# Rules vs conventions: what does 1/2t mean?

I have recently spotted that there is a lot of discussion on YouTube (and elsewhere) of the value of the expression

$6/2(1+2)$,

with rather heated arguments between those who say the answer is 1 and those who say the answer is 9. There are also  a few trolls and quite a lot of offensive comments out there!

I’m not going to enter that argument directly here, other than to say that I think that the expression looks slightly ambiguous to some professional mathematicians (including myself). Instead I’m going to look at a similar issue involving the expression

$1/2t$.

You might argue as follows :

$/$ means $\div$ and $2x$ means $2 \times t$ so the expression means $1 \div 2 \times t$. Since  division and multiplication have equal precedence, and we are supposed to work from left to right otherwise,  the result is $(1/2) \times t$ which is $t/2$.”

Yet I think that most professional mathematicians using or seeing the expression $1/2t$ would interpret it as meaning $\displaystyle \frac{1}{2t}$. (Am I right?) Nevertheless, I usually play it safe and avoid writing $1/2t$ (or similar) and write it some other way to avoid any issues.

So, are professional mathematicians getting it wrong, because they have forgotten the rules learned in school about how to evaluate expressions correctly? Or is mathematics sufficiently like a language that conventional usage can take precedence over rigid rules, and where expressions need to be interpreted in context?

I think that, in practice, the latter is what is actually happening in mathematics, but I can see some potential dangers there. If you work with ambiguous expressions, there is plenty of scope for proving $0=1$. On the other hand, my good friend Tony O’Farrell once told me that he counted the day as lost if he hadn’t proved that $0=1$ at least once! (I think that this is like saying that if you aren’t making mistakes then you aren’t trying hard enough.)

Here are some examples of expressions which could be seen as ambiguous, but where conventions or context would or at least might tell you what is intended.

• $1/2t$
• $1/t+2$
• $\cos 2t$
• $\cos t^2$
• $(0,1)$
• $(1,0)$
“Is it true or false that $1 \in 2$?”
“Is it true that $1 \subseteq \mathbb{N}$?”