign, which would make the absolute value "morally" more of a "property" of the number than "actangent = $\pi/4$" is a property of $1$.

But most of the time mathematics doesn't even care what numbers "really are". We work from descriptions of how they behave -- and surely one of the ways they behave is that for each number we can find a nonnegative number that we call its "absolute value". But what we do with that fact doesn't depend on whether we think of this as a "property" or a "function" -- it doesn't matter whether the absolute value was "inside" the number to begin with or we had to do some computations to find it. Just that the computation exists is all we need to know.

(Furthermore, even in the rare cases where we do care how numbers look "inside", the representation we choose for them is almost never a combination of a sign and a magnitude. This has technical reasons. We could choose a sign-magnitude representation, but the point here is that outside the formal construction of the reals we don't care whether one construction or the other was used).

Answer 2

Absolute value is commonly defined as a function $\mathbb R\to\mathbb R$ and in a wider sense also $\mathbb C\to\mathbb R$ and can indeed be described as "distance between argument and $0$". It takes nonnegative values, so also $[0,\infty)$ can be practicized as codomain.

A property can be interpreted as a function that takes values in $\{0,1\}$ or $\{\text{true},\text{false}\}$ if you like.

This cannot be said in general of the absolute value, so it is no property.