One of my tutoring students is taking an Introduction to Mathematical Proof course. During one of our recent sessions, he had a problem that challenged him to come up with a way of enumerating the rational numbers, thus showing that they can be mapped to the natural numbers, and thus are a countably infinite set. I saw the proof that the rational numbers are countable way back in undergrad, but I couldn’t remember the process. So, I tried to come up with a means of counting the rational numbers. The enumeration I came up with is solid, I’m pretty sure, and I can’t seem to find it replicated elsewhere on the web.
So, I’m putting it here for others to comment on and to see if anyone can point me to where this method may have been used before. Here is my method for enumerating the rational numbers.
Let , and , such that and is irreducible.
Now let refer to the th prime number.
The number . This value is unique in the natural numbers.
Thus, each positive rational number–and zero–maps onto a unique natural number. This same process can be duplicated for the negative rational numbers. Since both the positive and negative rational numbers can be shown to be countable in this way, the union of the two sets is also countable.
It’s now been well over a month since I last posted something. Jeez. And it’s not because I don’t have anything to say lately. I really do, actually.
I’ve just been letting myself get distracted.
So, there’s not much substance to this post. Mainly, I just want to say that there will be more words up here. I have a number of posts I’ve started but never took the time to properly write out, once-over, and publish.
In general, my life is going really well. I just finished a class–Elementary Analysis–that went well; I feel better prepared for when I eventually retake the GRE Mathematics Subject Exam. I have also decided to leave the farm I’ve worked at for the last year-and-a-half. While I love it out there, it has become a source of great frustration as I try to fit it in but also view it as a really low priority. And that’s not fair to them out there. I have more than enough students that want to hire me as a tutor to make up for the hours I spend on the farm.
So, my life is going well. I’m still fighting my ongoing battle against my inclination to passively consume with my desire to actively create. To expand on that topic my next post will be about the areas of my life I want to focus on and how I plan to make them my top priorities. I’d like to write that post while I’m back home visiting my family for the holidays.
I am in such a different mindspace now than I was two months ago. I have come off the biggest failure of my life, paused briefly from the stun, and then threw my momentum into new endeavors. Continue reading
It’s been a little bit since I’ve posted anything. But today has been a good day and those don’t get written about enough. Especially by me. Continue reading