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

Gelo Jacamile
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    The format $\sqrt{3}2$ is easily confused with $\sqrt{32}$. Indeed, when I saw the subject, my initial instinct was to correct it to $\sqrt{32}$. – Thomas Andrews Oct 12 '12 at 14:59
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    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 – Saharsh Oct 12 '12 at 15:03
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    @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
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    The proper name for the 'overline' is 'vinculum' – user50229 Apr 12 '13 at 12:09

4 Answers4

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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.

Ilmari Karonen
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Thomas Andrews
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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
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    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 Is Fiddling By Firelight Oct 12 '12 at 17:47
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    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
  • @ThomasAndrews there's also $x_2$ to confuse it with especially when you pronounce $x_2$ as "ecks two" – ratchet freak Oct 13 '12 at 15:38
  • To be even more explicit, we sometime end the overline on the square root by a small "leg". – Joel Cohen Oct 14 '12 at 12:10
  • @StevenStadnicki What do you mean by "operator" here? – Doug Spoonwood Oct 14 '12 at 13:50
  • @DougSpoonwood http://en.wikipedia.org/wiki/Operator_%28mathematics%29 - loosely, a function; in this case, a function from mathematical terms to mathematical terms that represents the 'multiply this term by 2' operation. – Steven Stadnicki Oct 14 '12 at 19:58
  • @StevenStadnicki If that holds for 2x, then you have an infinity of functions behind ordinary multiplication. You have the multiply x by 1 operation, the multiply x by 3 operation, you have the multiply 1 by x operation, the multiply 4 by x operation, and so on. Isn't multiplication just one operation instead of an infinity of operations? Or is multiplication both, and thus 1 equals infinity? – Doug Spoonwood Oct 14 '12 at 20:29
  • @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. – Steven Stadnicki Oct 15 '12 at 17:01
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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?"

Ross Millikan
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adam W
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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
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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.

Ilmari Karonen
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Michael Hardy
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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.

Bob Happ
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