That’s not how it’s works. Being “infinite” is not enough, the number 1.110100100010000… is “infinite”, without repeating patterns and dosen’t have other digits that 1 or 0.
If it’s infinite without repeating patterns then it just contain all patterns, no? Eh i guess that’s not how that works, is it? Half of all patterns is still infinity.
Computer programs exist that can tell you what the next digit is. That means it’s deterministic, and running the program will give you a prediction for each digit (within the memory constraints of your computer).
The fact that it’s deterministic is exactly why pi is interesting. If it was random it would typically be much easier to prove properties about it’s digits.
There’s no way to predict what the next unsolved pi digit will be just by looking at what came before it. It’s neither predictable nor deterministic. The very existence of calculations to get the next digit supports that.
Note: I’m not saying Pi is random. Again, the calculations support the general non-randomness of it. It is possible to be unpredictable, undeterministic, and completely logical.
Note Note: I don’t know everything. For all I know, we’re in a simulation and we’ll eventually hit the floating point limit of pi and underflow the universe. I just wanted to point out that your example doesn’t quite fit with pi.
As I said, you can’t predict the next number simply based upon the set of numbers that came before. You have to calculate it, and that calculation can be so complex that it takes insane amounts of energy to do it.
Also, I think I was thinking of the philisophical definition of “deterministic” when I was using it earlier. That doesn’t really apply to pi… unless we really do live in a simulation.
This might just be my computer-focused life talking, but I’ve never heard of deterministic meaning anything but non-random. At best philosophic determinism is about free will and the existence of true randomness, but that just seems like sacred consciousness.
I also don’t know why predictability would be solely based on the numbers that came before. Election predictions are heavily based on polling data, and any good CEO will prepare for coming policy changes, so why ignore context here? If that’s a specific definition in math then fair enough, but that’s not a good argument for or against the existence of arbitrary strings in some numbers. Difficult is a far cry from impossible.
This might just be my computer-focused life talking
I’m a software eng too, but I have broad interests. Like I said, the philosophic use doesn’t really have a place in this discussion and I messed up by bringing it in. The only way it would be relevant is if the universe is a simulation because, as you guessed, then free will itself becomes part of the equation.
I also don’t know why predictability would be solely based on the numbers that came before
There’s a miscommunication happening here, and I’m wondering if I’m not explaining myself well. Election predictions use polling as their dataset, and there are no calculations that really go into predicting the results other than comparing the numbers within those sets. That’s why they’re notoriously garbage (every single pollster had Hillary winning in late October 2016, for example). Also, there aren’t any calculations that go into a CEO/Boardroom’s intuitions on how shareholders will react to policy changes, so I’m not sure about the relevance here. In the case of pi, there is no dataset that you can use that tells you what the next unknown number in pi is. The only way to get that number is to run a very complex calculation. Calculations are not predictions.
If a new government makes it known that they’ll increase tariffs, any company dealing with international trade can prepare for increased costs sooner. Or even better, if there’s a movement for emissions regulations, they can get lobbyists and lawyers to find or add loopholes nice and early, long before cars would actually need to be more efficient.
Anyway, the miscommunication here seems to be what you mean by nondeterministic and unpredictable. We’ve been through deterministic, and that doesn’t perclude unpredictable.
For example, cryptographic hashes are completely deterministic yet impossible to predict. The determinism allows easily checking for the correct string, but the unpredictability makes guessing the correct string impossible beyond brute force. Yet if a security protocol used π to seed it’s hashes, it would be way more predictable than most methods. Even if your psudorandom table is indistinguishable from noise, if the table is known the whole system can be cracked relatively easily. Thus π would make that method predictable.
Now you could mean that each character or string says very little about each other character or string, but that’s a different claim; that you can’t predict one part of the number using another part. For example, if you say the code to your luggage is a five digit string starting at the 49702nd digit of π, that’s easy to lookup. But no amount of digits will help you figure out that this string really is from π and not something else. I’d call that chaotic rather than unpredictable, as unpredictable makes me think of probability more than calculability. π is found in so many places that many sci-fi stories use it in first contact scenarios, alongside e, the hydrogen line, Fibonacci numbers, and c or sometimes hbar. Dependable is hardly unpredictable.
If we go back to your original reason for describing the predictability of some numbers, we find a simple nonrepeating number (101100111000… let’s call it b for binary) and π. With b, any string can at least narrow down it’s location in the number, and if a string contains both 10 and 01 we can positively fix it’s location, even to a place that no one has calculated before. This is impossible with π, we can positively rule out many positions, but no position can be confirmed for any string, and any string may appear further in the digits as well, giving multiple possible positions for any string.
However we can still compute π, and thus can know (even better than predict!) any arbitrarily precise digit in finite time. There are numbers where that’s not possible, so-called non-computable numbers (For example). This number cannot to computed in finite time, only approximated. This sounds more unpredictable.
Predictability could be seen as a function of the ease of calculability, especially when time is a limited resource, but why not just say that π is more complex to predict than b, or that the existence of b doesn’t disprove that π can contain all finite strings? That was the original issue after all.
Sorry, TL;DR, I don’t think unpredictable is a good word to use outside of probability, and even so an easily predictable number is enough to prove that not all irrational numbers are normal numbers (not all numbers with infinite digits contain all finite strings).
However, as the name implies, this is nothing special about pi. Almost all numbers have this property. If anything, it’s the integers that we should be finding weird, like you mean to tell me that every single digit after the decimal point is a zero? No matter how far you go, just zeroes forever?
In some encoding scheme, those digits can represent something other than binary digits. If we consider your string of digits to truly be infinite, some substring somewhere will be meaningful.
One of the many things I loved about Sagan’s Contact is that, at the end, they found a pattern in pi when put into base 13. He didn’t really go into it as it was the end of the book, but I really wish he’d survived to write a sequel.
to be fair, though, 1 and 0 are just binary representations of values, same as decimal and hexadecimal. within your example, we’d absolutely find the entire works of shakespeare encoded in ascii, unicode, and lcd pixel format with each letter arranged in 3x5 grids.
No, because you can’t mathematically guarantee that pi contains long strings of predetermined patterns.
The 1.101001000100001… example by the other user was just that - an example. Their number is infinite, but never contains a 2. Pi is also infinite, but does it contain the number e to 100 digits of precision? Maybe. Maybe not. The point is, we don’t know and we can’t prove it either way (except finding it by accident).
Actually, there’d only be single pixels past digit 225 in the last example, if I understand you correctly.
If we can choose encoding, we can “cheat” by effectively embedding whatever we want to find in the encoding. The existence of every substring in a one of a set of ordinary encodings might not even be a weaker property than a fixed encoding, though, because infinities can be like that.
Still not enough, or at least pi is not known to have this property. You need the number to be “normal” (or a slightly weaker property) which turns out to be hard to prove about most numbers.
> natural numbers
> rational numbers
> real numbers
> regular numbers
> normal numbers
> simply normal numbers
> absolutely normal numbers
Have mathematicians considered talking about what numbers they find okay, rather than everyone just picking their favorite and saying that one’s the ordinary one?
I mean, unironically yes. It seems the most popular stance is that all math regardless of how weird is Platonically real, although that causes some real bad problems when put down rigorously. Personally I’m more of an Aristotelian.
In the case of things like rational or real numbers, they have a counterpart that’s weirder (irrational and imaginary numbers). For the rest I’m not sure, but it’s pretty common to just pick an adjective for a new concept. There’s even situations where the same term gets used more than once in different subfields, and then they collide so you have to add another one to clarify.
For example, one open interval in the context of a small set of open intervals isn’t closed analytically under limits, or algebraically closed, but is topologically closed (and also topologically open, as the name suggests).
“Nearly all real numbers are normal (basically no real numbers are not normal), but we’re only aware of a few. This one literally non-computable one for sure. Maybe sqrt(2).”
We’re so used to dealing with real numbers it’s easy to forget they’re terrible. These puppies are a particularly egregious example I like to point to - functions that preserve addition but literally black out the entire x-y plane when plotted. On rational numbers all additive functions are automatically linear, of the form mx+n. There’s no nice in-between on the reals, either; it’s the “curve” from hell or a line.
Hot take, but I really hope physics will turn out to work without them.
Simply containing each number sequence is a significantly weaker property than having them all occur at the right frequency. Still, while nobody has proven it, it’s generally expected to be true.
“I may be a staunch atheist,” said Richard Stallman, creator of the GNU + Linux operating system and self-proclaimed architect of the modern world, “but any decent analysis in comparative religion would conclude that the universe is a copyleft creation, thereby pi should automatically fall under the terms of the GNUv3 license.”
Is there an algorithm or number such that we could basically pirate data from it by saying “start digit 9,031,643,679 with length 5,345,109 is an MP4 of Shrek”? Something that we could calculate in a day or less?
An example I found: the string of digits 0123456789 occurs at position 17387594880. In this case, it took 11 digits to describe where to find a 10-digit number.
So I think such an algorithm would technically work, but your “start digit” would be so large it would use more data than just sending the raw file data. Not to mention the impossible amount of computing power needed.
What if instead we utilized an algorithm, some code, that would ultimately generate the file? I could imagine a program that generates a number which ultimately is more dense than the program. For example, if we just-so-happened to need a million digits of Pi the program would be shorter than the number. Is there a way to tailor an algorithm to collapse down to any number? As an example, what if we needed a million digits of Pi but the last 10 digits need to be all 9s?
Conceptually this is basically just standard encryption: some math that spits out gibberish unless you have the info to make that gibberish become something useful.
The short answer is no, and even if we could, the digit index you’d start at would have a larger binary representation than the actual data you were trying to encode.
Yep. Not always larger necessarily, but close to the same size on average, or maybe a little smaller if the domain is limited and compression can be applied. Not really useful.
Similarly: if you write a program to randomly run through all the combinations of pixels on a decently large screen (say, 1080p) you will eventually see every important question and answer that can be expressed on a screen.
Welp, time for quectoquectoquectoquectoquectometers.
Actually, a plank length seems to be 10 microquectometers, so my first guess might only be necessary for interpretation of the world, and not physical accuracy.