I divide a number by a prime number and I observed the division in the below picture. I suspect that the result, quotient, goes to infinity. But how can I check if so? The big calculator showed the same repetitive pattern goes until 90000 digit only as it was apparently limited by the program.
How to check if the number after decimal point of the result of a division goes to infinity?
Also, as you see there is a repetitive pattern in the quotient, which is 473684210526315789. Is there anything mathematically special about this repetition?
Is there any other similar example division that gives a result going to infinity with a pattern like this?
Thank you.
Best Answer
It seems like the question is something like "how can I be sure that the pattern that seems to repeat will actually repeat forever?".
Every fraction of positive integers will have a decimal portion that repeats forever, and the length of the repeating portion will be less than the denominator (it will be exactly one less when the denominator is prime). For an in-depth explanation of that fact and more about the length of the repeating part, see answers on MSE like this one, which get into why Euler's Totient Theorem is relevant.
The only remaining question is, what if we could be misled by a portion of decimals before the repeating portion? Well, the initial non-repeating part after the decimal point is caused by factors of 2 or 5 in the denominator, which don't affect the repeating business. For example, $\dfrac{1}{24}=0.0416666\ldots$, and we have three extra digits ($041$) before the repeating part, which has length 1 ($6$ repeats), which is certainly less than $24/2^3=3$.
So a general rule which will work okay for small denominators (but isn't best-possible), is this: throw away as many digits as there are factors of 2 and/or 5 in the denominator, and then the true repeating part will be less long than the denominator (even after you divide out by those 2s and 5s), so you can figure out the real answer.
For example, say you're calculating $\dfrac{4235}{38}$ and your calculator says $111.4473684210526315789473684210526315789473$ and you want to know if the $473684210526315789$s really repeat forever. Well, 38 is divisible by just one factor of 2 and no factors of 5, so we can throw away the first one digit after the decimal point (4) and worry just about the 473… 38/2=17, so the repeating part has length at most 16. $473684210526315789$ has length 16 and appears to repeat, so it does.
These methods are not efficient for certain larger fractions. For instance, $\dfrac{42815303527}{2499999750}$ has a repeating part of length only 7, but you'd need more clever techniques to be sure of it without calculating 10 million digits (or ~6.6million if blindly using Euler totient function).