For example, if you were given 0.428571..., and you wanted to convert it into a fraction, then you could immediately write it as 428571/999999, and then you could either leave it like that if you were lazy, or simplify it by taking out a bunch of common factors.

428571/999999

142857/333333

47619/111111

15873/37037

1443/3367

111/259

3/7

OK, so that took a while, but a computer could have done that very quickly as long as the loop of reccuring decimals wasn't very huge. It might take a bit longer to analyse 1/19, but it still could eventually.

This can be extended to decimals that have a terminating bit before they start recurring as well.

Again, this might not be the quickest way, but this is how I do it:

Let's take 0.27888... as a random example. The recurring part of that is 888..., which would mean 8/9 if it was there from the start. But we can work out that this number is exactly 0.61 less than 0.888... and so now we've got it as a terminating decimal and a purely recurring one.

These can both be written as fractions, -61/100 and 8/9, which when added is 251/900 and that is the fraction in its simplest form.

So by combining these two methods, it's possible to work out what the fractional form of any rational number is from its decimal.

]]>On the other hand, infinite decimals are defined by guessing, too. Because we can only work out or deduct finite decimals.

Wow, there isn't a method yet? Could be something we could work on

]]>so you have to do 1/9 times 9 to get 1.0 . so now you have a 1 and a nine! then i think you have to put time the numbers together or something. i cant remember but you end up with 1/9]]>

for example,

0.3+0.33+0.333+... approaches 1/3 when the amount of addee increases, or put another way, you can get the sum as close to 1/3 so long as you make enough additions.

So far it is still in finite sum and finite framework. Then how do we know the final result of the infinite sum? Simple, we assume it, we define it. (That is what Ricky has admitted) We just assume 0.3+0.33+0.333+...=1/3 on finite experiences and deductions. (I call this kind of assumption "guessing", "derivative", or whatever)

On the other hand, infinite decimals are defined by guessing, too. Because we can only work out or deduct finite decimals.

]]>0.052631578947368421...

]]>I can tell 0 is present if you see 7/7 as 1.0

I don't know, I study it in primary school just for arithmetric convenience. We pupils also memorized 11²=121, 12²=144, 13²=169, 14²=196, 15²=225, 16²=256, 17²=289, 18²=... I forgot.

These are actually quite handy to remember, and not too hard to learn. First of all becuase they're not too hard to calculate, 18*18=10*18+8*18=180+80+64. It also helps, though, that 18*18 and 8*8 will have the same last digit. If you only have to remember so many squares, the last digit can help you remember the full number. That's how I remember them atleast.

]]>I don't know, I study it in primary school just for arithmetric convenience. We pupils also memorized 11²=121, 12²=144, 13²=169, 14²=196, 15²=225, 16²=256, 17²=289, 18²=... I forgot.

]]>Why are 0,3,6 and 9 missing?

]]>So then 2/7 = 0.285714... and so on.

If I remember rightly, there was a different set of fractions that had decimal forms that involved the loop 758241 as well, but I've forgotten what the divisor was.

]]>