Rational Numbers from Repeated Fractions

Factual Questions
1

We were presented with a problem today in Calculus class.

Problem:
There is a number, with an infinitely repeating trailing digits. Evidently it can be any number but we were given 7.14362 14362 14362 14362 14362… etc. to work with.

After some Googling, I found the Rational Numbers from Repeated Fractions theorem, which at first I thought would be the solution, except I think it only works where the repeating decimal < 1, because I did the formula out and got 714355/714361, which is obviously wrong (maybe I did the formula wrong.)

Any ideas?

2

714355/99999

3

Intriguing and impressive. May I ask how?

4

Well, I figured 1/9 = 0.1111…

So then I thought 1/99999 = 0.0000100001…

And then 14362/99999 = 0.1436214362…

Then add 7, which is equal to 699993/99999

Which gives you a grand total of 714355/99999

5

Why not just go:

X = 7.14362 14362 14362 14362 14362…

So, 100000X = 714362.14362 143362…

Subtracting, we get 99999X = 714355

So, X = 714355/99999

6

Hey, why not? :slight_smile: