# Ambiguous Absolute Value Expressions

*Is there a standard "order of operations" for parallel vs nested absolute value expressions, in the absence of clarifying notation?*

*Cross-posted from here.*

Consider the expression

One way to interpret this is that there are two products being added together:

But you could also interpret it as the absolute value of an expression that itself contains an absolute value:

These two interpretations are not equivalent. For instance, substituting $x=0{:}$

Granted, this is a contrived example and I’ve never actually seen an ambiguous case come up in real life (or in any math textbook). I only stumbled upon this while developing algorithms to handle edge cases in a free response grader a couple years ago. (And even then, behavior on this edge case doesn’t make a difference in practice since none of the correct answers that would be graded against involve ambiguous notation.)

I also realize that the expression could be made un-ambiguous by explicitly writing multiplication symbols, or by tweaking the absolute value notation to distinguish between left bars and right bars (e.g. via sizing/spacing/padding, or by writing $\textrm{abs}(\cdot)$ instead of $\vert \cdot \vert$). And I realize that there are obvious ways to solve this issue in the context of software (e.g. designing a user interface that avoids ambiguous notation, or using heuristics like choosing the interpretation with the lowest nesting depth).

But I’m still curious to know: **given an ambiguous absolute value expression, is there a standard convention for interpreting it?** In other words, loosely speaking, is there a standard “order of operations” for parallel vs nested absolute value expressions, in the absence of clarifying notation? (The answer may be that there is no agreed-upon rule.)

So far, the most convincing argument I’ve seen was Dave L Renfro’s comment on this StackExchange answer:

*"If there is [a standard convention], then it almost certainly would only be applied in a computer coding (or calculator) setting, and it would not generally be known in the mathematical community"*

In other words, this comment suggests that there is no agreed-upon rule: mathematicians use symbol sizing (or other notational means) to avoid ambiguity and therefore have no need for a rule to interpret ambiguous cases (since ambiguous cases shouldn’t exist in a mathematical text).