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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 $$ 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 $$ 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}

$$ = 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 $$ can be neatly expressed by the formula exp(|log(x)|).

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    I don't think $\max\{x/y,y/x\}$ has popped up often enough to get its own name or notation.2011-08-24
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    I suspect when it has popped up it has been in the guise of "*Assume without loss of generality that $0 < x < y \,\ldots$*"2011-08-24
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    Terminological: Maybe "extreme ratio" or something similar would be a better term in the end. The phrase "extreme value" comes up in many other contexts, including extreme value theory in probability and statistics (which concerns e.g. distributions of maximum values of sequences of IID variables).2014-02-07
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    You can always write $\exp(|\ln(x/y)|)$, though you might not find that appealing.2014-06-17

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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.

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    So, it seems that your answer, which I have upvoted and accepted, is confirming the observation that this notion has not yet been deemed important/useful enough to be treated except on the fly.2011-08-26
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    @Mike Jones: That's my guess. While there may be a locally used name in the case of bi-Lipschitz functions, I don't think there is a sufficiently widely used name that people in general would recognize. By the way, I really like your analogy making use of max and min at the end. It may be of interest to see if this analogy can be extended in some way to include the less precise equivalence notion "order of an entire function" (and the various iterated orders of entire functions).2011-08-26
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    For "$x_n$ and $y_n$ are asymptotically equal up to a constant" one writes $x_n=\Theta(y_n)$ (as $n\to\infty$), as in [Big O notation](https://en.wikipedia.org/wiki/Big_O_notation#Family_of_Bachmann.E2.80.93Landau_notations)2014-06-17