In recent years, I've been on this kick of understanding mathematical things we are taught early in life, but which are typically never explained or proved. In these pages, I've written out my notes about rationals and repeating decimals.
We're taught that rationals repeat or terminate, and that irrationals do neither. I don't know about your
education, but in mine it never really explored the subject much beyond that. Finding what a rational looked
like involved doing long division until you ran out of remainder, or until you spotted a pattern. Going from
a repeating decimal to a rational wasn't covered at all, unless for simple cases like 1/3
(0.3).
I think I started out trying to figure out how to go from a repeating decimal to a rational. In the process, I figured out a lot of (what I considered to be) neat tricks. But really, I was just learning about how repeating decimals "work" in general. Once you know how they work, going from a repeating decimal to a rational is trivial!
Soon after that, it became apparant that the long division form of repeating decimal discovery is pretty lame.
It's so ad-hoc! What guarantee do I have that it really is going to repeat or terminate anyway? This led to
two very related explorations. For one, I needed to be sure that rationals really do
all repeat or terminate. But what I really wanted was a better method for finding the repeating decimal.
Now I knew how repeating decimals work, so I knew that I needed to find a way to get
a relatively prime number into [base]x-1.
The key is related to the totient function. And once you know that, it's easy to see that rationals repeat or terminate.
Once you have a grip on these things, it's easy to make a mechanical process for going from rationals to repeating decimals and back. Sometimes you can use a few tricks to take a shortcut, as well.
At any rate, here are the pages completed so far:
And now for some brain-dumping that should really have it's own page, but doesn't yet.
Another thing I realized while working on all these things, especially when you start messing around with other bases, is that repeating decimals (and decimals in general) are just an artifact of how we represent these numbers "on paper". Of course, given our education, we all tend to think about numbers in decimal in our heads as well. It's hard to have some abstract notion of every single number, after all! So we tie it to a base.
I've found that most of these "things we're taught but are never explored" issues link back up to how we do things mechanically, on paper. That is, they're intimately related to representing numbers in a specific base. "Repeating decimal" has no meaning with no base! Writing a number down is simply converting the "abstract" number into powers of your base. Repeating decimals are a side-effect; many rationals can only be represented as an infinite sum of powers of 10 (or any other base).
Other issues, like "why does the on-paper method of adding, with these carries, work?" also tie back to representing things in a specific base. A given digit is really that digit (capped to be less than the base) times some power of the base. Carrying is what happens when you exceed the "cap" -- your multiplier for this power of your base has grown bigger than your base, so you "carry" part of it into the next power.
It's interesting to ponder how these methods came to be, before the abstraction of different bases.
At any rate, number theory is well suited to exploring these issues. Every digit is a sort of modulo of the base.
And at this point, my brain dump peters out. Hopefully to be rounded out later...