fractions inside of a decimal? $\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.
Thanks.
 A: 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.
A: 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...
A: $\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.
A: The British used this notation for the decimal halfpenny when it was in use (1971-1983).
