Does pi follow a pattern?
…no, but maybe mathematicians do. I recently met someone, and when he asked me what I did my Ph.D. in (math),
he said “math…like just straight numbers, huh?”
“yup” (Not really. Low dimensional topology actually, but I wasn’t about to bring that up.)
He followed up quickly with “How many digits of pi do you know?”
Here’s the embarassing thing: my reply, “3.141592653589793”
That’s about as far as I can get these days. But he stopped me anyway. He was convinced. I was a bonified mathematician.
Contrast this story with one a friend in grad school told me. He met a man on a train in Switzerland, who happened to also be a mathematician. What did the Swiss man ask my friend to verify that he was a mathematician?
“Prove that if p is a prime number, then either p+1 or p-1 is a multiple of 6.”
Note the difference. The non-mathematician asked me something about what I know. The mathematician asked my friend something about how he thinks.
One of my points is that becoming a mathematician is more about developing a thought process than learning facts.
My other point is that I’m embarrassed to have so many digits of pi memorized. Which brings me to how and when I memorized it in the first place: the 11th grade, precalculus. (Hey, there was nothing else interesting going on in the class.) The text book had pi written out to several hundred places. I memorized maybe 40 or 50.
Later that school year in a health class we answered a questionnaire that categorized us according to some color scheme. I was a green. I was the green, the only one in the class! Then we divided up into our groups (hah) and collaborated (hah hah) to make posters describing characteristics shared by the group (hah hah hah).
One thing that I put on my poster (besides that I prefer to walk rather than drive) was as many digits of pi as I could remember off the top of my head (which was probably at least 30).
I should have put on the poster why either p+1 of p-1 is a multiple of 6, but I didn’t know any better at the time.
The teacher asked whether pi follows any pattern.
I said “no.”
Of course it does, actually. It follows the pattern that it is pi: the circumferance of a circle divided by it’s diameter. But I wasn’t about to bring that up. That’s not really what he meant any way.
But what did he mean?
Maybe he meant this: “OK, I know that pi goes on forever (the decimal expansion does, anyway). But does it repeat itself?”
The answer to that is “no,” which is the same as saying that pi is irrational.
But I don’t think that he meant exactly “does it repeat?” I thought that was what he meant at the time, but not anymore.
Here is an example of an irrational number (I’ll call it y) that does follow a pattern:
See what I mean? It doesn’t repeat itself, but it does follow a pattern.
I once had a professor who called this irrational number y “stupid,” which almost offended me (the context was that my friend who met the Swiss man gave the professor y as an example of an irrational number in the Cantor set if you interpret y as being written here in base 3, but I”m not about to bring that up here).
I prefer to call irrationals that follow a nice easy pattern like this “fun.”
I’ll have more to say about fun irrationals in another post, or at least irrationals that toe the line of being fun (I haven’t given “fun irrationals” any kind of hard and fast definition).
How about you? Are you a mathematician?
Even if you don’t think you are, can you think of any fun irrationals?
Can you tell me why p+1 or p-1 is a multiple of 6?
Try it, and let me know, especially if you think of any interesting (and/or) fun irrationals.