Do values attached to integers have implicit parentheses?

Solution 1:

I am going to play Devil's Advocate on this one, because I've given it a little bit of thought after a tangentially related question on MathOverflow about bad notation. There is a good lexical argument to be made for the "intuitive" reading of $5x/30x^2$ as $\frac{5x}{30x^2}$.

Disclaimer. The following argument is not intended to communicate why expressions such as $a/bc$ are unambiguous. By definition, if there is no strong socially agreed upon convention, it is therefore ambiguous. However, I will argue for the hypothesis that almost anyone raised from an early age to read a European language, who quickly reads such an expression without reflecting too much, will probably understand $a/bc = \frac{a}{bc}$. Therefore, there exists a reasonable opportunity to establish such a convention.

One of the responders to the question of bad notation complained about the usage of juxtaposition for multiplication: he thought it introduced too much ambiguitiy. (For instance, does $f(x+y)$) refer to the product of a scalar $f$ with $x + y$, or the value of a function $f$ evaluated at $x + y$, or something else?) In the comments to that answer, I speculated that the reason for this convention is that it reduces the problem of parsing a mathematical expression such as $ax^2 + bx + c$; to the previously-solved problem (for Europeans, anyway) of parsing a written sentence in a European language, which breaks sentences into words. Juxtaposed variables form nice little cohesive "words". This hypothesis can even be extended to account for exponents and subscripts — more notation in which we don't use operator 'symbols' — to play the role of intra-word apostrophes or other diacritical marks.

Under this hypothesis, I can make a strong case for prefering the reading $a/bc$ as $\frac{a}{bc}$. Unlike multiplication or exponentiation, we introduce an actual symbol — a prominent punctuation mark, if you will, as intrusive as the $+$ symbol — into the expression. This divides the expression into two "words", $a$ and $bc$, upon which we perform the operation of division; thus leading to the reading $a/(bc)$. This is especially likely in the case where $a$ is effectively a place-holder, as in $1/2x$, where for "fluent speakers of mathematics" the $1$ is only there as a placeholder to allow the concept of multiplicative inverse to be expressed. Add to this the fact that it makes no sense at all to write $a/bc$ if you wish to convey the meaning $ac/b$, and one can almost safely say that anyone who writes $a/bc$ means $\frac{a}{bc}$.

The above is meant as a descriptivist (as opposed to prescriptivist) argument for how people read and write mathematics. If this is going to happen anyway, and if (as appears to be the case) it doesn't entail any heavy modification of our existing conventions, why not accept it in order to allow more flexibility in our notational repertoire? We can cut down on parentheses, and get rid of this phony ambiguity between something which can be better expressed anyway and an expression which at present we can only confidently convey with parentheses or with extra vertical space.

Now, of course, there is no solid existing agreement about whether to read $a/bc$ as $\frac{a}{bc}$ or as $\left(\frac ab\right)c$, except to accept that latter because of BEDMAS or what-have-you. So if your job is to evaluate your student's ability to communicate, then you should conclude that they aren't doing such a good job. But if you are trying to evaluate what they mean — and if you believe that they learned to read a European language from a tender age — then chances that when they write $5x/30x^2$, they mean $\frac{5x}{30x^2}$.

Solution 2:

The programming languages I have used give equal priority to multiply and divide, and do them from left to right. This would support the $5*\cfrac{x}{30}*x^2$ assertion. As Wikipedia says, the acronym BEDMAS may be misleading as it implies that division precedes multiplication and addition precedes subtraction, while each pair is of equal priority. However, I suspect most people writing $5x/30x^2$ do mean $\cfrac{5x}{30x^2}$. I would say there is a tendency in people's minds to make the division slash lower priority than multiplication. But this is an excellent reason to put in parentheses when you are writing, and to check carefully what the author means when reading.