Absolute value notation is ambiguous

Question

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 $\sqrt{x^2}$ ?

Answer 1

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.

Otherwise, one really can't help it being confusing at first glance. It's a notation that several mathematicians have agreed upon and are using for quite a long time, and is highly unlikely to change now.

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.