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There are 4 division symbols that I have learned/taught. Below is 18 divided by 3, shown with 4 different symbols.

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This question was sparked by the comments on my answer to the question on examples of mathematical slang. I wrote about the second symbol and it seemed that there were mathematical professionals who hadn't seen it before. I know that it was useful for teaching long division.

I am wondering when teachers introduce the 4 symbols in the US and other countries. I am also wondering the value of teaching each symbol.

Edit: To clarify, I was hoping for anecdotal evidence of when teachers introduce different symbols.

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    $\begingroup$ I suspect the first symbol was introduced for typographical reasons, while the second symbol was introduced for ease of implementing the long division algorithm, but those are both unresearched speculations on my part. $\endgroup$ – Michael Joyce Jan 19 '16 at 14:34
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    $\begingroup$ For historical information, you might check here; the main reference of which I am aware is Cajori's A History of Mathematical Notations (the link goes to a search within the book for the word division; you may wish to organize it by "pages"). Alternatively, you may wish to ask this question on HSM. $\endgroup$ – Benjamin Dickman Jan 19 '16 at 17:27
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1$\;$ When students begin learning arithmetic, they first learn how to do $18+3$, then $18-3$, and $18*3$. Rather than using fraction form to teach division, it's useful to have a symbol that can be used like $+, -,$ and $*$ to teach division. This is where $÷$ comes into play. It's easier for them to see $18÷3$ than $\frac{18}{3}$.

2 $\;$The second method isn't used so much as to represent division as it is to perform division. This is used to teach students how to divide larger numbers.

3/4$\;$These are the same symbol, really, but often the fourth is used because it's difficult to type the third without MathJax. The third is obviously more useful as students progress beyond arithmetic to topics like algebra and calculus.

As far as the origin of these symbols, most came about simply because different individuals across the planet had to generate notation on their own to work mathematics, and naturally they didn't all choose the same notation.

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    $\begingroup$ Also the second symbol is simply the analogue to stacked multiplication, addition, or subtraction. $\endgroup$ – Aeryk Jan 19 '16 at 15:22
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    $\begingroup$ I think it's important to stress the fact that $18/3$ is a representation of the number you get when diving 18 by 3 (which is incredibly important as the maths you are going get more advanced), as opposed to 1 and 2, which are effectively equations. Learning 1 (and generally 2) make it easier to understand 3/4, but ultimately their prime use is as a gateway to fractions (the latter). It's not so much that "different people have different notations" as "the best notation is difficult to understand unless you've seen simpler versions first" $\endgroup$ – Ieuan Stanley Jan 19 '16 at 17:02
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    $\begingroup$ @IStanley - With the advent of computers, / is used as the operator ÷ much more than it is used to indicate the forming of an arbitrarily precise ratio. So I don't think teaching this nuance is likely to help more than confuse. If you want to indicate a symbolic ratio, you really have to stack them to be clear. Granted, that's hard to do in a comment. $\endgroup$ – Rex Kerr Jan 19 '16 at 19:06
  • $\begingroup$ The slash / is really meant to be a solidus, and only gives an approximation to the real thing: tex.stackexchange.com/a/33047/141 $\endgroup$ – David Roberts Jan 29 '16 at 23:17

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