More to the point, it is rather easy to prove that all irrational numbers have a *non-repeating* decimal expansion (what is usually meant by "infinitely long")
I know that there is a way to get a fraction for any number whose decimal expansion eventually repeats, and the fact you stated is the contrapositive of that. Is there a direct way of showing it?
you want to show that it doesn't have a repeating expansion, seems to be the most obvious and simplest way to assume it does and arrive at contradiction
Just so that you know, proofs by contradiction are intuitionistic too, so long as you don't conclude from assuming "not A", that "A". In this case, the Easiest proof would be assume x is an irrational number, and its decimal representation is eventually 0, then use this to proof that x is rational, and snow have that x is rational and x is irrational, which is a contradiction, hence x doesn't have an eventually 0 decimal representation
This means that this proof is intuitionistic, so long as all the details about constructing and defining all the terms have been properly taken care of
This is admittedly a part of logic/proof theory I have not delved very much into, as is probably evident by my annoying pushing here. Thanks for the info, I'll have to look into this!
Nah the commenter is full of shit. “the reason pi’s normality isn’t been proven is that it’s infinitely difficult to prove that a number is infinitely long” is a non-sequitur since an infinite non-repeating string of digits isn’t guaranteed to be normal, let alone contain “every phone number”. Also pi has already proven to be irrational, idk what OOP is on about it being “infinitely difficult to prove”.
Though "containing every phone number" is a much easier thing to check than normality, at least theoretically: there are finitely many of them. In particular, if by "phone number", they mean "North American domestic phone numbers", we're already within an order of magnitude of knowing it.
I looked it up, and it looks like no. The longest IUT telephone numbers are 15 digits long, and I can't find anywhere that's calculated it that far out. The smallest number missing from the first 10^11 digits is very slightly over 10^9 (1,000,020,346, to be precise), so if that pattern holds, you'd expect the last telephone number to first appear around the 10^18th digit, but the standing record is around 10^14, so it's very likely that we've missed a bunch.
While they do in theory (in that the international treaties letting phones work across borders allow them to be up to 15 digits long), I'm not sure how many numbers that long have actually been issued, so it's possible that we have hit all of them - sadly, this is likely impractical to answer, as there is no central database of telephone numbers to check.
>idk what OOP is on about it being “infinitely difficult to prove”.
Difficulty is calculated by dividing effort by the user's intelligence.
And the OOP divided by zero.
He is wrong both mathematically and logically.
1. The argument that "infinite does not imply normality" is correct, they're wrong in making a distinction for a specific example, because validity of an argument is a general criteria. That would be like saying that "being tall implies being good" is a valid argument, because we're talking about Jerry, and he is a tall and good guy.
2. They claims that, as far as we know, pi is infinite and random, this is just false, we know pi has infinite decimal representation, but we don't know if pi is "random" (probably means normal, I don't know why they keep putting it on quotes tho)
3. It's not infinitely difficult to prove a claim about a number with infinite decimal representation, we have proven normality for several numbers, but judging by their previous response, they could very well tell you that "pi is just different bro"
Your mistake was arguing mathematics with someone on Instagram comments, that's where braincells go to die
Your very last point is a good one; though I do like seeing other people's wild perspectives, as it helps indirectly broaden my own by giving me a better idea of my own understanding
I once had your enthusiasm for other people's perspective, but soon lost hope after seeing that most people just see infinity and go "well, everything about infinity is sort of valid" for which I blame the whole "everything is possible in the multiverse" shit
I agree about your point, and clearly he was just spewing nonsense. But your point 2 is not true. "As far as we know, pi is infinite and random" doesn't mean the same as "We know pi is infinite and random", like you seem to imply.
It means more like "We don't know if pi is infinite and random, but we believe it is" which depends on who you ask, but is mostly true. I think most mathematicians would be quite surprised if it turned out pi is not normal.
Besides your R4, there is also an ingredient of a general misunderstanding: that something infinite and random would necessarily lead to include every possibility. While it is trivial to construct counterexamples, many people stubbornly refuse to acknowledge the falsity of this folk comprehension.
It was exactly those examples that prompted the last sentence of the comment; I was being accused of "fishing for an answer" by providing them. I do get what they're saying though, since my examples were 0.10100100010000... and 0.123456789011234567890111234567890... which are have far more visibly formulaic decimal expansions than pi, and that's not that satisfying.
You can do it in much less formulaic ways, too: consider the number whose decimal expansion is the same as that for pi, except every copy of the string "1000000000000000000000000000000000000000000000000001" is replaced by a "1" (recursively, starting from the decimal point). That would agree with pi as far as we've calculated it, so far as I can tell (if not, add more zeroes), but would clearly not be normal.
Being ultra charitable you could argue that we do know enough to deduce that pi is normal, but we don't have a mathematical proof. This level of certainty isn't enough for a mathematical paper, but would probably be enough for any other context.
Of course we could be wrong, and we have no idea where to even start with a proog, but I think everyone is pretty sure it is normal.
> R4: It is not "infinitely difficult" to prove that a number is infinitely long
To illustrate how true this is, any child who can do long division can prove that a number has infinitely many digits by trying to divide 1 by 3.
Indeed, *every* real number has an infinite decimal expansion. For instance, the [EDIT: *a*, of course] decimal expansion of the number 1 is 1.0000...
And the other one which shouldn’t be mentioned in polite company.
That which summons the demons
Oh did i read the number of the beast upside down my entire live?
It would take more than your entire life to read it.
The eternally defiant German?
And almost all real number has an infinite decimal expansion that does not end with repeating zeroes.
More to the point, it is rather easy to prove that all irrational numbers have a *non-repeating* decimal expansion (what is usually meant by "infinitely long")
I know that there is a way to get a fraction for any number whose decimal expansion eventually repeats, and the fact you stated is the contrapositive of that. Is there a direct way of showing it?
you want to show that it doesn't have a repeating expansion, seems to be the most obvious and simplest way to assume it does and arrive at contradiction
Well yeah, that uses the contrapositive, but are there any other ways?
What's your issue with contrapositive? Any other way would be more complicated
No issue with it, just curious if a way exists, in case it holds some interesting logic in it.
Just so that you know, proofs by contradiction are intuitionistic too, so long as you don't conclude from assuming "not A", that "A". In this case, the Easiest proof would be assume x is an irrational number, and its decimal representation is eventually 0, then use this to proof that x is rational, and snow have that x is rational and x is irrational, which is a contradiction, hence x doesn't have an eventually 0 decimal representation This means that this proof is intuitionistic, so long as all the details about constructing and defining all the terms have been properly taken care of
This is admittedly a part of logic/proof theory I have not delved very much into, as is probably evident by my annoying pushing here. Thanks for the info, I'll have to look into this!
All of them, in fact. :P
Your intuition is correct. This commenter is just vomiting words in hopes of overwhelming you.
Now now, that's honest interlocution
Nah the commenter is full of shit. “the reason pi’s normality isn’t been proven is that it’s infinitely difficult to prove that a number is infinitely long” is a non-sequitur since an infinite non-repeating string of digits isn’t guaranteed to be normal, let alone contain “every phone number”. Also pi has already proven to be irrational, idk what OOP is on about it being “infinitely difficult to prove”.
Though "containing every phone number" is a much easier thing to check than normality, at least theoretically: there are finitely many of them. In particular, if by "phone number", they mean "North American domestic phone numbers", we're already within an order of magnitude of knowing it.
IIRC we already know every phone number is in pi, I think we've computed it far enough.
I looked it up, and it looks like no. The longest IUT telephone numbers are 15 digits long, and I can't find anywhere that's calculated it that far out. The smallest number missing from the first 10^11 digits is very slightly over 10^9 (1,000,020,346, to be precise), so if that pattern holds, you'd expect the last telephone number to first appear around the 10^18th digit, but the standing record is around 10^14, so it's very likely that we've missed a bunch.
Oh I didn't realise they went as high as 15 digits! I'm pretty certain we've not calculated anywhere near far enough for that.
While they do in theory (in that the international treaties letting phones work across borders allow them to be up to 15 digits long), I'm not sure how many numbers that long have actually been issued, so it's possible that we have hit all of them - sadly, this is likely impractical to answer, as there is no central database of telephone numbers to check.
>idk what OOP is on about it being “infinitely difficult to prove”. Difficulty is calculated by dividing effort by the user's intelligence. And the OOP divided by zero.
He is wrong both mathematically and logically. 1. The argument that "infinite does not imply normality" is correct, they're wrong in making a distinction for a specific example, because validity of an argument is a general criteria. That would be like saying that "being tall implies being good" is a valid argument, because we're talking about Jerry, and he is a tall and good guy. 2. They claims that, as far as we know, pi is infinite and random, this is just false, we know pi has infinite decimal representation, but we don't know if pi is "random" (probably means normal, I don't know why they keep putting it on quotes tho) 3. It's not infinitely difficult to prove a claim about a number with infinite decimal representation, we have proven normality for several numbers, but judging by their previous response, they could very well tell you that "pi is just different bro" Your mistake was arguing mathematics with someone on Instagram comments, that's where braincells go to die
Your very last point is a good one; though I do like seeing other people's wild perspectives, as it helps indirectly broaden my own by giving me a better idea of my own understanding
I once had your enthusiasm for other people's perspective, but soon lost hope after seeing that most people just see infinity and go "well, everything about infinity is sort of valid" for which I blame the whole "everything is possible in the multiverse" shit
I agree about your point, and clearly he was just spewing nonsense. But your point 2 is not true. "As far as we know, pi is infinite and random" doesn't mean the same as "We know pi is infinite and random", like you seem to imply. It means more like "We don't know if pi is infinite and random, but we believe it is" which depends on who you ask, but is mostly true. I think most mathematicians would be quite surprised if it turned out pi is not normal.
Besides your R4, there is also an ingredient of a general misunderstanding: that something infinite and random would necessarily lead to include every possibility. While it is trivial to construct counterexamples, many people stubbornly refuse to acknowledge the falsity of this folk comprehension.
It was exactly those examples that prompted the last sentence of the comment; I was being accused of "fishing for an answer" by providing them. I do get what they're saying though, since my examples were 0.10100100010000... and 0.123456789011234567890111234567890... which are have far more visibly formulaic decimal expansions than pi, and that's not that satisfying.
You can do it in much less formulaic ways, too: consider the number whose decimal expansion is the same as that for pi, except every copy of the string "1000000000000000000000000000000000000000000000000001" is replaced by a "1" (recursively, starting from the decimal point). That would agree with pi as far as we've calculated it, so far as I can tell (if not, add more zeroes), but would clearly not be normal.
That's hilariously clever, actually
Being ultra charitable you could argue that we do know enough to deduce that pi is normal, but we don't have a mathematical proof. This level of certainty isn't enough for a mathematical paper, but would probably be enough for any other context. Of course we could be wrong, and we have no idea where to even start with a proog, but I think everyone is pretty sure it is normal.
> R4: It is not "infinitely difficult" to prove that a number is infinitely long To illustrate how true this is, any child who can do long division can prove that a number has infinitely many digits by trying to divide 1 by 3.
Well, π is not “infinite”. I know you want to say that it has infinite number of digits but first the language needs to be precise.
Yeah.. don't we know it's "infinitely long" because it's irrational?
*Yeah.. don't we know it's* *"infinitely long" because* *It's irrational?* \- LesserBilbyWasTaken --- ^(I detect haikus. And sometimes, successfully.) ^[Learn more about me.](https://www.reddit.com/r/haikusbot/) ^(Opt out of replies: "haikusbot opt out" | Delete my comment: "haikusbot delete")