At work, to help cure my boredom, I play around with different number. Figuring out what I make on a per minute basis, and other useless stupid stuff like that. Now sometimes, to make my calculations easier, I convert decimals into fractions. But I haven’t figure out, if it’s possible, how to convert repeating decimals into fractions. For example, .6 is 6/10 which recuses to 3/5. OK, easy enough.
But how do you change .666666… into 2/3. Or .11111… to 1/9? Is there a way to do it? Thanks.
An easy rule of thumb if the decimal repeats itself entirely is to take the repeating part as the numerator and put it ‘over’ as many 9’s as decimal places as it’s repeating. For instance…
.2727272727…
27/99 which reduces to…
9/33 and…
3/11
Hope that helps!!
The Reason This Works:
If your number is, say 0.27272727…, then call that x
x = 0.2727272727…
**100x = 27.27272727…
100x - x = 27** (gets rid of that nasty, repeating part)
**99 x = 27
x = 27/99
x = 3/11**
It’s fairly easy to prove that:
0.11111… = 1/9
0.010101… = 1/99
0.001001001… = 1/999
0.000100010001… = 1/9999
(Hints for a proof: these are decimal fractions, so the associated series converge in a very well-behaved way, and 0.999999… converges to (i.e., is equal to) 1.)
More advanced exercise: prove that any decimal number expressible as the ratio of two natural numbers is eventually periodic – its decimal expansion consists of a number of digits, followed by another sequence of digits which repeats itself from then on.
Trivial clarification, which I hesitate to make out of simple politeness but feel compelled to out of simple anal-retentiveness : any non-terminating decimal expressible as a ratio is evenutally periodic. One-seventh (0.142857142857…) is non-terminating and repeats, but one-half (0.5) is terminating and doesn’t.
Strictly speaking, .5 should be written as .50000…
Wow, an easy and simple solution. Thank you all.
In other words, in the decimal .5, nothing is repeating.
Indeed. It’s just not too easy to prove to those who cannot, or will not, abandon their preconceived notions about limits (“but it never quite gets there …”). I mention this because of:
and
and
and even
and, of course,
and
this.
Oh, no… Now somebody’s going to bump one of those, and we’ll have to go through all that again…
Mathochist may have been thinking of .5 = .4999… (not that there’s anything wrong with that)
Even .499… isn’t periodic, surely?
Why not? A bunch of nines repeating with a period of one?
In either case, there’s a finite aperiodic tail, and then a bunch of repeating digits.
In “.5”?
In .500000000…
I accept your proof is true. I just don’t believe it. 
Maybe my problem is with infinity itself. I mean, I understand taking, say, an inch long bar, and cutting it in half, and then you can take that half inch and cut it in half again, and so on. But don’t you eventually just run into a point where you can’t cut it in half anymore? I know this is a stupid question, but I really don’t understand it.
Yes, but only if you’re dealing with an actual bar.
Analogies with physical processes in mathematics are a double-edged sword because they introduce real-world limitations. For a theory that exists to model the real world, that’s a good thing, but for an abstract, idealized theory (such as decimal representations), that’s a very bad thing.
I could not even read your question, because, before reading all of it, I would have to read the first half of it. And before I could do that, I would have to read the first half of that half. But first I would have to read the first half of that. And so on.
