# Why not write $\sqrt{3}2$?

Is it just for aesthetic purposes, or is there a deeper reason why we write $2\sqrt{3}$ and not $\sqrt{3}2$?

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The format $\sqrt{3}2$ is easily confused with $\sqrt{32}$. Indeed, when I saw the subject, my inoitial instinct was to correct it to $\sqrt{32}$. –  Thomas Andrews Oct 12 '12 at 14:59
I suspect it was also common for early typesetters to skip the overline, and just typeset $\sqrt{3}$ as $\sqrt{}3$, which would then be clearly ambiguous. –  Thomas Andrews Oct 12 '12 at 15:02
We write $2\sqrt{3}$ to just to simplify the number, while $\sqrt{3}2$ will make confusion –  Alpha Oct 12 '12 at 15:03
@ThomasAndrews : Writing √3 is not an instance of typesetters skipping the overline; rather it is a case where no overline is called for. The overline in $\sqrt{3x}$ indicates that the whole $3x$ is within the radical, rather than just the $3$. –  Michael Hardy Oct 12 '12 at 23:50
The proper name for the 'overline' is 'vinculum' –  user50229 Apr 12 '13 at 12:09

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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A related question might be, why do we usually write "2x" rather than "$x2$." Perhaps trying to avoid ambiguity with $x^2$, but that seems unlikely. –  Thomas Andrews Oct 12 '12 at 15:11
I suspect that we write '$2x$' because the '2' there is secretly acting as an operator, and operator composition is usually written as a left action; $2x$ is actually $2(x)$, where $2()$ is the 'multiply by $2$' operator applied to the unknown $x$. (This is related, obviously, to adam's answer) –  Steven Stadnicki Oct 12 '12 at 15:19
Yeah, I was about to post that point. In general, we put operators to the left. –  Thomas Andrews Oct 12 '12 at 15:21
I suspect the fact that we can read the former aloud as "...two ecks" while the latter has to be read as "...ecks times two" in order to be unambiguous with ".... multiplied by two" is a factor as well. –  Dan Neely Oct 12 '12 at 17:47
Writing $\sqrt{}\,3$ is not an instance of typesetters skipping the overline; rather it is a case where no overline is called for. The overline in $\sqrt{3x}$ indicates that the whole $3x$ is within the radical, rather than just the $3$. –  Michael Hardy Oct 12 '12 at 19:03

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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Does it matter? –  user50229 Apr 12 '13 at 12:11
I for one would rather count naturally, but I would agree that beyond that it does not matter, since either way involves something radical. –  adam W Apr 12 '13 at 13:39
It's interesting because with $i$ for example, I think people tend to write $2i$, but $i \sqrt{2}$ instead of $\sqrt{2}i$. In this case do you consider $i$ as a 'counting number' or $\sqrt{2}$? –  user50229 Apr 12 '13 at 14:39
counting = 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 $e^{i\sqrt{2}}$. –  adam W Apr 13 '13 at 14:12

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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The proper name for the 'overline' is 'vinculum' –  user50229 Apr 12 '13 at 12:13