Tag Archives: philosophy

What’s so special about prime numbers?

If you’ve ever watched Numberphile, you’ve probably heard a *lot* about prime numbers. In school prime numbers are mostly just curiosities. They’re numbers that can only be (cleanly) divided by 1 and themselves, so you hate getting them in a fraction. But the further you go in higher math, the more prime numbers seem to show up *everywhere* even in places you wouldn’t expect them.

My new favorite Numberphile video is on the reciprocals of prime numbers. A “reciprocal” of a number is just 1 divided by that number. So the reciprocal of “10” is “1/10” or in decimal form 0.1 . The video shows off the work of a 19th century mathematician named William Shanks who exhaustively catalogued the reciprocals of primes.

Because you see, prime numbers are special this way. Prime numbers don’t make “clean” reciprocals like 1/10 . The reciprocal of a prime tends to be made up of infinitely repeating digits instead. 1/7 is equal to 0.142857142857142857142857142857142857142857142857142857 with the “142857” part repeating infinitely. In math class we represented this with a line over the repeating digits. But I’m having trouble getting wordpress to properly display bars over numbers, so I’ll use “…” to represent repeating digits instead. So 1/7 would be 0.142857… in my decimal notation.

Now back to primes. What Shanks did was he took the reciprocal of larger and larger prime numbers and counted how many digits it took before the the numbers start repeating. So 1/7 repeats after 6 digits while 1/11 repeats after just 2 digits (0.09…). Shanks catalogued these repeating digits all the way up to prime numbers in the 80,000 range, whose reciprocals don’t start repeating until 60,000 digits or more.

The video is well worth a watch, and it’s fascinating to wonder if there’s any pattern to the data. But what struck me was a question from the host Brady near the beginning of the video: “do the reciprocals of all primes repeat?” The mathematician Matt Parker answered “yes” and continued the math lecture, but this got me thinking.

As soon as I told this question to a friend, they immediately said what many of you are probably thinking: “what about 1/5?” 5 is a prime number itself, but 1/5 is a nice, clean, non-repeating number of 0.2 . 2 is also a prime number and makes a clear 0.5 with its reciprocal. Maybe Matt Parker just wasn’t so attentive when he answered “yes” but it seems that not all reciprocals of primes repeat.

But then why are 2 and 5 so special? Why, out of every single prime number, are they the only ones with non-repeating reciprocals? Again I think everyone knows the answer: it has to do with Base 10, but I wanted to study this phenomenon a bit more so I did some math myself.

First, a quick note: we say our counting system is “base 10” because when writing a large number, each position in the number corresponds to units of 10 raised to some power. You may remember from school writing a number like 435 and being taught that is has “4 hundreds,” “3 tens,” and “5 ones.” AKA 435 is (4*100) + (3*10) + (5*1). It’s important that all the positions in a base 10 counting system correspond to 10x for some value X. The hundreds place represents 102, the tens place represents 101, and the ones place represents 100.

Now what about a base 12 counting system instead? What does the number 435 mean in base 12? Just like before, each position corresponds to some power of 12. So 122 is 144, meaning that 4 is in the “144s” place. The 3 is in the “12s” place and the 5 is still in the “1s” place because 60 and 100 both equal 1. So a 435 in base 6 is equal to (4*144) + (3*12) + (5*1), which would be 617 in base 10.

Now my question: do the reciprocals of primes still repeat the same way in a base 12 counting system as they do in base 10? We already know that 2 and 5 are special primes in base 10, their reciprocals don’t repeat. How about in base 12?

Well the reciprocal of 2 still works, it’s just equal to 0.6 instead of 0.5. But the reciprocal of 5 suddenly becomes madness

Here I did the long division for 1/5 in base 12. To keep myself on track I wrote a base-10 version of the subtractions I was doing at each step of the long division. And I don’t know how real mathematicians do it, but since I don’t have a number to represent “10” and “11” as single digits, I used “A” and “B”.

As you can see, *now* this prime’s reciprocal *does* repeat, even though it didn’t in base 10!

I think the mathematician was getting at something deeper when he said all reciprocals of primes repeat, but I’ll have to save it for another post as I had wanted to publish this one on Sunday and I’m already 3 days late.

Some philosophers are just preachers

The word for “preach” in English comes from the word “proclaim” in Latin. It did not necessarily have religious connotations in that language, you could proclaim just about anything. And yet today the word “preach” in invariably tied to a specific type of proclamation who’s connotation cannot be separated from the word. “Don’t preach at me,” “he’s preaching to the choir,” preaching connotes a way of speaking that accepts no argument and engenders no debate. What is preached is correct (as believed by the preacher), regardless of whether you like it.

That’s why it’s amusing how many philosophers I’ve seen who are just preachers and not, you know, philosophers. Not all of them mind you, some philosophers are ready willing and able to dive into the weeds of actually proving their conclusions (or at least trying to). But I’ve been to church enough times to know when I’m being preached at, so the proliferation of dime-store philosophers online are to me no more worthwhile than the doomsday preachers on the street corners.

Today’s preacher de jure is whoever the hell wrote this which was linked to me as a “compelling argument” in favor of utilitarianism and effective altruism. It includes this lovely passage:

You know what? This isn’t about your feelings. A human life, with all its joys and all its pains, adding up over the course of decades, is worth far more than your brain’s feelings of comfort or discomfort with a plan. Does computing the expected utility feel too cold-blooded for your taste? Well, that feeling isn’t even a feather in the scales, when a life is at stake. Just shut up and multiply.

I’m sure this sounded good as an imaginary debate in the author’s head, “FACTS DON’T CARE ABOUT YOUR FEELINGS” and all, but it gets to the type of anti-utility preaching that I’m surprised a supposedly utilitarian author is falling into. Anti-utility preaching is what I would define as the type of preaching done not for others but for oneself. Even if you are religious, even if you do think preaching can change people’s minds for the better, there are some preachers who have no desire to do that and just want to feel righteous by screaming at all the “sinners.” These preachers aren’t changing minds, they aren’t being productive in any way, and in fact are clearly driven by vanity, which most Christians and other religions think is a sin. In the same respect, if the utilitarian who wrote the above passage were really serious about changing minds, they should probably have had the self-awareness to realize that this method isn’t effective and is probably just turning more people off of their philosophy because they’re being a dick.

Just for fun, this leads me to a thought experiment: effective altruism of the kind this person is advocating for tries to use a kind of “ethical calculus” to calculate the exact goodness or badness of any action, and thus the correct action is the one that maximizes goodness and minimizes badness. Is it worth mutilating a child in order to perform an experiment which will cure cancer? Add up the goodness and badness of each scenario and find out that yes, that is perfectly valid. Even stranger, is it worth killing one person to cure everyone’s hiccups forever? Strangely enough yes, yes it is, this utilitarian preacher reasons that the tiny amount of badness caused by a hiccup, multiplied by *everyone* is greater than the amount of badness from killing of a single person.

So let’s go a tiny bit further, if we accept that effective altruism is truly the best morality, then it must also be true that goodness is maximized by more and more people becoming effective altruists. This follows logically from the fact that effective altruists are going to be better than other folks at making “the right” choices and therefore increasing goodness and decreasing badness with their actions. So in keeping with the hiccup example above, decreasing the number of effective altruists in the world will decrease the amount of goodness and so there must be some reduction in effective altruists that adds up to being worse than murder on the balance of goodness and badness in the world.

Therefore I can confidently state that by its own logic this article is worse than murder. It tangibly reduced my desire (if there ever was any) to be an effective altruist just so I don’t become associated with people like the writer, and it probably did the same to most people that read it. The author can take a nice warm bask in their own vanity, while feeling happy that they got to preach to the sinners on the internet. No minds were changed, no one was “saved,” but the preacher felt damn good doing it and isn’t that really what’s important?

Final addendum, I just looked it up and it seems Eliezer Yudkowsky was the author of this trash. But I don’t feel like rewriting the above post to include that fact so I’ve tossed it here at the end.