Fram, you should talk to nap. He's in his own private math universe.
I was in there once. We picked up a couple of eigenvalues and went to see The Matrix.
I remember once the conversation had turned to big numbers and he threw out something and I just came right back with either g_64 or A(g_64,g_64), I can't remember which, but he was just like "Damn, you don't fuck around, do you?"
(I do not.)
But yeah, I get the feeling we're in about the same boat, finding really it fascinating but not being very good at actually working through it all.
Sometimes I think Graham brought attention to that number just to spite the future. Seems to be the growing trend. Another 500 years from now, math and sadism will be considered the same fetish.
At what point in the future does math become a fetish?
One ‘googol’ years, is the official word for that number. It’s the current age of the Universe, one billion billion billion billion billion billion billion billion billion billion times over. Squeeze the entire history of our Universe into the thickness of a dollar bill, and one googol years would give you a pile of money that reaches one hundred quadrillion quadrillion quadrillion quadrillion light years high. It wouldn’t even fit in our Universe.
Fuck that noise.
I don't want to even try to think about g_64, just in case I accidentally manage to. Somebody finds me in my apartment days later. Brains, hair, and tiny little digits plastered to all the walls and ceiling.
What's the problem with big numbers. They're just big. Who cares how often something would fit into the universe were it made of something you can imagine?
Personally I'm not a fan of ONE specific mathematical discipline, but I like sequences and series
il faut calculer pour vivre et non vivre pour calculer
also, my old phone number almost fits perfectly at a certain point in pi
I'm pretty sure all phone numbers fit perfectly at some point in pi
in sequence?
6 numbers in a row is pretty cool
especially since they're pretty close to the decimal mark
i just finished my second engineering math. fuck cycloids
well, close to the decimal is a different matter
but I mean you've got an infinite number of digits to work with back there if you count far enough, so every 6 digit sequence kind of has to appear somewhere
our math professor is the worst. he just reads his lectures he wrote thirty years ago on a typewriter and always does the same exams, only with slightly different numbers.
It's a joke really, and hadn't I taken some other math courses before I wouldn't learn a thing
But this way when a problem comes up, I just take one of my trusty books and learn it myself
i remember reading about random phenomenae
from what i remember (Science, 2004ish so probably outdated), it's hardly random.
a few are statistically close, such as movement of alpha particles, i think.
Things like linear algebra and calculus and geometry are means to an end. They are called elementary maths because they form the basic foundation for understanding and applying even more interesting and complicated things.
Set theory is where I'm resting my head at the moment. All math is encompassed by set theory. Ofcourse, I've only got experience with undergrad math.
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FramlingFaceHeadGeebs has bad ideas.Registered Userregular
One ‘googol’ years, is the official word for that number. It’s the current age of the Universe, one billion billion billion billion billion billion billion billion billion billion times over. Squeeze the entire history of our Universe into the thickness of a dollar bill, and one googol years would give you a pile of money that reaches one hundred quadrillion quadrillion quadrillion quadrillion light years high. It wouldn’t even fit in our Universe.
Fuck that noise.
I don't want to even try to think about g_64, just in case I accidentally manage to. Somebody finds me in my apartment days later. Brains, hair, and tiny little digits plastered to all the walls and ceiling.
Don't worry, you can't. It's hopelessly huge. a googol you can actually write out. You can fit that shit on a piece of paper. g_64, no way. You can't even write out the number of digits in it. You can't write out the number of digits in the number of digits in g_64. You can't even write down the number of levels of digit-counting you would have to go through to get to something you could write down.
If you were to turn the entire universe into a computer capable of representing a digit with each individual particle, and were to crank the speed up so that you were refreshing every Planck time (the theoretical smallest possible interval at which time can be distinguished from space, something like 5*10^-44 seconds), over the entire lifetime of the universe, you wouldn't even have covered a perceptible fraction of g_64.
Here's the mind-fuck, though:
The last digit is 7.
Framling on
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your = belonging to you
their = belonging to them
there = not here
they're = they are
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StraightziHere we may reign secure, and in my choice,To reign is worth ambition though in HellRegistered Userregular
edited May 2008
Jesus Christ Fram I hate you.
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Tossrocktoo weird to livetoo rare to dieRegistered Userregular
il faut calculer pour vivre et non vivre pour calculer
also, my old phone number almost fits perfectly at a certain point in pi
I'm pretty sure all phone numbers fit perfectly at some point in pi
Well, it goes further than that
everything is somewhere in pi
Think about it: Because it never repeats, and continues infinitely, it must at some point take every possible combination of numbers
which means everything, everything is represented somewhere in it's digits. Hamlet in standard ASCII. The numbers one through four in such a sequence so that, if mapped to the letters TAGC, they would represent your genetic code. A jpeg of your face. A binary video file of your entire life, from birth to death.
Now that's a mindfuck.
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Tossrocktoo weird to livetoo rare to dieRegistered Userregular
Also, musaman, arbitrarily saying things like "you must argue this point using no mathematics, in order to be interesting" is ridiculous. I'll argue things how they need to be argued.
honestly coming in and demanding that mathematical facts be proven without math is a bit like
going to church and asking the preacher to explain everything without god
or even speaking
he's just gotta wave his dick around
ok I'll show my hand...
I teach algebra and statistics and both of these subjects involve this discussion. You can't argue with 9th graders using mathematics like a gamma function. I am not satisfied saying "because"
I would like to keep the conversation going, so more points of view are better.
Honestly, even the definition of the simple factorial function is defined from 1 to infinity and just sets 0! as 1.
My lining up way works as one of the simplest reasons to explain why it's gotta be 1 is in probability calculations.
There really is no "simple" and sound mathematical proof.
Your students need to learn that sometimes they just gotta accept something till they get smarter
I'd argue it on the aesthetics. With 0! = 1 you get a nice, simple primitive recursive function, namely
f(0) = 1
f(n+1) = f(n) * n
If 0! were something other than 1, you'd define !n as f(n) everywhere except 0, where it would disagree. If your students argue about the natural numbers starting at 1, tell them about Peano's axioms.
Think about it: Because it never repeats, and continues infinitely, it must at some point take every possible combination of numbers
Sorry, pet peeve:
Pi is, in fact, sufficiently random that any string of digits appears within its decimal expansion with probability one. However, never repeating is not sufficient for this amount of randomness.
Consider: Let alpha be the sum of 10^-n! as n goes from 0 to infinity (for those who read TeX, /alpha = /Sum_{n=0}^{/infty} 10^{-n!}). It can be shown that alpha is not merely irrational, alpha is transcendental, (i.e. There's no polynomial with rational coefficients such that alpha is the root of that polynomial) however irrationality is enough for alpha to consist of an infinite string of digits that never repeats. But the only digits in the decimal expansion of alpha are 0 and 1. Thus, it cannot, e.g., contain your mom's phone number.
il faut calculer pour vivre et non vivre pour calculer
also, my old phone number almost fits perfectly at a certain point in pi
I'm pretty sure all phone numbers fit perfectly at some point in pi
Well, it goes further than that
everything is somewhere in pi
Think about it: Because it never repeats, and continues infinitely, it must at some point take every possible combination of numbers
which means everything, everything is represented somewhere in it's digits. Hamlet in standard ASCII. The numbers one through four in such a sequence so that, if mapped to the letters TAGC, they would represent your genetic code. A jpeg of your face. A binary video file of your entire life, from birth to death.
Now that's a mindfuck.
I'd just like to point out that it's not necessarily the case that a sequence that never repeats and continues infinitely will at some point contain every possible combination.
For instance, if you were to take the digits of pi pi and replace each 8 with a 4, you'd still have a non-repeating infinite sequence of digits, but it will never contain Jenny's phone number. Similarly, there are a lot of patterns that will never appear in 1.101001000100001000001000000100000001... even though it never repeats and continues infinitely.
Now, this does indeed seem to be the case with pi, and its digits do conform to roughly the same distribution as would be expected by random noise in the long term, but that's not a consequence of its infinite non-repetition.
Here's an interesting question to which I do not know the answer: is there a point in the decimal expansion of pi where it repeats everything that has gone before? I don't mean repeating in the standard sense, I mean, say it started 3.14159265358979314159265358979323846, i.e. it repeats everything that has come before and then continues with other, non-repeated digits. It seems like this would have to happen, but I can't say with certainty.
I would also asume that if this could occur at all, it would occur an infinite number of times.
Framling on
you're = you are
your = belonging to you
their = belonging to them
there = not here
they're = they are
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SmasherStarting to get dizzyRegistered Userregular
edited May 2008
I'm no math major, but I'll give this a shot:
For a given digit index n, the probability that this digit ends a string of length n (ie, everything up to that digit) which is then repeated by the next n digits is 1/(10^n). The sum of this series as n goes to infinity is simply .1111..., or 1/9. In other words, the chance of there being any such sequence at all is 1/9; Thus I would conclude that there is almost surely not an infinite number of such sequences.
This of course assumes the digits of pi are randomly distributed; if somehow it turns out they're not then who knows.
This is sorta related to math
The autistic savant who sees numbers as images in his mind amazes me
He can do these incredibly complex calculations simply by describing the picture he sees in his head
Also the dude recited pi to an obscene number of places
This is sorta related to math
The autistic savant who sees numbers as images in his mind amazes me
He can do these incredibly complex calculations simply by describing the picture he sees in his head
Also the dude recited pi to an obscene number of places
You mean the Rain Man guy?
Or the guy who isn't actually autistic but still has the mind like a savant?
The second one
The scary part about him reciting pi is, he didn't memorize it or anything
No he just kept describing the picture in his head
The mind is a strange thing
It was previously thought inordinate strength in one area of the brain necessarily meant retardation in another
But that guy proved that wrong
For a given digit index n, the probability that this digit ends a string of length n (ie, everything up to that digit) which is then repeated by the next n digits is 1/(10^n). The sum of this series as n goes to infinity is simply .1111..., or 1/9. In other words, the chance of there being any such sequence at all is 1/9; Thus I would conclude that there is almost surely not an infinite number of such sequences.
This of course assumes the digits of pi are randomly distributed; if somehow it turns out they're not then who knows.
It's been so long since I've done anything math-related that I'll pretty much agree with anything that remotely makes sense, but that seems right (at least for the first occurrence, because once it happens and takes up a span of 2n digits, it's impossible for it to happen again until at least 4n; and even though pi is infinite I'm sure that still slightly affects the probability of it reoccuring...god, I hate statistics).
But if pi truly is random and infinite and, for any sequence of infinitely large ns the probability of this occurring doesn't ever reach/approach 0 (and this is what I'm trying to think about now), then Fram's probably right and it would be expected to occur an infinite number of times, however infinitely spaced out the n-point of each occurrence would be from one another.
Don't worry, you can't. It's hopelessly huge. a googol you can actually write out. You can fit that shit on a piece of paper. g_64, no way. You can't even write out the number of digits in it. You can't write out the number of digits in the number of digits in g_64. You can't even write down the number of levels of digit-counting you would have to go through to get to something you could write down.
If you were to turn the entire universe into a computer capable of representing a digit with each individual particle, and were to crank the speed up so that you were refreshing every Planck time (the theoretical smallest possible interval at which time can be distinguished from space, something like 5*10^-44 seconds), over the entire lifetime of the universe, you wouldn't even have covered a perceptible fraction of g_64.
I once read that there are varying degrees of infinity.
Such as there is an infinite amount of numbers between 0 and 1. However, there are the exact same amount of infinite numbers between 0 and 2.
The weird part is, if you take the amount of rational real numbers between 0 and 1 (1/2, 1/3, 1/4, 1/5, etc.), they are dwarfed by the amount of irrational real numbers between 0 and 1.
Or something like that.
I never really paid attention, I just seemed to recall the jist of it
This is sorta related to math
The autistic savant who sees numbers as images in his mind amazes me
He can do these incredibly complex calculations simply by describing the picture he sees in his head
Also the dude recited pi to an obscene number of places
Was that the guy who learned icelandic in like a week? That shit is just ridiculous.
The weird part is, if you take the amount of rational real numbers between 0 and 1 (1/2, 1/3, 1/4, 1/5, etc.), they are dwarfed by the amount of irrational real numbers between 0 and 1.
Or something like that.
You should look into countable vs uncountable sets. The most common example is the rational vs irrational sets. It's pretty neat...I'd explain it, but I'm not Framling. I can't remember shit.
Countable infinite sets are sets that can be mapped one-to-one onto the natural numbers {0, 1, 2, 3...}
The Integers {...-3, -2, -1, 0, 1, 2, 3...} are a countable infinite set, because you can map them to the natural numbers; one such mapping can be achieved by mapping 0 to 0, positive numbers p to 2p and negative numbers n to -(2n - 1). So there are the same number of positive integers as there are positive and negative integers.
The set of real numbers between 0 and 1 is an uncountable infinite set. The easiest proof of this is by a simple contradiction: If every real number is accounted for, then there must be a real number whose first digit after the decimal is different than the real number mapped to 0, whose second digit after the decimal is different than the real number mapped to 1, whose third digit after the decimal is different than the real number mapped to 2, and so on for every decimal after the digit (the number of digits after the decimal in any real number is a countable set). The real number we just defined is not represented by a natural number since it's different from every real number that's being represented by a natural number, therefore it is impossible to map all decimal numbers between 0 and 1 to the set of natural numbers. So there are more real numbers between 0 and 1 than there are integers.
surprisingly, the newest number of all is seven hundred and twenty one, which, despite its rather innocuous standing in the grand scheme of things, was only introduced a couple of weeks ago
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Fuck that noise.
I don't want to even try to think about g_64, just in case I accidentally manage to. Somebody finds me in my apartment days later. Brains, hair, and tiny little digits plastered to all the walls and ceiling.
also, my old phone number almost fits perfectly at a certain point in pi
Personally I'm not a fan of ONE specific mathematical discipline, but I like sequences and series
I'm pretty sure all phone numbers fit perfectly at some point in pi
in sequence?
6 numbers in a row is pretty cool
especially since they're pretty close to the decimal mark
i just finished my second engineering math.
fuck cycloids
well, close to the decimal is a different matter
but I mean you've got an infinite number of digits to work with back there if you count far enough, so every 6 digit sequence kind of has to appear somewhere
It's a joke really, and hadn't I taken some other math courses before I wouldn't learn a thing
But this way when a problem comes up, I just take one of my trusty books and learn it myself
from what i remember (Science, 2004ish so probably outdated), it's hardly random.
a few are statistically close, such as movement of alpha particles, i think.
Set theory is where I'm resting my head at the moment. All math is encompassed by set theory. Ofcourse, I've only got experience with undergrad math.
Don't worry, you can't. It's hopelessly huge. a googol you can actually write out. You can fit that shit on a piece of paper. g_64, no way. You can't even write out the number of digits in it. You can't write out the number of digits in the number of digits in g_64. You can't even write down the number of levels of digit-counting you would have to go through to get to something you could write down.
If you were to turn the entire universe into a computer capable of representing a digit with each individual particle, and were to crank the speed up so that you were refreshing every Planck time (the theoretical smallest possible interval at which time can be distinguished from space, something like 5*10^-44 seconds), over the entire lifetime of the universe, you wouldn't even have covered a perceptible fraction of g_64.
Here's the mind-fuck, though:
your = belonging to you
their = belonging to them
there = not here
they're = they are
Well, it goes further than that
everything is somewhere in pi
Think about it: Because it never repeats, and continues infinitely, it must at some point take every possible combination of numbers
which means everything, everything is represented somewhere in it's digits. Hamlet in standard ASCII. The numbers one through four in such a sequence so that, if mapped to the letters TAGC, they would represent your genetic code. A jpeg of your face. A binary video file of your entire life, from birth to death.
Now that's a mindfuck.
INAPPROPRIATE APOSTROPHE
I hate being jailed
I'd argue it on the aesthetics. With 0! = 1 you get a nice, simple primitive recursive function, namely
f(0) = 1
f(n+1) = f(n) * n
If 0! were something other than 1, you'd define !n as f(n) everywhere except 0, where it would disagree. If your students argue about the natural numbers starting at 1, tell them about Peano's axioms.
I love math.
Sorry, pet peeve:
Pi is, in fact, sufficiently random that any string of digits appears within its decimal expansion with probability one. However, never repeating is not sufficient for this amount of randomness.
Consider: Let alpha be the sum of 10^-n! as n goes from 0 to infinity (for those who read TeX, /alpha = /Sum_{n=0}^{/infty} 10^{-n!}). It can be shown that alpha is not merely irrational, alpha is transcendental, (i.e. There's no polynomial with rational coefficients such that alpha is the root of that polynomial) however irrationality is enough for alpha to consist of an infinite string of digits that never repeats. But the only digits in the decimal expansion of alpha are 0 and 1. Thus, it cannot, e.g., contain your mom's phone number.
I'd just like to point out that it's not necessarily the case that a sequence that never repeats and continues infinitely will at some point contain every possible combination.
For instance, if you were to take the digits of pi pi and replace each 8 with a 4, you'd still have a non-repeating infinite sequence of digits, but it will never contain Jenny's phone number. Similarly, there are a lot of patterns that will never appear in 1.101001000100001000001000000100000001... even though it never repeats and continues infinitely.
Now, this does indeed seem to be the case with pi, and its digits do conform to roughly the same distribution as would be expected by random noise in the long term, but that's not a consequence of its infinite non-repetition.
Here's an interesting question to which I do not know the answer: is there a point in the decimal expansion of pi where it repeats everything that has gone before? I don't mean repeating in the standard sense, I mean, say it started 3.14159265358979314159265358979323846, i.e. it repeats everything that has come before and then continues with other, non-repeated digits. It seems like this would have to happen, but I can't say with certainty.
I would also asume that if this could occur at all, it would occur an infinite number of times.
your = belonging to you
their = belonging to them
there = not here
they're = they are
For a given digit index n, the probability that this digit ends a string of length n (ie, everything up to that digit) which is then repeated by the next n digits is 1/(10^n). The sum of this series as n goes to infinity is simply .1111..., or 1/9. In other words, the chance of there being any such sequence at all is 1/9; Thus I would conclude that there is almost surely not an infinite number of such sequences.
This of course assumes the digits of pi are randomly distributed; if somehow it turns out they're not then who knows.
The autistic savant who sees numbers as images in his mind amazes me
He can do these incredibly complex calculations simply by describing the picture he sees in his head
Also the dude recited pi to an obscene number of places
You mean the Rain Man guy?
Or the guy who isn't actually autistic but still has the mind like a savant?
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The scary part about him reciting pi is, he didn't memorize it or anything
No he just kept describing the picture in his head
The mind is a strange thing
It was previously thought inordinate strength in one area of the brain necessarily meant retardation in another
But that guy proved that wrong
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It's been so long since I've done anything math-related that I'll pretty much agree with anything that remotely makes sense, but that seems right (at least for the first occurrence, because once it happens and takes up a span of 2n digits, it's impossible for it to happen again until at least 4n; and even though pi is infinite I'm sure that still slightly affects the probability of it reoccuring...god, I hate statistics).
But if pi truly is random and infinite and, for any sequence of infinitely large ns the probability of this occurring doesn't ever reach/approach 0 (and this is what I'm trying to think about now), then Fram's probably right and it would be expected to occur an infinite number of times, however infinitely spaced out the n-point of each occurrence would be from one another.
I think my brain just exploded.
The string 5318008 occurs at position 13,809,596... so, clearly the universe got bored one day and decided to screw around with its calculator.
Such as there is an infinite amount of numbers between 0 and 1. However, there are the exact same amount of infinite numbers between 0 and 2.
The weird part is, if you take the amount of rational real numbers between 0 and 1 (1/2, 1/3, 1/4, 1/5, etc.), they are dwarfed by the amount of irrational real numbers between 0 and 1.
Or something like that.
I never really paid attention, I just seemed to recall the jist of it
Was that the guy who learned icelandic in like a week? That shit is just ridiculous.
kpop appreciation station i also like to tweet some
You should look into countable vs uncountable sets. The most common example is the rational vs irrational sets. It's pretty neat...I'd explain it, but I'm not Framling. I can't remember shit.
What I find more interesting is botched math proofs.
The Integers {...-3, -2, -1, 0, 1, 2, 3...} are a countable infinite set, because you can map them to the natural numbers; one such mapping can be achieved by mapping 0 to 0, positive numbers p to 2p and negative numbers n to -(2n - 1). So there are the same number of positive integers as there are positive and negative integers.
The set of real numbers between 0 and 1 is an uncountable infinite set. The easiest proof of this is by a simple contradiction: If every real number is accounted for, then there must be a real number whose first digit after the decimal is different than the real number mapped to 0, whose second digit after the decimal is different than the real number mapped to 1, whose third digit after the decimal is different than the real number mapped to 2, and so on for every decimal after the digit (the number of digits after the decimal in any real number is a countable set). The real number we just defined is not represented by a natural number since it's different from every real number that's being represented by a natural number, therefore it is impossible to map all decimal numbers between 0 and 1 to the set of natural numbers. So there are more real numbers between 0 and 1 than there are integers.
surprisingly, the newest number of all is seven hundred and twenty one, which, despite its rather innocuous standing in the grand scheme of things, was only introduced a couple of weeks ago
Botched math proofs are not new.
Transfinite numbers are not new.
Most of the shit in this thread is not new.
Except to the people in the thread who've never heard of them.
Which is kinda the point of the fucking thread.
So, who wants to hear about Turing's incompleteness theorem and Busy Beaver functions?
your = belonging to you
their = belonging to them
there = not here
they're = they are
imo
seems you and math just might hit it off after all