Enumerating the Rational Numbers

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 m, n \in \mathbb{Z} and m \ge 0, n > 0 such that \frac{m}{n} \in \mathbb{Q} and is irreducible.

Now let p_n refer to the nth prime number.

The number (p_n)^m \in \mathbb{N}. 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.

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Criteria for a Facebook Response

Someone I respect commented on an article I posted on Facebook. I disagreed with some of his points. He also offered a point of clarification. The conversation I wanted to have was then derailed by my own defensiveness. I let my disagreement with a couple of his points overflow onto his whole post. Reflecting on the conversation, I really wish I had engaged with his points. I could have learned much more and the conversation could have been worth following.

I don’t really know why I responded the way I did. I’ve since apologized to him, after he called me out in the course of the comment thread. It made me realize, though, that I’m not living up to my ideals when it comes to discourse. Many of the conversations that I want to have are happening via Facebook now. And I’ve let myself slip into the same entrenchment that I claim to abhor.

I value argument, discussion, conversation, and learning. It is important to me that my point of view and position on issues are malleable. So, I’m making a change. I’ve decided to create some criteria for responding to posts and comments on Facebook (and other social media).

My criteria for a Facebook response: Continue reading

This is my anxiety

It’s a tingle, like an itch. From my shoulder blades down to the tips of my fingers and back. Then it reverberates in my chest. My stomach clenches, then my head swims. I remember to breathe Continue reading

Listen, the First Amendment guarantees you freedom of speech, certainly.

That freedom it’s protecting is freedom from *government* censorship. You are free to say whatever you want without *government* persecution. You are not protected from the criticisms of others though. And yes a person or company can refuse you based on what you say because they are not the government.

The First Amendment gives you a broad right of expression, but that means it also bestows a great *responsibility* to self-regulate the ideas and thoughts you express.

So, please stop claiming that this is America and you can say whatever you want and expect that to protect you from being held responsible for the words that tumble from your slack jaw.

I’m around, I just haven’t been writing

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.

A Faith Unshaken

click image for source: Amnesty International

On October 9th Malala Yousafzai, a 15-year-old girl living in Swat, Pakistan, was shot in the head by members of Tehreek-e-Taliban Pakistan. Malala has already become celebrated in Pakistan for speaking out against Taliban rule and for fighting for education for women.

That morning, as I drove to work, I listened to an NPR broadcast about the shooting and I was sickened and angered and scared all at the same time. Still am, really, when I see her story trickle up in my various feeds.

She has such huge strength. And she was shot with deliberate intent. They stopped her bus, asked where she was, and shot her. Then left her. It is one of the most cowardly, evil acts I have ever heard of.

And yet, my faith is not shaken. It is not the Taliban’s evil that we should focus on, but on Malala’s courage and determination in the face of their terrorism. This story strengthens my faith in humanity because Malala exists and her spark is one that will never be stifled.

news sources:
Aljazeera, NPR

Thinking Out Loud: Faith

Love depends intimately and completely on faith. When I am confronted with no evidence, or counter-evidence, for the love that is expressed for me, I have to take the words on faith. That faith has to be more powerful than my own insecurities, than the apathy or disinterest that threatens to undermine the expression.

It seems like it is harder to have faith in a person than in divinity. People will challenge my faith in them, will disappoint me, push me away. Divinity, though, is only what a person wants it to be. How can one’s faith be challenged if it is all in one’s head?

At the same time, I’ve always held that what people credit to divine influence is really their own unacknowledged inner-strength. So, having faith in a deity I would interpret as faith in the self. And I know just how hard it is to maintain faith in yourself. While others may disappoint us, it is easy to forgive when we find out their perspective. I know when I disappoint myself, though, it takes me a long time to forgive. I hold myself to a higher standard.

Hmm, whether it is in myself or someone else, faith in people is hard.

Faith is a struggle I understand only too well.
When she says “I love you,”
The words are only as true as I believe.
Some days, such words mean more than nothing
Due only to my investment in their truth,
I can find no evidence to prove them.
Ain’t that the case with any love?
More than we see it, we feel it, believe it.
Love can only exist through Faith.

This is an early draft of a portion of a verse to a larger poem I’m working on. The whole poem is about how I, as an atheist, experience Faith.