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$\frac{1}{3} = 0.333333.... $

$\frac{1}{3} = 0.33\frac{1}{3} $

I ran into this fraction-in-a-decimal notation in a course I'm helping somebody with. I have never seen this before, and google results simply yield "how to turn fractions to decimals" and vice-versa results.

Has anybody else seen this notation? Does anybody know what, if anything, is a standard regarding it?

More bluntly, this is being taught and I was wondering if it's legal.


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It is always legal to "define" anything you want and to use your preferred notation. Another thing is if what you "define" makes sense or is useful at all. But I guess that if it is notation being used by some professor, it should have being defined at some point during class, no? I personally don't remember seen such notation before. – Adrián Barquero Apr 16 '11 at 2:38
Logically unimpeachable, but truly weird! The instructor is presumably trying to remind the students about the meaning of the last remainder in the "long division" algorithm. – André Nicolas Apr 16 '11 at 2:44
Yes, very weird indeed. The student actually seems to prefer this, but my sentiments reflected that of many responses: makes sense, but easy to lead to confusion. – emragins Apr 16 '11 at 23:29
up vote 6 down vote accepted

Note that $\displaystyle\ \frac{1}3\: =\: \ 0.33\frac{1}3\ $ means $\displaystyle\ \frac{1}3\: =\: \frac{3}{10} +\: \frac{3\frac{1}3}{100}\ $ which, times $100\:,\:$ becomes $\displaystyle\ \frac{100}3\: =\ 33\frac{1}3\:.\:$

So the notation is "legal". Whether or not it is advisable depends on the context. Certainly it could lead to confusion if not well-explained. It does prove handy as a notational way to represent recursive computations of infinite digit "streams" in functional programming languages. Here the $\:1/3\:$ in the final "digit" represents the continuation function that computes the remaining digits in the tail of the stream. Analogous ideas are sometimes employed in computer algebra systems for representing similar objects e.g. power series and $\rm p$-adic numbers.

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I also took it to represent the recursive nature of it. I hadn't thought of breaking it down to a place-value problem, though, so that was interesting in how it works out. – emragins Apr 16 '11 at 23:28

I have seen it. My thought would be that it was seen primarily in the 19th century. And early 20th century. But nowadays not taught that way.

Don't know what you mean by "legal". As far as I know, not even Indiana has outlawed this...

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Math is outlawed! (I just hate these idiots... it makes me want to go on a rant. :) ) – Mateen Ulhaq Apr 16 '11 at 4:20

$\frac{1}{3} = 0.33\frac{1}{3} $ is the equivalent of saying $100 \div 3 = 33 \text{ remainder } 1$. Try a different example, such as $3000 \div 7 = 428 \text{ remainder } 4$ and this might encourage you to write something like $\frac{3}{7} = 0.428\frac{4}{7} $.

I would not encourage this as it mixes two different representations and might confuse, but with care it can be done unambiguously.

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interesting... but also well to note that 3000/7 = 428.571428; 3/7 = .428571428; and 4/7 = .571428. So, while I agree there's a little ambiguity, it still works out with the initial intention of the notation. – emragins Apr 16 '11 at 23:26

The British used this notation for the decimal halfpenny when it was in use (1971-1983).

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