Just as when x and y are arbitrary real numbers, we often wish to consider their distance apart, and use the absolute value function to do so (namely, by means of the expression |x – y|), so also when dealing with inherently positive quantities, such as age, mass, size, etc, we often wish to consider, say, the larger of the two ratios x/y and y/x, but it seems that in the curriculum and literature there is no standard function to do this. So, analogous to “absolute value”, I am going to call this function, for the purposes of this question, the “extreme value” function. So my question is: Why hasn’t the extreme value function ever been defined? Or, perhaps it has, and I’ve just overlooked it. Perhaps someone, like John Kelley, has buried its definition in a problem that is meant to extend the theory in the text (which Kelley does a lot in his General Topology).
Anyway, for the purpose of this question, I will use the notation $
An example of usage: For a computer to be able to store a positive number x, a necessary (but not sufficient) condition is that $
Notice that for positive x and y, there is a nice parallelism between absolute value and extreme value, namely:
|x – y| = max{x, y} – min{x, y}
$
Also, just as |x| can be neatly expressed by the formula the square root of x squared, so also $