# Absolute value notation is ambiguous

What is the meaning of $$|a|b|c|\space?$$ It could be $$\text{abs}(a\cdot\text{abs}(b)\cdot c),$$ or $$\text{abs}(a)\cdot b\cdot\text{abs}(c),$$ which differs in sign if $$b$$ is negative. Or, for a more natural example, $$|x+2|y+3|z|$$ could be either $$\text{abs}(x+2)\cdot y+3\cdot\text{abs}(z)$$ or $$\text{abs}(x+2\cdot\text{abs}(y+3)\cdot z).$$ The vertical bar notation is ambiguous; it doesn't tell whether it's a left or right parenthesis. Is there a better, commonly used notation for absolute value, other than $$\text{abs}(x)$$, or $$\sqrt{x^2}$$ ?

• Use parentheses to disambiguate $\,\left|a \left(|b|\right) c\right|\,$, or \big|\,a |b| c\,\big| for nested $\,\big|\,a |b| c\,\big|\,$.
– dxiv
Feb 9, 2018 at 1:24
• I think that to write $\vert a\vert b\vert c\vert$ is to sow intentional confusion, to commit a mathematical solecism. Feb 9, 2018 at 1:24
• What would be the point of the first use, however, given that $\lvert (a\cdot\lvert b\rvert\cdot c)\rvert = \lvert abc\rvert$? Feb 9, 2018 at 1:31
• Nice observation - I never noticed that. Feb 9, 2018 at 1:35
• @Lubin Or even $\,\{(a,b,c) \in \mathbb{Z}^3 \,\mid\, |a|\mid |b| \mid |c|\,\}\;$ ;-)
– dxiv
Feb 9, 2018 at 1:52

If you wish to resolve the ambiguity in the vertical bar notation, you add multiplication signs to make it clearer - $|a|\cdot b\cdot|c|$. You may also try to read the expression - $|a|b|c|$ - from left to right, mentally pairing up the bars as they come.
Though you can surely define the notation $\text{abs}(x)$ for yourself, and use it in your text (having clearly defined it in the preface of the book!). It will only depend on if other mathematicians also adopt it.
PS: While I have answered your question, I find the example a bit contrived. It's highly unlikely one will even want to write such an expression as $|a|b|c|$, as it can easily be rewritten in its much clearer form $b|ac|$, thus eliminating the confusion entirely.
• Additional ideas: use longer bars—$$\bigl|a|b|c\bigr|$$—or parenthesize—$$(|a|)b(|c|)$$ Feb 9, 2018 at 3:27