Is it just for aesthetic purposes, or is there a deeper reason why we write $2\sqrt{3}$ and not $\sqrt{3}2$?
Why not write $\sqrt{3}2$?
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2The proper name for the 'overline' is 'vinculum' – 2013-04-12
4 Answers
The format $\sqrt{3}2$ is easliy confused with $\sqrt{32}$.
I also suspect that many early typesetters would skip the overline, so that $\sqrt{3}$ would be typeset as $\sqrt{\vphantom{3}}3$. In that case, $2\sqrt{\vphantom{3}}3$ is unambiguous but $\sqrt{\vphantom{3}}32$ highly ambiguous.
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0@DougSpoonwood In fact, it's often useful to consider a multiplication operator for each number (and even an addition operator for each number, e.g. the 'add$3$to my argument' operator). Multiplication can be treated as a single function/operator with multiple operands or a plethora of operators each with a single operand, depending on context, and both are completely consistent; look up the concept of _currying_ functions. – 2012-10-15
One possibility - would you rather think of the number as "two of the thing known as $\sqrt3$," or as "$\sqrt3$ many of the number two?"
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0counting = natural in my example. But if I had to choose one as the "counting number" between the two, I would choose $i$, and I always write it first, at least for example with unitary numbers such as $e^{i\sqrt{2}}$. – 2013-04-13
Certainly one can find old books in which $\sqrt{x}$ was set as $\sqrt{\vphantom{x}}x$, and just as $32$ does not mean $3\cdot2$, so also $\sqrt{\vphantom{32}}32$ would not mean $\sqrt{3}\cdot 2$, but rather $\sqrt{32}$. An overline was once used where round brackets are used today, so that, where we now write $(a+b)^2$, people would write $\overline{a+b}^2$. Probably that's how the overline in $\sqrt{a+b}$ originated. Today, an incessant battle that will never end tries to call students' attention to the fact that $\sqrt{5}z$ is not the same as $\sqrt{5z}$ and $\sqrt{b^2-4ac}$ is not the same as $\sqrt{b^2-4}ac$, the latter being what one sees written by students.
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2The proper name for the 'overline' is 'vinculum' – 2013-04-12
It's simply a matter of clarity. If you write $\sqrt 3 2$ meaning $2 \times \sqrt 3$ rather than $\sqrt{32}$, it would be clearer to write $(\sqrt 3) 2$ or $\sqrt 3 \times 2$, but then you have to say: oh, what the heck, just go with $2 \sqrt 3$.
Another thing to consider is that neglecting to properly extend overlines is a tell-tale sign of a TeX novice. As you are already aware, to get $\sqrt{32}$ you need to write \sqrt{32}
in your source.