If we’re measuring a length of pi centimeters, why does it look finite but the number of digits is going on forever? It looks like that it’s moving small amount of time every time. Same with 0.5555555… or any number with infinite decimal places. I can’t wrap my head around it.
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In Ancient Greece there was a guy named Zeno and he came up with an interesting paradox:
To get from one point to another you have to cross half the distance. Then you have to cross half *that* distance. Then again. Then again.
But no amount of halvings will ever get you completely to the other point. It would take an infinite number of these half-steps to get there, and since you can’t do an infinite number of things in a finite amount of time, getting from one point to another is impossible.
I bring up the paradox to illustrate this concept of infinite halvings. Let’s convert that concept into mathematics. We take a half, 1/2. Then half of that, 1/4. Then half of that, 1/8. And so on, and add them together:
1/2 + 1/4 + 1/8 + 1/16 …
What’s great about math is we *can* do an infinite amount of things in finite time. Specifically, we can add (some) sequences of infinite numbers together. In fact, if you add the above infinite sequence the total is *exactly* 1:
1 = 1/2 + 1/4 + 1/8 + 1/16 …
What does this have to do with numbers whose decimal representations go on forever? Well, just remember that each place in a decimal representation of a number is simply some multiple of a power of ten, and the value of the number is those multiples added together:
325 = 3*10^(2) + 2*10^(1) + 5*10^(0)
And this is true for fractional pats as well:
0.325 = 3*10^(-1) + 2*10^(-2) + 5*10^(-3)
Or, written another way:
0.325 = 3/10 + 2/100 + 5/1000
Some numbers, however, can only be written this way if you have an infinite number of terms. Consider:
1/3 = 3/10 + 3/100 + 3/1000 + 3/1000…
Why? Well this is simply a quirk of our base-ten writing system. Only numbers that share a factor with 10 (2 or 5) can be written using a finite number of decimal places. Because only terms that share a factor with 10 can be evenly divided by it. All other numbers will always have a remainder that must also be divided, which itself well have a remainder that must be divided, and so forth.
But, still, at the end of the day, all of those terms sum up to a single, finite number.
You might ask, how can an infinite amount of numbers sum to a finite number?
Well, to be fair, this is not always the case. But with the examples we’ve been using, they have a special property called *convergence.* You should notice that each term is smaller than the last. So while the sum is getting bigger and bigger, it does so in smaller and smaller increments. If the increments shrink in size fast enough, instead of the sum exploding to infinity, it instead will approach a number and get continually and continually closer without ever reaching it (using only a finite amount of terms), just like Zeno.
Imagine you have a pizza. Can you divide it in half? Sure. Can you divide it in thirds? Sure.
Ok, now imagine you have ten pizzas. Without cutting any of them, can you divide them in half? Sure, 5 and 5. Ok, now divide it in to thirds. 3, 3, 3…and 1. Hmmm.
Ok, what if I let you divide that left over pizza? Great, you say now its easy again, I can just cut it in thirds! Waaaaaait just a minute I say. You can divide that left over pizza, but it has to be in pieces of 10. So now you have 3 pizzas and 1 pizza in 10 slices. Now can you divide it in thirds? Ok, 3 pizzas + 3 slices of pizza 3 times, but you are still left with one extra slice of pizza. Ok, now divide THAT slice of pizza into 10 even smaller slices. Wash. Rinse. Repeat.
Thats how the decimal system works, each time you move down a decimal point you are dividing a smaller quantity by 10 and then figuring out how many you need. Any left over is divided into another 10 pieces, and you keep trying to get closer and closer and closer.
Since we have chosen 10 as the base (thanks to our 10 fingers and 10 toes most likely) we’ll run into lots of situations where whatever we are measuring can’t be represented cleanly. We could have chosen a different base, say 12, in which case its easy to represent 1/3 (and 1/4 and 1/6) but we can no longer represent 1/5 cleanly. No matter what base we choose there will always be some numbers which we can represent as finite and some which will be infinite.
But keep in mind that is just how we are representing those numbers, it doesn’t mean the actual quantity they represent is infinite, as each decimal we add is a smaller and smaller fraction of the overall length, it eventually reaches a point where the other digits are meaningless.
**Fun fact:** Using just the first 40 digits of pi you can accurately calculate the circumference of the ENTIRE VISIBLE UNIVERSE to a margin of error about the size of a hydrogen atom. In the real world will probably never need to worry about even 40 digits of precision because the tools we use to measure are going to be less accurate than whatever we are trying to measure.
It seems the confusion here is stemming from imagining that each new digit is adding on to the length. That would be incorrect. Each new digit is increasing the *accuracy*.
We measure a string with a length of pi. It’s a bit more than 3 but a lot less than 4 so we add a decimal. Now we measure and the same string is a more than 3.1 but less than 3.2. We still don’t accurately have the string measured, so we add another decimal. It looks like the *same string* is longer than 3.14 but less than 3.15.
We can continue this indefinitely, but the key is that the string we are measuring stays exactly the same length. Our ability to measure it accurately changes as we keep adding digits, but fundamentally it’s no different than just measuring something that is exactly 3.0000000…. We can keep measuring it and adding on zeros to confirm that it isn’t 3.00…1, but adding an infinite number of zeros (or other digits in the case of pi) doesn’t conflict with the thing we are actually measuring being finite.
You and your 2 friends want to split a square cake. You could try to cut it in three, but that is difficult. It is a lot easier to cut it in four, take one piece each, cut the remaining piece in four, take a piece each, cut the next piece in four, take a piece each, cut… and so on. Each time you get closer and closer to getting 1/3 of the full cake, but to hit exactly 1/3 you need to continue the process of cutting and dividing forever. Therefore the amount is finite but the dicimals are infinite
The length of something that you write down doesn’t have to be related to what it looks like.
For example, the word “big” is shorter than the word “small”, but a big thing is bigger than a small thing.
In English, we just happen to need more letters to say the word “small” than we do “big”.
The number 0.5555555… represents a very specific size. It needs infinite digits to spell out, but just because you need to write a lot of digits doesn’t mean it’s a bigger number. It just means you need to use more pen ink to write about it and more breath to talk about it.
So, just like in English where some words need more letters to say, in math, some numbers need more digits to say. But there isn’t really a link between the number of letters in a word and the word’s meaning, and there isn’t really a link between the number of digits in a number and the number’s meaning.
(Actually, we don’t even need to write out all the digits; just as you have done, putting “…” at the end of it is the same as writing all the 5s out infinitely. So actually, 0.555… is exactly the same number as 0.5555… which is exactly the same number as 0.55555… which is exactly the same number as 5/9. We have many ways of writing a number that all have the same meaning!)
Anyway, the reason the 5s go on forever is because, well, it’s a different number than if you stop the 5s at some point. 0.55555…5 with one hundred 5s is a liiiiiiiittle smaller than 0.55555… infinitely.
We write decimals in base 10. That means any number you write down is being represented as the sum of a bunch of powers of 10. For example, 0.426 is 4 x 1/10 + 2 x 1/100 + 6 x 1/1000.
Some (in fact most) numbers can’t be written exactly as the sum of powers of 10.
Someone else here mentioned 1/3. It’s a finite quantity, but you can never quite get there with powers of 10. If you try to estimate it with one decimal place, it’s bigger than 0.3 but smaller than 0.4. Likewise, it’s bigger than 0.33 but smaller than 0.34. It’s bigger than 0.3333333 but smaller than 0.3333334. In fact, no matter how many decimal places you take, 1/3 is always “in the crack” between the last digit being a 3 and a 4. That’s why you have to have infinite 3s in the decimal. The more places, the closer it gets… but it never reaches it in a finite number of places.
You need to realise that you are talking about two different numbers:
Pi, which is finite.
The number of digits in pi, which is infinite.
For any given base, there are some numbers that can not be expressed with a finite number of digits. Take 1/3, for example. In base 10, you need infinitely many digits to write the exact value of this number. Any finite length of 0.33333… will always be a tiny, tiny bit less than 1/3.
They’re not just numbers. They are ratios of something. Those ratios depending on your perspective are either perfect or imperfect.
But at the end of it, the simple answer is, they represent a relationship between something, those relationships dont always have clean edges.
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