Why has the extreme value function never been defined? –Or has it? 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 $<x>$ for the extreme value of a positive number x, and define it as the maximum of {x, 1/x}.
An example of usage: For a computer to be able to store a positive number x, a necessary (but not sufficient) condition is that $<x>$ be “sufficiently small”. (Of course, a “sufficiently small” extreme value will never be less than unity.)
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}
$<x/y>$ = 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 $<x>$ can be neatly expressed by the formula exp(|log(x)|).
 A: It may be of interest to note that this way of measuring how far apart two quantities are essentially shows up in the limit comparison test in calculus courses (for positive series and for improper integrals). Specifically, if $\{x_n\}$ and $\{y_n\}$ are two sequences of positive real numbers such that $0 < a \leq \max\{\frac{x_n}{y_n}, \frac{y_n}{x_n}\} \leq b < \infty$ for all sufficiently large $n$, then $\sum x_n$ and $\sum y_n$ both converge or both diverge. Of course, without loss of generality for the above notion, but not for the quantified versions you're asking about, we can use $b = \frac{1}{a}$. Sometimes one says (or maybe it's just me who says this) that $\{x_n\}$ and $\{y_n\}$ have the same order of magnitude as $n \rightarrow \infty$ or that $\{x_n\}$ and $\{y_n\}$ are asymptotically equal up to a constant as $n \rightarrow \infty$.
This notion of equivalence also comes up with bi-Lipschitz functions and Lipschitz equivalent metrics, and at least in the case of bi-Lipschitz functions, the "best" value(s) of $a$ and/or $b$ often play an important role, but I'm not aware of any generally accepted name for this "best" value that one could use for the situations you and I have given.
