The answer to this question seems obvious.
'An absolute value expresses the quantity of ones between any number and 0'.
But does that mean it must be positive?
I took a shot at answering my question:
- "Well, you can't have a negative quantity, in real life (but math doesn't depend on real life)."
- "Maybe somewhere in the inner workings of math it turns out that there is no 'negative-positive' distinction, and that distinction just gives us another way of thinking about the relative positions of two numbers on a number line."
- "Maybe expressing absolute values in positives is just the convention, and that's all there is to it (but that seems to suggest a kind of value where negative and positive don't matter)"
- "Maybe the kind of value where negative and positive don't matter is an absolute value, and that's why it's expressed |x| not -x or x (but then, why do we think of negative numbers as a kind of number, and positive numbers as a kind of number, but not 'absolute numbers' as a kind of number?)"
That's my best shot
So why can't absolute values be expressed with negative numbers?