Explain Like I’m 5: Why Pi never ends

Oh boy.

You just had to ask didn't you? Do you even know what's coming? You might think this is a simple question, with a simple solution, but that is most certainly not the case. I hope your body is sufficiently prepared, because your mind is certainly about to get utterly blown.

So here goes nothing.

Quick review: What is Pi? Pi the ratio of a circle's circumference to its diameter. That means if you were to measure perfectly a circle's circumference C and diameter D, and then divide C / D, you would get something around 3.14, or pi. Well at least in this universe. The answer you seek essentially has to do with the concept of countability.

Just what is countabilty? Well, basically, it means it's possible for you to count it. You can "count" apples, for example, but you can't count the political philosophy of democracy. There are things you can say "how much do I have?" and things you can't.

Oddly enough, although Pi is a number, it's not countable, and since we know it's not countable, we know that its decimal approximation continues on to infinity.

What?

I know, I’m trying to explain this to you like you’re 5. Let's back up a bit.

Let's start with the numbers you learned when you were 5. These are positive, whole numbers. Whole in the sense they don't get broken into fractions or decimals. These are called natural numbers, such as 1, 2, 3, ... up to infinity. These numbers are countable. That is, say I want to list every natural number on a piece of paper. Well, I would start by writing down 1, then write down 2, then 3, and so on until we listed all natural numbers! But wait you say! There's an infinite amount of natural numbers, you couldn't write down all of them, and I don't think a single piece of paper would cut it! Well, you're right. It's impossible to write them all down since they continue on forever. But, say you yourself had an infinite amount of time, energy, and pieces of paper. Well, in that case, it's *possible* to write them all down.

Don't worry if that seems weird. Let's look at negative whole numbers now. When we add negative whole numbers to the natural numbers we form a new set of numbers called the Integers. The big difference here is the additional negative infinity. So we have 1, 2, 3, ... up to infinity, but also -1, -2, -3, ... down to negative infinity. Again though, if we had infinite time, energy, and paper, we could list out all of the integers. We might do it differently, like alternating between positives and negatives, such as 1, then -1, then 2, then -2, etc. But we would eventually be able to list all of them given infinite time.

Continuing on, things are going to start to get a bit tricky now. Not all numbers are whole numbers. Some numbers are parts of whole numbers. Numbers like 1/3, 4.743, -2.5000, 5, are what we call "rational" numbers. Rational numbers include integers, so whole numbers are considered rational, but it also includes decimals, fractions, and other partial numbers too. So now comes the question, how do we list out every single rational number on paper? Or really, is it even possible? It seems like you can't. I mean, let's give it a try. I start with 1. Then I say 1.1. Then I say 1.01. But if I keep going in this pattern, there's always going to be a number less. I could have 1.00000001. But 1.0000000001 is less! It's impossible!

But you've got it all wrong, you see. You can still "count" them. The argument here is no different than the argument for listing out an infinite amount of natural numbers. This time however, the number can also be infinitely divided into parts.

It's better to think of rational numbers as fractions. Decimals will just make things more confusing, and you know you can always convert them fractions to decimals and vice-versa. A fraction represents a division of parts. 2/3's for example indicates a whole number is split into 3 parts, and you have 2 of those. Let's say we decide to keep "halving" the number 1. First we get 1/2. Then we get 1/4. Then 1/8. And so on and so fourth to infinity. Try to imagine actually "reaching" one of these numbers. For example, with 1/2, there are two possible "paths" to reach 1/2, and if we combined both of these "paths" we'd get 1, so there must be 2 paths. Same case with 1/4, except there are now 4 paths. 8 paths with 1/8, continuing on infinitely. The key thing here is that, no matter how high the number is, there is always a finite number of paths. That means we can count these paths. And by this logic, we can count the rational numbers. This doesn't just apply to 2's. We could do the same with 3's, like 1/3, 1/9, 1/27, etc, and get the same results. Does that make sense? I hope it does.

Now let's talk about Pi. Pi is what we call an irrational number. Irrational numbers, are any number that is NOT rational. You might think you can represent pi as a decimal, but can you represent it as a fraction? Go ahead, give it a try. The answer is you can't. Well, you *could* only list some of the decimal values of Pi, and as such get an approximate fraction. But this is not pi. It is only a rational number that is close to Pi. And that's the catch here! Let's start with 3 and divide it up into finite paths like we did before. No matter how many infinitely tiny paths we make, no matter how small the distinction is, pi will ALWAYS be between those paths, ever eluding out grasp.

And that’s that! If you're still confused, don't fret. This is not trivial stuff by any means, and I take no guarantees this would indeed make sense to a 5 year-old.

So to sum things up, Pi, along with all irrational numbers, are not countable. You cannot list all irrational numbers on a piece of paper, even if you had an infinite amount of time. And as from above, since you cannot count, or enumerate Pi, any attempts will always result in an infinite rational number approximation.

This post was inspired by a question on Reddit's "Eli5" subreddit. I may start giving detailed explanations to question I find interesting like I did here.

Shibes

Okay what is this

And who am I for that matter? Welcome to The Medium, a place where I'll be providing you thoughts and rants directly dissected from my head. And I mean that sentiment quite literally actually. You see, in the 22nd century, society popularized the idea of extracting depressed, overstrung, withered engineers' thoughts from their volatile minds and providing them to the masses through the advanced social networking technology of the 22nd century, and delivered via the brute, rod-to-brain electrodes of the 19th century, originally used on the analysis of neural networks in rodents. Unfortunately, I was not able to obtain the advanced social networking technology they had, because I am not capable of time travel. I did however, obtain the primitive electrodes, as they exist in the 21st century. After some debauchery, and a frivolous pursuit to replicate the idea, I came to a startling conclusion: It turns out sticking electrodes in your brain is not at all pleasant for a human body, nor is it natural. So I abandoned the notion, but I felt the experience was too visceral to describe as anything but "literal."

Science, technology, engineering, mathematics. That's a lot of words. Let's call it STEM instead. STEM happens to be a "thing." It's not a great definition, but it apparently means something. And like all "things" in society, it is something humans seem have difficulty coping with. There are just uncountably many issues associated with it: Just what is the best mobile computer to use when you're busy overthrowing your government in the Middle East? How many programs does it take to construct a human that is mentally and physically capable of screwing in a light bulb? And why are there so few transgendered kanurian cyborgs with a markov-chained CAPTCHA tranny mark to be found the scientific labor force? Like a concerned lioness tending to her kin cubs, I am here to caress these answers into the sugary, delicious taffy of exquisite treatise that the masses demand.