In my discrete maths class, I have come across an interesting phenomenon for which I can't find an explanation!
If we divide $1$ by $13$ we obtain $0.07692307\ldots$
If we divide $3$ by $13$ we obtain $0.23076923\ldots$
If we divide $4$ by $13$ we obtain $0.30769230\ldots$
As you can see, the digits are recurring in the same order but starting at a different point in the sequence.
Can someone explain this to me? What exactly is happening here?
Well, let me start by writing some equations which are boring: $$\frac{1}{13}=0.0769230769230$$ $$\frac{10}{13}=0.769230769230$$ $$\frac{100}{13}=7.69230769230$$ $$\frac{1000}{13}=76.9230769230$$ $$\frac{10000}{13}=769.230769230$$ $$\frac{100000}{13}=7692.30769230$$ Obviously, multiplying by $10$ just shifts the sequence over. However, suppose we subtract out the integer part of each of these equations. Then they're suddenly interesting as they each repeat the same pattern starting from a different place: $$\frac{1}{13}-0=\frac{1}{13}=0.0769230769230$$ $$\frac{10}{13}-0=\frac{10}{13}=0.769230769230$$ $$\frac{100}{13}-7=\frac{9}{13}=0.69230769230$$ $$\frac{1000}{13}-76=\frac{12}{13}=0.9230769230$$ $$\frac{10000}{13}-769=\frac{3}{13}=0.230769230$$ $$\frac{100000}{13}-7692=\frac{4}{13}=0.30769230$$ Hey, those equations are all pretty neat- they have the same property that the digits appear in a shifted order, and if you just wrote out the fraction without seeing where it comes from, that's not terribly obvious.
Notice that this only works for the numerators $1,\,3,\,4,\,9,\,10,$ and $12$. The reason these numbers are special is that they are the powers of $10$ mod $13$. More generally, we could replace $13$ by any number $n$ coprime to $10$ and say that this is true of the powers of $10$ mod $n$. This is particularly interesting when every integer between $1$ and $n-1$ is a power of $10$ mod $n$, meaning that every non-integer fraction would have this property (which is true of $n=7$ and other numbers listed here)
