Isn't there a non-zero chance that given an infinite number of digits, the probability of finding repeats of pi, each a bit longer, increases until a perfect, endless repeat of pi will eventually be found thus nullifying pi's own infinity?
No, because it would create a contradiction. If a "perfect, endless repeat of pi" were eventually found (say, starting at the nth digit), then you can construct a rational number (a fraction with an integer numerator and denominator) that precisely matches it. However, pi is provably irrational, meaning no such pair of integers exists. That produces a contradiction, so the initial assumption that a "perfect, endless repeat of pi" exists cannot be true.
Yes and that contradiction is already present in my premise which is the point. Pi, if an infinite stream of digits and with the prime characteristic it is normal/random, will, at some point include itself, by chance. Unless, not random...
This applies to every normal, "irrational" number, the name with which I massively agree, because the only way they can be not purely random suggests they are compressible further and so they have to be purely random, and thus... can't be.
It is a completely irrational concept, thinking rationally.
> Pi, if an infinite stream of digits and with the prime characteristic it is normal/random, will, at some point include itself, by chance.
What you are essentially saying is that pi = 3.14....pi...........
If that was the case, wouldn't it mean that the digits of pi are not countably infinite but instead is a continuum. So you wouldn't be able to put the digits of pi in one to one correspondence with natural numbers. But obviously we can so shouldn't our default be to assume our premise was wrong?
> It is a completely irrational concept, thinking rationally.
The belief that a normal number must eventually contain itself arises from extremely flawed thinking about probability. Like djkorchi mentioned above, if we knew pi = 3.14....pi..., that would mean pi = 3.14... + 10^n pi for some n, meaning (1 - 10^n) pi = 3.14... and pi = (3.14...) / (1 - 10^n), aka a rational number.
> The belief that a normal number must eventually contain itself arises from extremely flawed thinking about probability.
Yes. There is an issue with the premise as it leads to a contradiction.
> Like djkorchi mentioned above, if we knew pi = 3.14....pi..., that would mean pi = 3.14... + 10^n pi for some n, meaning (1 - 10^n) pi = 3.14... and pi = (3.14...) / (1 - 10^n), aka a rational number.
Yes. If pi = 3.14...pi ( pi repeats at the end ), then it is rational as the ending pi itself would contain an ending pi and it would repeat forever ( hence a rational number ). I thought the guy was talking about pi contain pi somewhere within itself.
pi = 3.14...pi... ( where the second ... represents an infinite series of numbers ). Then we would never reach the second set of ... and the digits of pi would not be enumerable.
So if pi cannot be contained within ( anywhere in the middle of pi ) and pi cannot be contained at the end, then pi must not contain pi.
> If that was the case, wouldn't it mean that the digits of pi are not countably infinite but instead is a continuum.
No; combining two countably infinite sets doesn't increase the cardinality of the result (because two is finite). Combining one finite set with one countably infinite set won't give you an uncountable result either. The digits would still be countably infinite.
Looking at this from another direction, it is literally true that, when x = 1/7, x = 0.142....x.... , but it is obviously not true that the decimal expansion of 1/7 contains uncountably many digits.
> Pi, if an infinite stream of digits and with the prime characteristic it is normal/random, will, at some point include itself, by chance.
A normal number would mean that every finite sequence of digits is contained within the number. It does not follow that the number contains every infinite sequence of digits.
In general, something that holds for all finite x does not necessarily hold for infinite x as well.
Exactly - and when you remove the assumptions, what's left?
Pi is assumed to be infinite, random, and normal. The point here is not these assumptions may be wrong. Underneath them may sit a greater point; that irrationality is defined in a contradictory way - which may be correct, or not, or, both.
Given proof Pi is infinite lay on irrationality, it is rather an important issue. Pi may not be infinite, and a great place to observe that may be Planck.
> A normal number would mean that every finite sequence of digits is contained within the number.
Is that true? I don't see how that could be true. The sequence 0-9 repeated infinitely is, by definition, a normal number (in that the distribution of digits is uniform)
...and yet nowhere in that sequence does "321" appear ...or "654" ...or "99"
There are an infinite number of combinations of digits that do not appear in that normal number I've just described. So, I don't think your statement is true.
> I don't see how that could be true. The sequence 0-9 repeated infinitely is, by definition, a normal number (in that the distribution of digits is uniform)
Well, your first problem is that you don't know the definition of a normal number. Your second problem is that this statement is clearly false.
Here's Wolfram Alpha:
> A normal number is an irrational number for which any finite pattern of numbers occurs with the expected limiting frequency in the expansion in a given base (or all bases). For example, for a normal decimal number, each digit 0-9 would be expected to occur 1/10 of the time, each pair of digits 00-99 would be expected to occur 1/100 of the time, etc. A number that is normal in base-b is often called b-normal.
Your "counterexample" is not a normal number in any sense, most obviously because it isn't irrational, but only slightly less obviously because, as you note yourself, the sequences "321", "654", and "99" do not ever appear.
> Your "counterexample" is not a normal number in any sense, most obviously because it isn't irrational, but only slightly less obviously because, as you note yourself, the sequences "321", "654", and "99" do not ever appear.
lol. Your counterargument is a tautology because it contains "the sequences "321", "654", and "99" do not ever appear."
It's like if you claim, "A has the property B" then I say, "based on this definition, I don't think A has property B"
Then you say, "if it doesn't have property B, then it's not A"
...okay, but my point is, the definition that I had (from wikipedia) doesn't imply B. So for you to say, "if it doesn't have B, then it's not A" is just circular.
Now, you can point out that the definition I got from wikipedia is different from the one you got from wolfram. That's fine. That's also true. And you can argue that the definition you used does indeed imply B.
But what you cannot do is use B as part of the definition, when that's the thing I'm asking you to demonstrate.
You: all christians are pro-life
Me: I don't see how that's true. Here's the definition of christianity. I don't see how it necessarily implies being against abortion.
You: your """"counterexample"""" (sarcastic quotes to show how smart I am) is obviously wrong because, as you note yourself, that person is pro-choice, therefore, not a christian.
^^^^^ do you see how this exchange inappropriately uses the thing you're being asked to prove, which is that christians are pro-life, as a component of the argument?
Again, it's totally cool if you fine a different definition of christian that explicitly requires they be pro-life. But given that I didn't use that definition, that doesn't make it the slam dunk you imagine.
> But given that I didn't use that definition, that doesn't make it the slam dunk you imagine.
You might have a better argument if there were more than one relevant definition of a normal number. As you should have read in the other responses to your comment, the definition given on wikipedia does not differ from the one given on Wolfram Alpha.
> And you can argue that the definition you used does indeed imply B.
Given that the implication of "B" is stated directly within the definition ("For example, ..."), this seemed unnecessary.
> but my point is, the definition that I had (from wikipedia) doesn't imply B. So for you to say, "if it doesn't have B, then it's not A" is just circular.
Look at it this way:
1. You provided a completely spurious definition, which you obviously did not get from wikipedia.
2. You provided a number satisfying your spurious definition, which - not being normal - didn't have the properties of a normal number.
3. I responded that you weren't using the definition of a normal number.
4. And I also responded that it's easy to see that the number you provided is not normal, because it doesn't have the properties that a normal number must have.
Try to identify the circular part of the argument.
And, consider whether it's cause for concern that you believe you got a definition of "normal number" from wikipedia when that definition of "normal number" is not available on wikipedia.
It depends on your definition of "normal number". You seem to be using what wikipedia[1] calls "simply normal", which is that every digit appears with equal probability.
What people usually call "normal number" is much stronger: a number is normal if, when you write it in any base b, every n-digit sequence appears with the same probability 1/b^n.
IIRC the property ‘each single digit has the same density’ is the definition for a ‘simply normal number’ (in a given base), while ‘each finite string of a particular length has the same density as all other strings of that length’ is the definition for a ‘normal number’ (in a given base). And then ‘normal in all bases’ is sometimes called ‘absolutely normal’, or just ‘normal’ without reference to a base.
"Firey but peaceful" were the immortal words uttered by on-location CNN anchor Don Lemon before being hit by a bottle, as arson raged behind him and dozens of people were killed.
Conversely, one protestor who
was unarmed was killed on J6.
(And before someone "corrects" that - four people died at the capitol, but, two were natural causes and one was from drugs. Given the number of people was easily in the high hundreds K, statistically that's less than should generally be expected.)
I've seen estimates as low as 6 and as high as 36. Given deaths attributable to the protests is an extremely contentious issue, with significant sociopolitical implications, a bias in numbers reported is a given, as chaotic environments give chaotic data.
Ie it's possible to make a case for some of the deaths occuring during the time as "caused by", or "happened during". And those wanting to spin it one way will as suits their particular goal.
I'm going with a middle ground, which is against what I'd think is likely, given what I watched unfolding on livestreams - from people there, versus minimising reporting from the media. Major US cities being randomly set on fire is not "peaceful protest".
Perhaps one day, we can return to the days when a KB was a KB and a MB was a MB. Those grand old days, when we all accepted kilo and mega stretch a little more for computers. Because in binary, base10 metric is a wee bit of a shoehorn. Just a bit.
I agree. I don't care how technically correct they are if I sound like an idiot when I'm saying it.
The best I've seen is just to have the base as a subscript, like `kB_2` (2 is subscript) or `kB_10`. Though in practice I have yet to come across a situation where the difference a) matters and b) isn't clear from the context.
You're just used to the common prefixes. Kibi is not any weirder than yotta, pico, or deci. They all sound silly if you think about it - so we just don't.
No, its definitely not silly. Why make comments like this? I'll take the repercussions, but passing judgement on language by how things sound is doodoo behavior.
Not really. It's important for introduced language to sound appealing and appropriate for its use, or it will fail to catch on. If Apple has called the iPhone the gibiFon it'd have been far less likely to turn into the success it was.
Do you not remember all the people making fun of eye-phone? And all the sanitary pad memes for iPad? There were so many people who thought those were silly - they got over it with time.
Another route might be inspiration from exponential math notation. Traditional kilo/mega/giga/tera-bytes are just 2 to the power of 10, 20, 30, 40, etc.
So perhaps a terabyte could be a "bin fourty", or a "two-to-fourty", etc. (Although as it linguistically relaxes into Tootafortie, it'll sound goofy too.)
You're saying that units-of-10 in English (and using Arabic numerals) will "not work" for other languages, when the international status-quo we're complaining about is already powers-of-1000 in Greek which are then mutated with Latin?
No it all really changed when storage service manufacturers realized that they could market 1,000,000,000 bytes as "1 gigabyte", to people who then saw their computer tell them that there was about 7% less than a gigabyte in there.
Can't say when they started using it, but gigabyte external hard drives would be about when the gap got large enough normal people started to notice it.
Maybe they'd have gone more times, more safely, or more quickly if they hadn't been befuddled by having to deal constantly with archaic and nonsensical imperial units?
Quick, how many blocks will 4096 bytes use on my storage device?
The argument is that the base10 interval makes no sense with computers, because they're physically base2.
You can't really have 10 without wasting 2, and that's why it made sense to use 1024 instead of 1000.
Personally I feel the pushback against gibi/mibi/kibi overblown. It's ultimately better to be coherent everywhere and always specify everything with decimals/rounded over random context dependent decisions.
But still, the original argument for 1024 made sense too.
Unless you have some well known powers of 2 you have to calculate anyway.
To be able to convert easily from byte to gibibyte, mebibyte, gibibyte, etc. is a bigger benefit. Just moving the decimal sperator to convert the units is big advantage of the metric system.
I know about 1 KB = 1024 bytes, sometimes. I'm a computer nerd, grew up playing on computers and hacking on them, and I'm a programmer now.
But, if someone asks me for a good explanation why 1 KB != 1000 bytes, I don't have a good answer. I know about powers of 2, but why are powers of 2 more important than "kilo" meaning 1000 like it does in every other context?
It's like if a kilometer wasn't 1000 meters, because of the way car odometers worked, or the shape of the tires or something. Why would technical details about a car change the meaning of "kilometer"?
Addressing, at some point, always ends up with physical wires representing bits, so chips are manufactured with power-of-two sizes. It's like asking why we measure crude oil in barrels.
That's actually a better point than you realize because crude oil is another special case! Typically, the steel drum barrel that we're all familiar with is a 55-gallon (208L) drum, except that crude oil barrels are 47 gallons (159 L).
So clearly the right thing to do here to clear up any confusion is to introduce the concept of computer-sized bytes, and metric bytes. Metric bytes would be 0.9765625 of a regular computer byte, so 1000 MB would be 1000 Metric Bytes, or 1024 * 0.9765625 = 1024 Bytes.
Thus hard drives could be rated at 1,000 GMB, for 1,000 giga metric bytes, which would really be a 1 TMB drive or 1 tera metric bytes, which is the same as 1024 giga regular-computer-sized-bytes, or 1024 GRCSB.
Totally straightforwards and not confusing to anybody.
Yes. I know. I've taken an architecture course in university, and I've completed the nand2tetris course and have conceptually build a computer from nand gates up. I ask again:
> why are powers of 2 more important than "kilo" meaning 1000 like it does in every other context?
It's not that powers of 2 are more important. It's that there will never be, for example, a RAM chip that has 32GB of RAM. They will have 34.36GB, which is an ugly number. But, they happen to have a very nice, round number of bytes if you look at them otherwise - they have 32GiB. And since these two numbers are pretty close, and the clean power of two one is far more natural for humans than the SI one in this context, it was natural to just call it GB.
Does that hold up in practice though? Last I checked my USB drives and RAM bytes were not perfect powers of 2. One clear example that comes to mind is my GPU with approximately 12 GB of RAM. That's no power of 2.
These numbers being a power of two seems pretty important, important enough that we redefine words to match powers of 2. Then, when we look at the exact number of bytes, it's not a power of 2.
The important point is that they are multiples of powers of two, instead of multiples of powers of ten. Your RAM has 12GiB of RAM, but in SI GB it has 12.884 GB of RAM.
Flash gets fussier especially when there are reserved sections.
But come on, are you really saying that 1100 0000000000 0000000000 0000000000 bytes of RAM isn't close enough to being a power of two to prove the same point?
Our starting point is that a kilobyte is 1000 bytes, but then people say "that's not close enough to 1024, which is a power of 2", and so we redefine the word "kilobyte" to mean 1024, etc. Then I buy a device with a gigabyte and it doesn't have 1,000,000,000 bytes, and it doesn't have exactly 1,073,741,824 (2^30) bytes either, it has some other random number.
So we started with Système International units and a common understanding of what they mean. Computer people said, "that's not close enough, let's redefine standardized words so they will be exact", and then they use those redefined words in an inexact way.
And for the sane normie people, a kilobyte is still 1000 bytes.
But no, being a few percent off is very different from saying "it's not a pure factor of two, it's a very small number multiplied by a very large power of two".
Your GPU has an exact multiple of 2^30 bytes of memory.
If you want to talk about a USB drive, then to do that properly we need the size and count of chips inside a real model.
In this analogy, it would be more like if "barrel" was a standardized unit of volume that everyone understood and used, but then in the oil industry specifically they used a slightly different volume and still just referred to it as a "barrel" because it's what they're used to.
And, whenever pressed for clarification, the oil people admitted "yes, technically our unit should be noted as 'oil barrels' which are different from the normal kind, but we like to just say 'barrels' because it's easier".
Real-world example: What weighs more, a pound of feathers or a pound of gold?
Reflexive answer: gold (well obviously gold is heavier than feathers)
Logical answer: neither (1 pound = 1 pound)
Actual trick answer: feathers (precious metals used troy weights instead of the one just about everything else used, and 1 pound in the troy system weighs less than 1 pound in the other one)
I thought we were taking about SI units, their general meaning, and the technical details of computers. Barrels seem completely unrelated to those things, being neither a SI unit, nor having to do with computers.
Like a lot of arguments, we're arguing over the definition of a word here ("kilobyte"), nothing more. I'm asking why technical details about a computer are so important they can override the generally understood (and well defined) meaning of that word.
> I'm asking why technical details about a computer are so important they can override the generally understood (and well defined) meaning of that word.
Because the technical details about a computer are important when describing its technical characteristics.
In short, context matters, and we adapt the meaning of words by the context they're used in all the time. It's ordinary.
In fact, it's so ordinary in this particular case, that all we humans did it for decades, before a weird group not representing the existing organic consensus came along and decided the terms absolutely must be changed, and presented us with extremely silly-sounding ones to replace the existing ones, that of course few adopted, leading to the situation we have today where the existing terms are used interchangably to mean both things, and there is now a greater ambiguity around them than existed before.
It wasn't perfect before, but the "solution" made it worse.
Therefore, it sucks in practice at meeting its goal, no matter how much sense it may make to the minority that thinks "gibibyte" is something anyone would ever want to say in public, other than in a funny voice to a dog or a baby.
Because everything (except SSDs now a days) in a computer, on a fundamental level is either 0 or 1. So when you want something that maps to that, 2 to the power of 10 is exactly 1024 bits. Somewhere along the line, someone decided that accuracy of that mapping was more important than adherence to the exact meaning of kilo.
The alternative, would have been to use something else than kilo, mega ect., that represented the base 2 magnitudes. It would be awkward to say you have 8.306.688 bytes of ram if you need to be exact.
It's a conflict between communications and storage. If you are doing data communications, you are probably dealing with phenomena measured in hertz. Those use SI prefixes, so it's natural to use them with bits as well.
But if you are doing data storage, there are many natural power-of-two structures. Using 1024-based prefixes with them often leads to more convenient numbers.
I remember in the 90's we used the prefix case to differentiate between SI (kB) and powers of 1024 (KB). Not sure how widespread it was though; no Internet to poll at the time :)
Where is the best place to find up-to-date information on stuff like layouts, and how to manage crossbrowser/accessibility etc?
It's a minefield when trying to self-learn these things as it's hard to tell when info is wrong or bad practice, outdated, etc.
For example - is this CSS grid-generation/layout current best-practice for building the base, foundational layer for a simple static website? Say, with three columns, one centered and wider with content, the others narrow and empty - serving as margins?
And how would one arrive there, as a solution? Searching online turns up an infinity of options such that it's difficult if not impossible to figure out how to do things in if not maybe "the" right or best way, at least "a" right way.
Grid has been around for a while now, together with flex it’s the way to go to start building layouts. But like everything it requires practice to get the nuances down and learn about the pitfalls. You won’t experience many cross browser issues these days with either of them. And in terms of accessibility you mostly need to consider that visual order does not necessarily match tab order. Especially for grid where you can arbitrarily place elements in the grid.
Caniuse (https://caniuse.com/) aggregates data from MDN with their own data and has reasonably good search. I see browser and standards people linking to it from time-to-time.
You wouldn't want to use a single `margin: auto`, because that sets the top and bottom margins to `auto` as well. For example if the parent container is flex, you'd then end up with a container that is centered vertically as well, which is not what you wanted. What you're probably looking for is `margin: 0 auto`.
Why is it "nearly all"? Which customers didn't have their data stolen and why were they magically left aside of this? It's obvious the data theives had complete dominance in the system so what query did they run to get only "nearly all"?
The ideal solution is for the astronauts to drink their own urine. Apparently this can be quite healthy - in moderation like occasional spacewalking, and it's not as if astronauts in general aren't used to all manner extreme limit-pushing.
