It's too early on a Monday morning for this hot math nonsense, come on
December 8, 2014 9:12 AM Subscribe
showbiz_liz: "My immediate thought before clicking the link is "oh well FUCK YOU TOO, THEN""
no spoilers please
posted by boo_radley at 9:17 AM on December 8, 2014 [2 favorites]
no spoilers please
posted by boo_radley at 9:17 AM on December 8, 2014 [2 favorites]
So if you ask someone to think of any positive whole number, and it can be any size they like, then you are guaranteed whatever number they choose will have the digit "3" in it. Right?
As well as every other digit.
posted by iotic at 9:30 AM on December 8, 2014 [1 favorite]
As well as every other digit.
posted by iotic at 9:30 AM on December 8, 2014 [1 favorite]
Am I missing something? Could this be applied to any of the digits 0-9? It's seems 3 is simply a re-occuring symbol. What's so magical about this?
posted by Alt255 at 9:31 AM on December 8, 2014 [1 favorite]
posted by Alt255 at 9:31 AM on December 8, 2014 [1 favorite]
Infinity is huge. Small numbers, in the sense of numbers that we would have have anything to do with are essentially 0% of the integers. Count up all the atoms in the universe. You've moved basically 0% of the way to infinity. Take that number, we'll call it N, raise it to the power of itself N times. You still haven't budged. That's why they essentially all contain 3. They're all inconceivably huge numbers.
posted by empath at 9:32 AM on December 8, 2014 [5 favorites]
posted by empath at 9:32 AM on December 8, 2014 [5 favorites]
This was posted to Youtube on Apr 1, 2012. Surely the Spolier Limit has passed?
(listen to the end. The presenter just thinks 3 is a funny number)
posted by achrise at 9:33 AM on December 8, 2014 [1 favorite]
(listen to the end. The presenter just thinks 3 is a funny number)
posted by achrise at 9:33 AM on December 8, 2014 [1 favorite]
So if you ask someone to think of any positive whole number, and it can be any size they like, then you are guaranteed whatever number they choose will have the digit "3" in it. Right?
No, people can't choose a number large enough to be represent anything like a truly random number from the integers. Hell, computers can't. God couldn't, probably.
posted by empath at 9:33 AM on December 8, 2014 [1 favorite]
No, people can't choose a number large enough to be represent anything like a truly random number from the integers. Hell, computers can't. God couldn't, probably.
posted by empath at 9:33 AM on December 8, 2014 [1 favorite]
achrise: " Surely the Spolier Limit has passed?"
well we're approaching it, but I don't think we've hit the spoiler limit yet.
Alt255: "Am I missing something? Could this be applied to any of the digits 0-9? It's seems 3 is simply a re-occuring symbol. What's so magical about this?"
Yes, you are missing something called watching the video.
posted by boo_radley at 9:35 AM on December 8, 2014 [14 favorites]
well we're approaching it, but I don't think we've hit the spoiler limit yet.
Alt255: "Am I missing something? Could this be applied to any of the digits 0-9? It's seems 3 is simply a re-occuring symbol. What's so magical about this?"
Yes, you are missing something called watching the video.
posted by boo_radley at 9:35 AM on December 8, 2014 [14 favorites]
Could this be applied to any of the digits 0-9? It's seems 3 is simply a re-occuring symbol.
It also applies equally if you choose some other base. Imagine base one million, or something, where every number from 0 to a million has its own individual symbol. Any one symbol would still appear in most numbers.
posted by empath at 9:35 AM on December 8, 2014 [1 favorite]
It also applies equally if you choose some other base. Imagine base one million, or something, where every number from 0 to a million has its own individual symbol. Any one symbol would still appear in most numbers.
posted by empath at 9:35 AM on December 8, 2014 [1 favorite]
This guy has a very charismatic way of explaining mathematical concepts. This would be a great video to show in a high school math course, I think. Nice find!
posted by sockermom at 9:38 AM on December 8, 2014
posted by sockermom at 9:38 AM on December 8, 2014
People *can* choose a random natural number (i.e. there exist probability distributions on the natural numbers) but people can't *uniformly* choose a random number i.e. some natural numbers will always have more chance of being chosen than others).
posted by Omission at 9:40 AM on December 8, 2014 [6 favorites]
posted by Omission at 9:40 AM on December 8, 2014 [6 favorites]
Over half of seven-digit base-10 positive-integer numbers has at least one 3 in them (Spoiler - or any other digit 0-9).
posted by achrise at 9:42 AM on December 8, 2014
posted by achrise at 9:42 AM on December 8, 2014
The claim sounds really remarkable at first, but if you consider even reasonably large numbers for a moment, it seems obvious. I think he does a really good job proving it and guiding the viewer to that conclusion, but I feel like he failed to really drive it home with a terse summary.
I might try something like this: There are so many more large numbers than small numbers that if you choose an integer at random from the entire space of integers, it will with overwhelming likelihood – as a very conservative lower estimate – have over a billion digits. You'd have to get very lucky to not pull a 3 out of a bag of 10 numbers a billion times in a row.
posted by WCWedin at 10:10 AM on December 8, 2014 [7 favorites]
I might try something like this: There are so many more large numbers than small numbers that if you choose an integer at random from the entire space of integers, it will with overwhelming likelihood – as a very conservative lower estimate – have over a billion digits. You'd have to get very lucky to not pull a 3 out of a bag of 10 numbers a billion times in a row.
posted by WCWedin at 10:10 AM on December 8, 2014 [7 favorites]
It was satisfying to realize the moment that he designated the variable 'T' that it could be any digit, and then wait for confirmation at the end. Nicely presented.
posted by OHenryPacey at 10:11 AM on December 8, 2014 [1 favorite]
posted by OHenryPacey at 10:11 AM on December 8, 2014 [1 favorite]
An old teacher of mine played on a soccer club called "Almost Everywhere."
posted by grobstein at 10:15 AM on December 8, 2014 [2 favorites]
posted by grobstein at 10:15 AM on December 8, 2014 [2 favorites]
I don't think this conclusion is strictly correct, though. The set of integers with the digit 3 is countably infinite, but so is the set of integers without the digit 3-- they're both countably infinite, and you can come up with a bijection between them (as you can between all countably infinite sets). If anything, there are "equally many" integers without the digit 3 as there are with it. And since you can't uniformly randomly pick a natural number, it also doesn't make sense to talk about the chance of a "random natural number" having the digit 3.
If you were to pick some probability distribution over the natural numbers, then you could compute a probability for getting a number with 3 in it, but that probability wouldn't be 1 (unless you cheated and picked a distribution that avoids numbers without 3s).
posted by Pyry at 10:24 AM on December 8, 2014
If you were to pick some probability distribution over the natural numbers, then you could compute a probability for getting a number with 3 in it, but that probability wouldn't be 1 (unless you cheated and picked a distribution that avoids numbers without 3s).
posted by Pyry at 10:24 AM on December 8, 2014
Yeah, but mine goes to 11.
posted by Chuffy at 10:29 AM on December 8, 2014 [4 favorites]
posted by Chuffy at 10:29 AM on December 8, 2014 [4 favorites]
Well I guess that's numberwang then.
posted by cmoj at 10:31 AM on December 8, 2014 [16 favorites]
posted by cmoj at 10:31 AM on December 8, 2014 [16 favorites]
The set of integers with the digit 3 is countably infinite, but so is the set of integers without the digit 3-- they're both countably infinite, and you can come up with a bijection between them (as you can between all countably infinite sets)
You can also come up with a bijection between the even numbers and the integers. That doesn't mean that if you pick a number at random, it's as likely to be an even number as it is to be an integer. His argument is correct - as you increase the range, the proportion of integers missing the digit three shrinks, and that proportion tends to zero as the range tends to infinity.
posted by iotic at 10:33 AM on December 8, 2014 [5 favorites]
You can also come up with a bijection between the even numbers and the integers. That doesn't mean that if you pick a number at random, it's as likely to be an even number as it is to be an integer. His argument is correct - as you increase the range, the proportion of integers missing the digit three shrinks, and that proportion tends to zero as the range tends to infinity.
posted by iotic at 10:33 AM on December 8, 2014 [5 favorites]
Hey somebody get all this damned Cantor dust off my desk now.
posted by Wolfdog at 10:36 AM on December 8, 2014 [6 favorites]
posted by Wolfdog at 10:36 AM on December 8, 2014 [6 favorites]
About 33% of the way into the video, I wanted to know "why?" but all I got was more math. Then at the end, I got "Almost all numbers contain almost all numbers! "
Six minutes is a long ride for a math joke.
posted by four panels at 10:38 AM on December 8, 2014
Six minutes is a long ride for a math joke.
posted by four panels at 10:38 AM on December 8, 2014
How many times... it's not 3, it's 23!
posted by fearfulsymmetry at 10:48 AM on December 8, 2014
posted by fearfulsymmetry at 10:48 AM on December 8, 2014
The set of integers with the digit 3 is countably infinite, but so is the set of integers without the digit 3-- they're both countably infinite, and you can come up with a bijection between them (as you can between all countably infinite sets)
It's actually questionable that it's possible to choose a random number at all from a countably infinite set, I think.
In this case, they create a probability function for a set of size N, and then increase N to some arbitrarily high number, and show that as N 'approaches' infinity, the probability of choosing a number with a 3 in it becomes closer and closer to 100%.
posted by empath at 10:53 AM on December 8, 2014 [4 favorites]
It's actually questionable that it's possible to choose a random number at all from a countably infinite set, I think.
In this case, they create a probability function for a set of size N, and then increase N to some arbitrarily high number, and show that as N 'approaches' infinity, the probability of choosing a number with a 3 in it becomes closer and closer to 100%.
posted by empath at 10:53 AM on December 8, 2014 [4 favorites]
See here for the problems of sampling from an infinite set.
posted by empath at 10:54 AM on December 8, 2014 [1 favorite]
posted by empath at 10:54 AM on December 8, 2014 [1 favorite]
But I think this is really just about percentages, not probabilities.
posted by empath at 10:54 AM on December 8, 2014
posted by empath at 10:54 AM on December 8, 2014
Can this be extended to arbitrary sequences of digits?
posted by motty at 10:58 AM on December 8, 2014 [2 favorites]
posted by motty at 10:58 AM on December 8, 2014 [2 favorites]
People *can* choose a random natural number (i.e. there exist probability distributions on the natural numbers) but people can't *uniformly* choose a random number i.e. some natural numbers will always have more chance of being chosen than others).
Somewhat related is Benford's Law. If you wanted to use a random number that seems more "realistic", you might want for it to start with 1.
posted by bonje at 11:01 AM on December 8, 2014 [5 favorites]
Somewhat related is Benford's Law. If you wanted to use a random number that seems more "realistic", you might want for it to start with 1.
posted by bonje at 11:01 AM on December 8, 2014 [5 favorites]
Maths is broken.
posted by popcassady at 11:05 AM on December 8, 2014
posted by popcassady at 11:05 AM on December 8, 2014
tut-tut! Maths are broken.
posted by boo_radley at 11:05 AM on December 8, 2014
posted by boo_radley at 11:05 AM on December 8, 2014
English are broken.
Next: Almost all the names of people in the universe contain the word "Nigel".
posted by weapons-grade pandemonium at 11:12 AM on December 8, 2014 [4 favorites]
Next: Almost all the names of people in the universe contain the word "Nigel".
posted by weapons-grade pandemonium at 11:12 AM on December 8, 2014 [4 favorites]
His whole notion is fundamentally flawed because he ignores the fact that 24 is the highest number.
posted by jbickers at 11:15 AM on December 8, 2014 [4 favorites]
posted by jbickers at 11:15 AM on December 8, 2014 [4 favorites]
So we live down at the numerical planck scale, where things are so small that the basic rules that cover 99.99999999% of mathematical space simply don't apply, and most numbers don't have a three in them!
posted by Naberius at 11:19 AM on December 8, 2014 [9 favorites]
posted by Naberius at 11:19 AM on December 8, 2014 [9 favorites]
empath -
It's actually questionable that it's possible to choose a random number at all from a countably infinite set, I think.
Interesting. Do you think it's equally questionable that you could have a way of choosing a random real number between say, zero and one? Because that's an uncountably infinite set ...
posted by iotic at 11:31 AM on December 8, 2014
It's actually questionable that it's possible to choose a random number at all from a countably infinite set, I think.
Interesting. Do you think it's equally questionable that you could have a way of choosing a random real number between say, zero and one? Because that's an uncountably infinite set ...
posted by iotic at 11:31 AM on December 8, 2014
This is super easy to see with base 2 (binary), if you want to consider how many numbers have a 1 in them (ie, all of them).
posted by empath at 11:32 AM on December 8, 2014 [3 favorites]
posted by empath at 11:32 AM on December 8, 2014 [3 favorites]
Do you think it's equally questionable that you could have a way of choosing a random real number between say, zero and one? Because that's an uncountably infinite set ...
You can assign the probability of selecting a number within a certain arbitrarily small interval, but that's not the same thing as selecting a particular real number, all of which have probability 0 of being selected.
posted by empath at 11:33 AM on December 8, 2014 [1 favorite]
You can assign the probability of selecting a number within a certain arbitrarily small interval, but that's not the same thing as selecting a particular real number, all of which have probability 0 of being selected.
posted by empath at 11:33 AM on December 8, 2014 [1 favorite]
It's actually questionable that it's possible to choose a random number at all from a countably infinite set, I think.
Right, it's not meaningful to talk about uniformly picking a random element from a countably infinite set-- that's my point. So when we talk about the 'size' of integer subsets, all we can say is whether the subset is itself countably infinite or not, we can't meaningfully talk about probabilities.
For example, odd numbers and numbers divisible by twelve are both countably infinite, but does that mean you're just as likely to pick a 12-divisible number as an odd one? The question itself is meaningless-- you can't pick a random integer with uniform probability.
If you believe that
(1) you can talk about fractions of natural numbers with some property and
(2) that these fractions can be computed as the limits of fractions over finite subsets,
then I will 'prove' that exactly 1/3 of natural numbers are odd.
Consider the set (odd numbers italicized for clarity):
X = {1, 2, 4, 3, 6, 8, 5, 10, 12, 7, 14, 16, ...} = N
Convince yourself that this set is in fact the natural numbers-- every natural number occurs once and exactly once in the set.
Let's denote the subset of X which is the first i elements by X[1:i]. So, for example, X[1,3] = {1,2,4}.
What fraction of the first 3k elements are odd? In other words, what is frac_odd(X[1:3k])? Convince yourself that frac_odd(X[1:3k]) = 1/3 for all k.
If you accept that we can compute the fraction of elements with some property of an infinite set as the limit of fractions on subsets (2), then this next step is legitimate:
frac_odd(N) = frac_odd(X) = lim_{k-->infinity} frac_odd(X[1:3k]) = lim_{k-->infinity} 1/3 = 1/3
In conclusion 1/3 of natural numbers are odd.
posted by Pyry at 11:40 AM on December 8, 2014 [3 favorites]
Right, it's not meaningful to talk about uniformly picking a random element from a countably infinite set-- that's my point. So when we talk about the 'size' of integer subsets, all we can say is whether the subset is itself countably infinite or not, we can't meaningfully talk about probabilities.
For example, odd numbers and numbers divisible by twelve are both countably infinite, but does that mean you're just as likely to pick a 12-divisible number as an odd one? The question itself is meaningless-- you can't pick a random integer with uniform probability.
If you believe that
(1) you can talk about fractions of natural numbers with some property and
(2) that these fractions can be computed as the limits of fractions over finite subsets,
then I will 'prove' that exactly 1/3 of natural numbers are odd.
Consider the set (odd numbers italicized for clarity):
X = {1, 2, 4, 3, 6, 8, 5, 10, 12, 7, 14, 16, ...} = N
Convince yourself that this set is in fact the natural numbers-- every natural number occurs once and exactly once in the set.
Let's denote the subset of X which is the first i elements by X[1:i]. So, for example, X[1,3] = {1,2,4}.
What fraction of the first 3k elements are odd? In other words, what is frac_odd(X[1:3k])? Convince yourself that frac_odd(X[1:3k]) = 1/3 for all k.
If you accept that we can compute the fraction of elements with some property of an infinite set as the limit of fractions on subsets (2), then this next step is legitimate:
frac_odd(N) = frac_odd(X) = lim_{k-->infinity} frac_odd(X[1:3k]) = lim_{k-->infinity} 1/3 = 1/3
In conclusion 1/3 of natural numbers are odd.
posted by Pyry at 11:40 AM on December 8, 2014 [3 favorites]
You can assign the probability of selecting a number within a certain arbitrarily small interval, but that's not the same thing as selecting a particular real number, all of which have probability 0 of being selected.
How about the real number that is the distance i throw an object from myself? The probability distribution isn't even, but there is a distribution. Assuming continuity of possible distances, and that the actual real number distance can be determined, that's a random process over an uncountable set, resulting in real number distances, the probability of getting any particular one being "zero".
posted by iotic at 11:42 AM on December 8, 2014
How about the real number that is the distance i throw an object from myself? The probability distribution isn't even, but there is a distribution. Assuming continuity of possible distances, and that the actual real number distance can be determined, that's a random process over an uncountable set, resulting in real number distances, the probability of getting any particular one being "zero".
posted by iotic at 11:42 AM on December 8, 2014
Picking a random real number from the range (0,1) has the probability density function f(x) = {1 if 0 < x < 1, 0 otherwise}. It qualifies as a probability density function because it is always non-negative, and because it integrates to one. The equivalent to the probability density function for discrete variables is the probability mass function, and it must sum to one.
In other words, if you think you can uniformly pick a natural number, you have the problem of coming up with a function p(x) such that p(x) = C for all x (every number x must have the same probability of being chosen) and also that sum_{x = 0-->infinity} p(x) = 1.
You can't come up with a function that satisfies these two constraints: if C>0 then the sum is infinite, and if C=0 then the sum is zero, and it must be 1 to qualify as a valid probability distribution.
posted by Pyry at 11:54 AM on December 8, 2014 [1 favorite]
In other words, if you think you can uniformly pick a natural number, you have the problem of coming up with a function p(x) such that p(x) = C for all x (every number x must have the same probability of being chosen) and also that sum_{x = 0-->infinity} p(x) = 1.
You can't come up with a function that satisfies these two constraints: if C>0 then the sum is infinite, and if C=0 then the sum is zero, and it must be 1 to qualify as a valid probability distribution.
posted by Pyry at 11:54 AM on December 8, 2014 [1 favorite]
Wolfdog: "Hey somebody get all this damned Cantor dust off my desk now."
Oh - I already did. I thought it was some coke. Shit, what's this gonna do to my brain??? I feel something splitting!
posted by symbioid at 11:54 AM on December 8, 2014 [2 favorites]
Oh - I already did. I thought it was some coke. Shit, what's this gonna do to my brain??? I feel something splitting!
posted by symbioid at 11:54 AM on December 8, 2014 [2 favorites]
Pyry - if you can choose a random real number between 0 and 1, it's pretty straightforward to find the nearest rational number, then use a bijection from the rationals in that range onto the integers to give you an integer, randomly selected, over all the integers, with some probability distribution.
posted by iotic at 11:59 AM on December 8, 2014
posted by iotic at 11:59 AM on December 8, 2014
For example, odd numbers and numbers divisible by twelve are both countably infinite, but does that mean you're just as likely to pick a 12-divisible number as an odd one? The question itself is meaningless-- you can't pick a random integer with uniform probability.
That question is totally meaningful: look at integers less than n, then let n go to infinity. The result is called natural density, and you'd get that density of the 12-divisibles is 1/12 and of the odds is 1/2. So picking a number divisible by 12 is one-sixth as likely as picking an odd number.
posted by madcaptenor at 12:01 PM on December 8, 2014 [2 favorites]
That question is totally meaningful: look at integers less than n, then let n go to infinity. The result is called natural density, and you'd get that density of the 12-divisibles is 1/12 and of the odds is 1/2. So picking a number divisible by 12 is one-sixth as likely as picking an odd number.
posted by madcaptenor at 12:01 PM on December 8, 2014 [2 favorites]
it's pretty straightforward to find the nearest rational number,
What's the nearest rational number to π/4?
posted by madcaptenor at 12:02 PM on December 8, 2014
What's the nearest rational number to π/4?
posted by madcaptenor at 12:02 PM on December 8, 2014
I don't think the natural density should be conflated with the density over the entire set, because the natural density depends on considering the set in a particular order, and so different orderings of the same set will give rise to different densities. In other words, the natural density is a property of a particular sequence and not of the underlying set itself. The question "what fraction of natural numbers 1 to k are odd, as k goes to infinity" is different from "what fraction of natural numbers are odd".
posted by Pyry at 12:13 PM on December 8, 2014 [1 favorite]
posted by Pyry at 12:13 PM on December 8, 2014 [1 favorite]
How about the real number that is the distance i throw an object from myself?
You can only measure it to finite precision, so you're back to the 'arbitrarily small interval'
posted by empath at 12:20 PM on December 8, 2014
You can only measure it to finite precision, so you're back to the 'arbitrarily small interval'
posted by empath at 12:20 PM on December 8, 2014
Nearly all (as in, essentially 100%) of real numbers are normal. They don't terminate or repeat. It's impossible to 'select' even one of those numbers.
posted by empath at 12:23 PM on December 8, 2014 [1 favorite]
posted by empath at 12:23 PM on December 8, 2014 [1 favorite]
because the natural density depends on considering the set in a particular order
Yeah, but it's the same order you use for everything else having to do with natural numbers.
posted by madcaptenor at 12:37 PM on December 8, 2014
Yeah, but it's the same order you use for everything else having to do with natural numbers.
posted by madcaptenor at 12:37 PM on December 8, 2014
Yeah it's kind of an essential characteristic of the natural numbers that they're ordered. If you mess with the order, you've created some other set.
posted by empath at 12:41 PM on December 8, 2014 [1 favorite]
posted by empath at 12:41 PM on December 8, 2014 [1 favorite]
Somewhat related is Benford's Law. If you wanted to use a random number that seems more "realistic", you might want for it to start with 1.
It's more than random or "realistic". It's very realistic and utterly fascinating.
posted by Tell Me No Lies at 12:52 PM on December 8, 2014 [1 favorite]
It's more than random or "realistic". It's very realistic and utterly fascinating.
posted by Tell Me No Lies at 12:52 PM on December 8, 2014 [1 favorite]
You know what makes me angry? This makes me angry. 1 + 2 + 3 + 4 + 5 + ... = -1/12
posted by Gordafarin at 1:49 PM on December 8, 2014 [4 favorites]
posted by Gordafarin at 1:49 PM on December 8, 2014 [4 favorites]
You don't even need to go to huge number fields to get a feel for how this works. Even for 7 digit numbers integers (as in, between 1,000,000 and 9,999,999) more than half of these contain the number three. For still relatively small 100 digit numbers (compared to infinty), 99.99 % of number contain the digit 3.
I can't imagine anyone would think that 99.99% is not "almost all"!
posted by doozer_ex_machina at 2:23 PM on December 8, 2014
I can't imagine anyone would think that 99.99% is not "almost all"!
posted by doozer_ex_machina at 2:23 PM on December 8, 2014
I find the Law of Fives to be more and more manifest the harder I look.
posted by vibratory manner of working at 3:14 PM on December 8, 2014 [2 favorites]
posted by vibratory manner of working at 3:14 PM on December 8, 2014 [2 favorites]
You know what makes me angry? This makes me angry. 1 + 2 + 3 + 4 + 5 + ... = -1/12
That makes me delighted, and not the slightest bit angry.
posted by vibratory manner of working at 3:19 PM on December 8, 2014 [2 favorites]
That makes me delighted, and not the slightest bit angry.
posted by vibratory manner of working at 3:19 PM on December 8, 2014 [2 favorites]
>Can this be extended to arbitrary sequences of digits?
Yes, just work in base 10^length where your sequence is a single digit.
almost all integers contain your cell phone number.
posted by doiheartwentyone at 3:28 PM on December 8, 2014 [2 favorites]
Yes, just work in base 10^length where your sequence is a single digit.
almost all integers contain your cell phone number.
posted by doiheartwentyone at 3:28 PM on December 8, 2014 [2 favorites]
For arbitrary finite sequences of digits that is (which is, of course, almost none of them)
posted by doiheartwentyone at 3:32 PM on December 8, 2014
posted by doiheartwentyone at 3:32 PM on December 8, 2014
Almost all integers contain every word you've ever spoken and ever will speak in order complete with ISO 8601 timestamps encoded as unicode characters in decimal.
posted by jason_steakums at 3:47 PM on December 8, 2014 [2 favorites]
posted by jason_steakums at 3:47 PM on December 8, 2014 [2 favorites]
In the same sense of "almost all" (which still leaves me a bit queasy when discussing the natural numbers**), almost every integer contains a flawless, uncompressed 1080P movie of your life, from the moment of your birth through to the moment you will die. Flawlessly captured, from whatever vantage point you wish to specify.
It also contains approximately
30(fps) x
3600(seconds) x
24(hours) x
365(days) x
80(years) x
1920(h) x
1080(v) x
16777215(wrong colours) =~
10^24 times as many movies in which a single pixel in a single frame of the movie is incorrect. And about 10^48 times as many in which two pixels are incorrect, and so forth. Good luck finding the right one! It's almost certainly there...
(( ** I prefer this discussion to be framed within the real numbers from 0 to 1. Maybe just for the fact that you have a slightly more plausible generating mechanism for random numbers in that realm, even if it relies on an infinite speed-up.
Just flip an unbiased coin repeatedly, use 1 for H and 0 for T, and record the resulting binary sequence. Now speed up the flipping and recording process by a factor of two each time and you've generated yourself a lovely normal number (almost certainly) in a couple of seconds. There: that was easy.))
posted by pjm at 4:51 PM on December 8, 2014 [3 favorites]
It also contains approximately
30(fps) x
3600(seconds) x
24(hours) x
365(days) x
80(years) x
1920(h) x
1080(v) x
16777215(wrong colours) =~
10^24 times as many movies in which a single pixel in a single frame of the movie is incorrect. And about 10^48 times as many in which two pixels are incorrect, and so forth. Good luck finding the right one! It's almost certainly there...
(( ** I prefer this discussion to be framed within the real numbers from 0 to 1. Maybe just for the fact that you have a slightly more plausible generating mechanism for random numbers in that realm, even if it relies on an infinite speed-up.
Just flip an unbiased coin repeatedly, use 1 for H and 0 for T, and record the resulting binary sequence. Now speed up the flipping and recording process by a factor of two each time and you've generated yourself a lovely normal number (almost certainly) in a couple of seconds. There: that was easy.))
posted by pjm at 4:51 PM on December 8, 2014 [3 favorites]
tut-tut! Maths are broken.
Which reminds me of a joke from an old friend of mine:
How do you calculate the area of a circle?
π r2
No, Pie are round. Cornbread are squared.
posted by AlonzoMosleyFBI at 5:02 PM on December 8, 2014 [2 favorites]
Which reminds me of a joke from an old friend of mine:
How do you calculate the area of a circle?
π r2
No, Pie are round. Cornbread are squared.
posted by AlonzoMosleyFBI at 5:02 PM on December 8, 2014 [2 favorites]
For more enumerated permutations of a finite multiset, visit your local library.
posted by jomato at 7:06 PM on December 8, 2014
posted by jomato at 7:06 PM on December 8, 2014
tut-tut! Maths are broken.
posted by boo_radley at 11:05 AM on December 8 [+] [!]
If "Maths" is plural, then what is one Math?
posted by device55 at 8:13 PM on December 8, 2014
posted by boo_radley at 11:05 AM on December 8 [+] [!]
If "Maths" is plural, then what is one Math?
posted by device55 at 8:13 PM on December 8, 2014
Yeah it's kind of an essential characteristic of the natural numbers that they're ordered. If you mess with the order, you've created some other set.
No, sets don't know about order. You have to add some additional structure (i.e. an ordering of some kind). The linear ordering on the natural numbers is an essential characteristic of...a set endowed with a particular linear ordering. It's not an essential characteristic of the set of natural numbers. Hence changing the order affects, rather tautologically, the order structure but it doesn't affect the underlying set.
posted by busted_crayons at 9:14 PM on December 8, 2014
pjm, your comment about lives embedded, as it were, in N, is rather beautiful, I think.
Very often, in the context of natural numbers, "BLAH for almost all n" means "BLAH for all but finitely many n". Or maybe there's an ultrafilter involved...
posted by busted_crayons at 9:24 PM on December 8, 2014
Very often, in the context of natural numbers, "BLAH for almost all n" means "BLAH for all but finitely many n". Or maybe there's an ultrafilter involved...
posted by busted_crayons at 9:24 PM on December 8, 2014
So there's this really simple explanation: "If you keep flipping a coin, it's basically guaranteed you'll hit tails at least once. Or heads, even. Digits are just a ten sided coin, so yeah, a long enough number is going to have a 3 somewhere in there."
But instead he smarms his way through five minutes of you wondering what's so special about 3, and he knows that's all you're thinking, so he buries the fact that there's nothing about 3 until the absolute end.
I'm not a fan. I know he's enjoying the opportunity to explain how this manual process of counting digits can be circumvented using an equation, but he's cheating. Not as bad as that silly -1/12th thing, which really serves to do nothing but make people think their natural mathematical aptitude is somehow flawed, but this was a job for statistical explication.
But heh. It's his shtick.
posted by effugas at 11:47 PM on December 8, 2014 [1 favorite]
But instead he smarms his way through five minutes of you wondering what's so special about 3, and he knows that's all you're thinking, so he buries the fact that there's nothing about 3 until the absolute end.
I'm not a fan. I know he's enjoying the opportunity to explain how this manual process of counting digits can be circumvented using an equation, but he's cheating. Not as bad as that silly -1/12th thing, which really serves to do nothing but make people think their natural mathematical aptitude is somehow flawed, but this was a job for statistical explication.
But heh. It's his shtick.
posted by effugas at 11:47 PM on December 8, 2014 [1 favorite]
If "Maths" is plural, then what is one Math?
It's an infinite set. The chances of ever having just one Math are vanishingly small.
posted by iotic at 2:54 AM on December 9, 2014
It's an infinite set. The chances of ever having just one Math are vanishingly small.
posted by iotic at 2:54 AM on December 9, 2014
In the same sense of "almost all" (which still leaves me a bit queasy when discussing the natural numbers**)
OMG thank you. There's no natural measure on the set of natural numbers, especially one that has interesting subsets of measure zero, so this has been driving me crazy all thread.
posted by Elementary Penguin at 3:23 AM on December 9, 2014 [1 favorite]
OMG thank you. There's no natural measure on the set of natural numbers, especially one that has interesting subsets of measure zero, so this has been driving me crazy all thread.
posted by Elementary Penguin at 3:23 AM on December 9, 2014 [1 favorite]
But instead he smarms his way through five minutes of you wondering what's so special about 3, and he knows that's all you're thinking, so he buries the fact that there's nothing about 3 until the absolute end.
I think that's a bit harsh. It's a parlor trick: he makes a statement that sounds unlikely on its face, and then proves it's true. Along the way he demonstrates how mathematicians incrementally approach problems: in this case by writing out a table of values, writing down a recurrence relation, logically deducing a closed-form solution, and applying basic calculus. He also touches on an interesting definition: how can we formalize the idea of a property holding for "almost all" numbers?
If your friend says: "if I drop this coin into my pocket then it will no longer be in my hand" you would say "so what?", but if she uses sleight of hand to make a coin disappear you might be entertained.
posted by jomato at 3:56 AM on December 9, 2014 [2 favorites]
I think that's a bit harsh. It's a parlor trick: he makes a statement that sounds unlikely on its face, and then proves it's true. Along the way he demonstrates how mathematicians incrementally approach problems: in this case by writing out a table of values, writing down a recurrence relation, logically deducing a closed-form solution, and applying basic calculus. He also touches on an interesting definition: how can we formalize the idea of a property holding for "almost all" numbers?
If your friend says: "if I drop this coin into my pocket then it will no longer be in my hand" you would say "so what?", but if she uses sleight of hand to make a coin disappear you might be entertained.
posted by jomato at 3:56 AM on December 9, 2014 [2 favorites]
You have to add some additional structure (i.e. an ordering of some kind). The linear ordering on the natural numbers is an essential characteristic of...a set endowed with a particular linear ordering. It's not an essential characteristic of the set of natural numbers.
The natural numbers are totally (linearly) ordered.
posted by empath at 7:50 AM on December 9, 2014
The natural numbers are totally (linearly) ordered.
posted by empath at 7:50 AM on December 9, 2014
Somewhat related is Benford's Law. If you wanted to use a random number that seems more "realistic", you might want for it to start with 1.
oh, bless your heart. Reading this, I was driving myself nuts trying to remember the name of that.
posted by Zed at 7:50 AM on December 9, 2014
oh, bless your heart. Reading this, I was driving myself nuts trying to remember the name of that.
posted by Zed at 7:50 AM on December 9, 2014
this was a job for statistical explication.
It's not statistical, it's calculus. For large N, the number approaches 100%.
posted by empath at 7:52 AM on December 9, 2014 [1 favorite]
It's not statistical, it's calculus. For large N, the number approaches 100%.
posted by empath at 7:52 AM on December 9, 2014 [1 favorite]
OMG thank you. There's no natural measure on the set of natural numbers, especially one that has interesting subsets of measure zero, so this has been driving me crazy all thread.
A couple of things: counting measure is a perfectly good -- and, if there's any justice in the world, "natural" -- measure on N, so you need some adjective to make what you're saying true. Maybe your notion of "natural" requires that all singletons have the same (positive) measure, or you want some kind of translation-invariance, in which case, yeah, you're not going to be able to make a (countably additive) probability measure on N with that property.
Second, there are plenty of nice, natural probability measures on N. For example, take your sigma-algebra to be all subsets of N, and for each subset S, let m(S) be the sum of X_S(n)/n^2, as n varies over N, with X_S the characteristic function of S. (If you want this to take values in [0,1] you should normalize by multiplying by 6/pi^2, i.e. 1 over the sum of 1/n^2.) This obviously gives measure 0 to the empty set, is countably additive, and assigns measure 1 to the whole of N. You can replace 1/n^2 by your favourite summable sequence (adjust your scaling if you care about full-measure sets having measure 1 instead of some other positive number). In particular, if you tell me your favourite interesting proper subset of N, I can tweak this construction so that that subset has measure 0.
Third, as alluded to earlier, measures are not the only reasonable way to formally say "almost every". For example, N has plenty of ultrafilters. The definition of an ultrafilter on N nails down the intuitive notion of "almost all natural numbers" in a nice way. It's a fairly intuitive generalization of the phrase "all but finitely many", which is, for many very practical purposes (convergence of sequences!) the right notion of "almost all" natural numbers.
The natural numbers are totally (linearly) ordered.
Yes, as a linearly ordered set, they are. As a set, they are not, i.e. {2,1,3,4,...} is the same set as {1,2,3,4,...}. Sets are completely determined by their elements. The natural numbers are not just a set, they are a set with additional structure, including a total order. (My complaint was with the assertion that messing with the order "creates some other set"; it does not: it creates some other ordered set.)
posted by busted_crayons at 8:00 AM on December 9, 2014 [1 favorite]
empath,
What could have made sense in ten seconds was made complicated over five minutes. Fun if you enjoy the complication, I guess.
posted by effugas at 8:26 AM on December 9, 2014
What could have made sense in ten seconds was made complicated over five minutes. Fun if you enjoy the complication, I guess.
posted by effugas at 8:26 AM on December 9, 2014
Yeah i wasn't being precise, of course the natural numbers are not only a set.
posted by empath at 9:10 AM on December 9, 2014
posted by empath at 9:10 AM on December 9, 2014
My apologies for pedantry! (Pedantry is how we stare into the Abyss with minimal abyssal staring-reciprocation or something.)
posted by busted_crayons at 9:11 AM on December 9, 2014 [1 favorite]
posted by busted_crayons at 9:11 AM on December 9, 2014 [1 favorite]
In particular, if you tell me your favourite interesting proper subset of N, I can tweak this construction so that that subset has measure 0.
Let's say I restrict you to measures where m({k}) ≥ m({l}) if k < l. Can you still do this? (My instinct is no, but my instinct is also to know that you can do weird shit when you allow arbitrary measures.)
posted by madcaptenor at 9:32 AM on December 9, 2014
Let's say I restrict you to measures where m({k}) ≥ m({l}) if k < l. Can you still do this? (My instinct is no, but my instinct is also to know that you can do weird shit when you allow arbitrary measures.)
posted by madcaptenor at 9:32 AM on December 9, 2014
It is because three is a magic number.
You really don't have to guess, when it's three you can see it's a magic number.
posted by theartandsound at 11:00 AM on December 9, 2014
You really don't have to guess, when it's three you can see it's a magic number.
posted by theartandsound at 11:00 AM on December 9, 2014
Let's say I restrict you to measures where m({k}) ≥ m({l}) if k < l. Can you still do this? (My instinct is no, but my instinct is also to know that you can do weird shit when you allow arbitrary measures.)
Just annihilate the interesting set by assigning measure 0 to each of its members.
Then if the remaining set of natural numbers is infinite, assign them the measures 1/2, 1/4, 1/8, and so on, in order.
Or, if the remaining set of natural numbers is finite (say, n of them) then assign each all measure 1/n.
-
Shoot, I've made six weirder measures than that before breakfast a lot of days.
posted by Wolfdog at 12:01 PM on December 9, 2014 [1 favorite]
Just annihilate the interesting set by assigning measure 0 to each of its members.
Then if the remaining set of natural numbers is infinite, assign them the measures 1/2, 1/4, 1/8, and so on, in order.
Or, if the remaining set of natural numbers is finite (say, n of them) then assign each all measure 1/n.
-
Shoot, I've made six weirder measures than that before breakfast a lot of days.
posted by Wolfdog at 12:01 PM on December 9, 2014 [1 favorite]
Well, I wasn't born yesterday. There's no 3 in 42.
Thanks Metafilter for helping me enjoy math(s)!
posted by sneebler at 12:21 PM on December 9, 2014
Thanks Metafilter for helping me enjoy math(s)!
posted by sneebler at 12:21 PM on December 9, 2014
Yup, what Wolfdog said.
There's no 3 in 42.
I count 14.
posted by busted_crayons at 12:22 PM on December 9, 2014 [2 favorites]
There's no 3 in 42.
I count 14.
posted by busted_crayons at 12:22 PM on December 9, 2014 [2 favorites]
The "nearly all" claim gets clearer and easier to believe as you realize that as numbers get longer, they get more likely to contain all the digits. There's no reason that "nearly all contain the digit 3" and "nearly all numbers contain the digit 5" are contradictory, because as you approach infinity, those two statements will both be true for an increasing proportion of the numbers.
I don't think this is wanking, or just a math joke. It strikes me as a good demonstration of the way that numbers don't behave the way we intuitively think they might, as we're wallowing around down in the numbers that are low enough for us to clearly conceptualize them. It's very brain-hurting to say, "There are infinite numbers that don't contain a 3, and infinite numbers that do, and somehow as you get closer and closer to infinity the infinite set of numbers that DO contain a 3 gets bigger than the infinite set that don't...." Ow. And yet I like this kind of thing because I have these moments when I've almost got it, and for that moment my brain is almost occupying the same sphere as the brains of my physicist and mathematician friends, who deal with these kinds of un-graspable abstractions all the time.
For the other link, the one where 1+2+3+4+...=-1/2, you have to accept on some level that the sum of an infinite series can be declared to be a number that never actually appears when you're doing the sum. The idea that 1-1+1-1+1...=1/2 is...I don't even know what to call it. If you keep adding and subtracting the numeral 1 forever, you're going to get a result that keeps toggling back and forth between 0 and 1 forever—it's a digital outcome, not an analog one; it doesn't ever actually move through the values between 0 and 1. But, as the physicist who narrates that video points out, the idea that this sum is 1/2, because 1/2 is the average of all those zeros and ones, has useful applications in the kind of physics that is also too rarefied for most of us. It's less a true outcome than a kind of accepted convention which turns out to be useful, and I sometimes think that what sets most of us apart from the mathematicians and the physicists is that they are comfortable with answers that make sense on this very abstract level while we're still trying to squeeze it all into the the kind of concrete "count the bottle-caps" math that makes sense to us. How can that series add up to 1/2, or the other one to -1/12, when those numbers never appear as you sum the numbers, if you can never actually get to those fractions by traveling along the equation? There are people who are comfortable saying, "it's true because the math works out," and there are people, like me, who are OK saying, "I can go with that because people who understand this a lot better than I do tell me it works out" and there are people who say, "No. Nope. Nopity-nope. No."
It's all kind of wonderful, I think. I have friends who are physicists, and a friend who is a mathematician, and a friend who does something with computer theory that only, like, 12 people on the whole planet understand, and sometimes they get talking on Facebook and none of it makes sense to me at all but I like to just "listen" to them and marvel at the wonder that is the human brain.
posted by not that girl at 5:48 PM on December 9, 2014 [2 favorites]
I don't think this is wanking, or just a math joke. It strikes me as a good demonstration of the way that numbers don't behave the way we intuitively think they might, as we're wallowing around down in the numbers that are low enough for us to clearly conceptualize them. It's very brain-hurting to say, "There are infinite numbers that don't contain a 3, and infinite numbers that do, and somehow as you get closer and closer to infinity the infinite set of numbers that DO contain a 3 gets bigger than the infinite set that don't...." Ow. And yet I like this kind of thing because I have these moments when I've almost got it, and for that moment my brain is almost occupying the same sphere as the brains of my physicist and mathematician friends, who deal with these kinds of un-graspable abstractions all the time.
For the other link, the one where 1+2+3+4+...=-1/2, you have to accept on some level that the sum of an infinite series can be declared to be a number that never actually appears when you're doing the sum. The idea that 1-1+1-1+1...=1/2 is...I don't even know what to call it. If you keep adding and subtracting the numeral 1 forever, you're going to get a result that keeps toggling back and forth between 0 and 1 forever—it's a digital outcome, not an analog one; it doesn't ever actually move through the values between 0 and 1. But, as the physicist who narrates that video points out, the idea that this sum is 1/2, because 1/2 is the average of all those zeros and ones, has useful applications in the kind of physics that is also too rarefied for most of us. It's less a true outcome than a kind of accepted convention which turns out to be useful, and I sometimes think that what sets most of us apart from the mathematicians and the physicists is that they are comfortable with answers that make sense on this very abstract level while we're still trying to squeeze it all into the the kind of concrete "count the bottle-caps" math that makes sense to us. How can that series add up to 1/2, or the other one to -1/12, when those numbers never appear as you sum the numbers, if you can never actually get to those fractions by traveling along the equation? There are people who are comfortable saying, "it's true because the math works out," and there are people, like me, who are OK saying, "I can go with that because people who understand this a lot better than I do tell me it works out" and there are people who say, "No. Nope. Nopity-nope. No."
It's all kind of wonderful, I think. I have friends who are physicists, and a friend who is a mathematician, and a friend who does something with computer theory that only, like, 12 people on the whole planet understand, and sometimes they get talking on Facebook and none of it makes sense to me at all but I like to just "listen" to them and marvel at the wonder that is the human brain.
posted by not that girl at 5:48 PM on December 9, 2014 [2 favorites]
Not that girl,
1+1 does in fact equal zero...mod 2. That the math works out, and is even useful, doesn't obscure the fact that you're operating in a separate mathematical domain that has additional freedoms and constraints. That's what bothers me about the -1/12 thing. Comparing infinities is as much a separate domain as life in mod 2. Yes, it's weird but don't act like it's weird in regular arithmetic. -1/12 is not a true outcome there.
By comparison this 3 thing is a nice way to show some equations. It actually is true, without cheating. Much preferred.
posted by effugas at 11:04 PM on December 10, 2014
1+1 does in fact equal zero...mod 2. That the math works out, and is even useful, doesn't obscure the fact that you're operating in a separate mathematical domain that has additional freedoms and constraints. That's what bothers me about the -1/12 thing. Comparing infinities is as much a separate domain as life in mod 2. Yes, it's weird but don't act like it's weird in regular arithmetic. -1/12 is not a true outcome there.
By comparison this 3 thing is a nice way to show some equations. It actually is true, without cheating. Much preferred.
posted by effugas at 11:04 PM on December 10, 2014
« Older Cooking isn't fun, but you should do it anyway. | Parable of the Polygons Newer »
This thread has been archived and is closed to new comments
posted by showbiz_liz at 9:16 AM on December 8, 2014 [1 favorite]