Absolute Value Cannot Be Negative

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Absolute value is an instance of what's chosen a 'metric'. These are function which measure 'distances' in mathematical spaces.The absolute value can exist idea of as a metric that measure the distance from zero of numbers (and then our mathematical space is $\mathbb{R}$ or some other 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 weather condition that take been called considering 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 the states already in some sense. But could we define a metric to be negative and still keep the other conditions intact? $$\\$$ Well let'southward run into... Suppose $|\cdot|$ is a metric satisfying the last three conditions and with $|x|<0$ for some x. Then; $$0 = |0| = |x-x| \leq |x|+|10|,$$ but since $|ten|<0$ and then certainly $2|ten|<0$, so the line above is a contradiction. So if nosotros have a negative accented value we can't likewise have the triangle inequality (this is the proper noun for the concluding condition we listed). But the triangle inequality is a status that is very natural, it's the idea that travelling along the hypotenuse of a triangle is quicker than travelling forth the ii other sides. Therefore to do useful things nosotros really need this to be the case for our metrics. And then that is why nosotros don't define negative absolute value. Non because it is forbidden, only because doing and so would mean we can't use it in a useful manner.

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answered May 10, 2014 at 16:08

EHHEHH

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Absolute Value Cannot Be Negative,

Source: https://math.stackexchange.com/questions/789192/why-cant-absolute-values-be-expressed-with-negative-numbers

Posted by: mcintireoppoichest.blogspot.com

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