# Why can't absolute values be expressed with negative numbers. [closed]

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?

## closed as unclear what you're asking by TMM, egreg, Jack M, Davide Giraudo, Alexander Gruber♦May 10 '14 at 17:02

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• i may be wrong, but i'd say that "mathematics don't care about real life" is perhaps stretching the truth a little bit? we're not restrained to "real life" - but we are inspired (up to certain point) by it. – essay May 10 '14 at 15:42
• @sylvia fair enough, – Hal May 10 '14 at 15:52

## 2 Answers

Absolute value is an example of what's called a 'metric'. These are function which measure 'distances' in mathematical spaces.The absolute value can be thought of as a metric that measure the distance from zero of numbers (so our mathematical space is $\mathbb{R}$ or another set of numbers). This 'metric' (our absolute value) satisfies the following properties (in this context anyway): $$|a|\geq 0 \\ |a|=0 \iff a=0 \\ |ab| = |a||b| \\ |a+b| \leq |a| + |b|$$ These are all conditions that have been chosen because they have proved, through experience, to be useful. So from the first condition we see that the absolute value being nonnegative has been chosen for us already in some sense. But could we define a metric to be negative and still keep the other conditions intact? $$\\$$ Well let's see... Suppose $|\cdot|$ is a metric satisfying the last three conditions and with $|x|<0$ for some x. Then; $$0 = |0| = |x-x| \leq |x|+|x|,$$ but since $|x|<0$ then certainly $2|x|<0$, so the line above is a contradiction. So if we have a negative absolute value we can't also have the triangle inequality (this is the name for the last condition we listed). But the triangle inequality is a condition that is very natural, it's the idea that travelling along the hypotenuse of a triangle is quicker than travelling along the two other sides. Therefore to do useful things we really need this to be the case for our metrics. So that is why we don't define negative absolute value. Not because it is forbidden, but because doing so would mean we can't use it in a useful way.

"'An absolute value expresses the quantity of ones between any number and 0'.

But does that mean it must be positive?"

Unless the number is zero, yes. Here is the definition of absolute value:

$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$