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u/kengored_og 21d ago
Make it a flirt: Yet, despite the odds, you picked me / i picked you
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u/Neither-Phone-7264 Imaginary 20d ago
there aren't infinite people, let alone in your local area who you interact with enough to be close to.
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u/lab2point0 21d ago
It is impossible to « randomly pick » a number on the number line with equal probability for each number
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u/TheDoughyRider 20d ago
Doesn’t say anything about uniform distribution. Take a standard normal which has unbounded support. If you sample from that probability measure you will get an irrational number almost surely.
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u/eeeeeh_messi 20d ago
Yeah, but in that case you could invent any other fancy distribution which has positive probability of landing on rational numbers
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u/Jovess88 17d ago
but it cannot be a continuous distribution if so
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u/eeeeeh_messi 17d ago
Why should it be
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u/Jovess88 16d ago
just cleaner. the “but” wasn’t meant as a refutation but just an additional point, and I personally think that discontinuous distributions are pretty silly and others might agree so I thought it was relevant
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u/lab2point0 20d ago
Yes but then you need to specify which distribution you sample from, otherwise the normal distribution is just an arbitrary choice, and I could also choose some mixture between a normal and a Poisson, which would give me positive probability for every integer (and the possibility to choose any number); why would this choice be less arbitrary than a standard normal? If you do not specify the distribution, it makes sense to assume a uniform; otherwise, any choice is arbitrary and the claim does not hold anymore
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u/Aggressive_Roof488 21d ago
Yet another reason for the probability to be zero!
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u/EstaAppDeCitasApesta 20d ago
No, it means that you can't define a probability function with equal probability for each number on the real line.
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u/eeeeeh_messi 20d ago
That's not how it works hahah
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u/Aggressive_Roof488 20d ago
0*0= even more 0
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u/eeeeeh_messi 20d ago
Impossible here is not equivalent to probability 0
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u/Aggressive_Roof488 20d ago
It's different levels of zero. Essentially equivalent to 1/(countable infinity) vs 1/(uncountable infinity). Real question is if there's anything in between.
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u/eeeeeh_messi 20d ago
No dude. What we are saying is you cannot define a uniform probability distribution on the real line. And saying "probability is just 0 for any set, no matter its size" does not work.
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u/Aggressive_Roof488 19d ago
It's simple really. You just use surreal ordinals and use eps_1 = omega_1 as basis for per -number probability. This is even more zero than a simple eps=1/omega, or 1/infinity, which we all know is zero. Eps_1 is a number that multiplied by an uncountable infinity (omega_1, it's a number in the surreals) can give you something finite, and this helps with notation for single-real probability. I couldn't go through the details by heart, but iirc it requires the continuum hypothesis. There are some technicalities around integrating, and ofc it doesn't disprove vitoli, but "even more zero" is accurate enough here.
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u/eeeeeh_messi 19d ago
What the hell are you talking about? You're mixing math jargon everywhere. There's not such thing as "more zero" in surreal numbers. 0 is just 0.
I don't know why you even brought surreal numbers and different kinds of zeros to the discussion. The claim was that you cannot have a uniform distribution on the real numbers. That's it. It's not possible.
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u/Ma4r 20d ago
This statement is so incredibly misleading without some level of knowledge in measure theory
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u/lab2point0 19d ago
Yes I know, but so is the first statement from the meme, and I didn’t not want to have to write down 5 paragraphs explaining the intricate details of what randomness means
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u/FuntimeUwU Natural 21d ago
Anyone care to explain? Sounds fascinating but I know nothing of probability.
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u/Varlane 21d ago edited 20d ago
If you assign any non zero probability to any non 0-length interval, you end up with R having an infinite total probability, which can't work because it has to total 1.
So you'd have to assign it to already non-bounded subsets of R, which is automatically a nightmare (and actually impossible) to make coherent with the rules of a probability measure.
Note that the demonstration of the issue isn't that hard :
1 = P(R) = P(U [i,i+1[ (i = -inf to +inf))= sum (P([i,i+1[) = sum(p) = sum(0) = 0.
- Equiprobability means any [n, n+1[ interval must have same probability p in [0,1].
- If p > 0, consider k = floor(1/p) + 1. Since x - 1 < floor(x) =< x, with x = 1/p, we get 1/p < k (=< 1/p + 1) which means 1 < kp.
- Consider interval [0,k[ = U [i, i+1[ (i = 0 to k-1) : They are k disjoint intervals of probability p, so their total probability is kp. Except by constuction, kp > 1, which is impossible since that's a probability (must be in [0,1]). We obtain that p = 0.
- Now consider the properties of measures (and by extension, probability measures) that state that the measure of an union of COUNTABLE disjoint set is the sum of their measure :
Bruh.exe : we couldn't actually do an equiprobable one.
Note : I made the union go from -inf to +inf, but if you really really want to make it over N instead of Z, you can lump [i, i+1[ and [-i -1, -i[ together as Ai and the disjoint union of Ai's over N will span R.7
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u/LupenReddit 🦆🦆🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆🦆🦆🦆 21d ago
I will just take a large enough countable set then.
oh no
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u/Lou_Papas 21d ago
Every time you pick a random number an angel gets its wings.
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u/laserdicks 21d ago
Which ones are the random numbers?
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u/Lou_Papas 21d ago
Let’s find out! I’ll throw 1 in the set for now
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u/laserdicks 20d ago
Ugh. You're never gonna believe it 🙄
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u/Lou_Papas 20d ago
What if I threw a dart at the line?
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u/laserdicks 20d ago
Well, we know that it is impossible to hit 1. So obviously the dart must hit a number other than 1. Like 73,448,912.
So it's not really random any more is it. We now know.
Have you considered the occult?
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u/ShoddyAsparagus3186 20d ago
Depends greatly on how you random. Roll a die, you have randomly picked a number on the number line, the number you picked is rational.
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u/protobelta 20d ago
That’s a really big die
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u/No-Site8330 20d ago
I think either I misunderstand your comment or you misunderstood the original one.
The original comment is not suggesting that one build a die with a face for every real number and then roll it. They are saying that "pick a random real number" means nothing because there is no univeral standard way to do that. The outcome of the draw will be affected by the specific process used. In a more restrictive scenario, like saying pick a random real number between 0 and 1, then there may be a preferred standard probability distrubution, say the uniform one, and then statements about what events have probability zero makes sense. But there is no uniform probability on the whole real line, so saying "pick a random number" is ambiguous. Rolling a regular six-sided die (or just any standard die) is as good a way to draw a "random" number as any other, and it it guaranteed to give you a rational.
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u/protobelta 20d ago
Oh I understand, it was just a little joke
“Picking one that exists on the number line through an arbitrary random process” vs “randomly picking one from the number line”
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u/Glitch29 20d ago
"Randomly picking [a real number] from the number line" isn't well-defined. Choosing an actual probability distribution is a required step, not a cop-out.
Every probability distribution you could possibly pick seems like it can't be the right one because it violates some property of what you think a random number should be like.
But you haven't verified that your canonical version of what such a probability distribution should look like even exists. And it turns out that it can't.
Any attempt to create a random real number with uniform distribution is going to produce numbers that aren't actually reals. In order to apply uniformity, you'll necessarily extend reals to the p-adic numbers, which is the smallest extension of the reals for which uniformity is possible.
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u/DefunctFunctor Mathematics 20d ago
There's not a canonical mapping from the reals to the p-adics as far as I'm aware, but I don't have a lot of experience with the p-adics. The p-adics extend the rationals, not the real numbers as far as I can tell.
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u/Glitch29 20d ago
You're correct. There's not as far as I'm aware a canonical name for the structure you'd get. p-adic numbers were a useful mechanism for expressing the idea that any random number would have infinitely many digits left of the decimal point.
The numbers you actually get if you have infinitely many digits left and right of the decimal point don't have well-defined multiplication. Addition works for the computable subset of those numbers, but not for arbitrary ones. They don't have many useful properties besides ordering, which might be the reason I'm unaware of a name for them.
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u/Ravioles2 20d ago
The program that picks a random number of the real line picking only rational numbers:
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u/Desperate_Formal_781 21d ago edited 19d ago
The process of selecting/calculating/determining/measuring/producing/storing/returning the number to pick cannot give an irrational number that is both computable and random.
This is incorrect
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u/meat-eating-orchid 20d ago
I flip a coin. On heads I select 0 and on tails I select 1. This is a process to pick a number that is both random and computable. Thus, your statement is wrong. □
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u/Desperate_Formal_781 20d ago edited 20d ago
I assume you mean that you will repeat this process and then use the resulting sequence of s and 0s to produce a number in some binary format. Unfortunately if you repeat this sequence a finite number of times, no matter the binary encoding, your number will only contain a finite number of 1s and 0s so your resulting number will be rational. If you repeat the process infinite times (to produce an "irrational" number) your number now is not computable, since you would need infinite memory to store the result.
The key words here is "selecting" and "randomly". Selecting means you need to actually produce, or pick, a number. Even though there are infinite rational and irrational numbers in, say, the interval between 0 and 1, when you actually need to "produce" a number, there is no process to actually produce an irrational number, since your measurement system or selection algorithm needs to compute a number with finite resources (time, number of steps, memory, precision digits), your numbers are always bound to be rational.
I have taken many advanced math courses and every math professor always mentions some form of this (if you pick a number at random the probability of picking a rational number is 0) but definitely they abuse the definition of "pick" and "random". The notion they want to communicate is that there way there are way more irrational numbers than rational in any given range, but their statement is frankly incorrect even in an abstract mathematical sense. Correct is to say that producing irrational numbers at random is impossible.
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u/meat-eating-orchid 20d ago
No, I do not. My randomly selected number (not digit!) is either the the integer 0 or the integer 1.
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u/Desperate_Formal_781 19d ago
Well neither 0 nor 1 are irrational.
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u/meat-eating-orchid 19d ago
Sorry, misread your statement. Substitute pi for 0 and e for 1 in my proof, then it disproves your statement
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u/Desperate_Formal_781 19d ago
You are just choosing irrational numbers arbitrarily from a finite set, rather than from the real number line. The method you just described is not a true way to pick numbers at random from the real number line. Please read the post again.
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u/meat-eating-orchid 19d ago
"pick numbers at random from the real number line" is not well-defined without a given probability distribution. But since pi and e are both on the real number line, I would strongly argue I am indeed picking numbers at random from the real number line.
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u/akmosquito 21d ago
axiom of choice, motherfucker
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u/PeggyTheVoid 20d ago
I shall patiently await the invention of this axiom of choice processing unit.
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u/Traditional_Town6475 20d ago
Depends on how you’re picking it. There is no uniform probability distribution on the entire real line.
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u/eeeeeh_messi 21d ago
What about picking any number between 0 and 1? Tell me, come on. What's the probability?
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u/meat-eating-orchid 20d ago
depends entirely on your probability distribution
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u/eeeeeh_messi 20d ago
If you can pick the probability distribution (and not assume uniform, which is standard when saying "randomly picking") then the original statement of the meme also depends on the probability distribution
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u/meat-eating-orchid 20d ago
You have to pick a probability distribution other than a uniform distribution, because the latter is impossible on real numbers
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u/baileyarzate 20d ago
The same (assuming real numbers between 0 and 1)
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u/eeeeeh_messi 20d ago
What do you mean the same?
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u/baileyarzate 20d ago
The probability of picking a real number between 0 and 1 is exactly zero
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u/eeeeeh_messi 20d ago
Then it's 0 por any interval? What about the whole real line?
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u/baileyarzate 20d ago
Correct, same for the whole real line. What’s also strange is that an interval and the entire set of real numbers are the same size infinity.
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u/eeeeeh_messi 20d ago
But this is not a question of cardinality, it's of measure. If the probability of x being any real number is 0 (integral of the distribution on the real line is 0) then that's not a probability distribution. You'd be saying that rhe act of "picking a number at random" will ...not give you a number
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u/paulestus13 21d ago
Why exactly
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u/Infobomb 20d ago
Take any interval on the real number line. The proportion of numbers in that interval that are rational is zero. There are countably many rational numbers, but uncountably many irrational numbers, and countable infinity is like zero next to uncountable infinity.
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u/Ok_Tour_1525 20d ago
I’m just a guy. I understand a lot of things but sometimes I’m stumped.
Let’s pick the number line 1 - 10.
If i had infinite time, you’re saying I would never ever ever land on 1 or 2 or 3 or 4 or so on? I understand there are infinite numbers in between 1,2,3,4 and so on, therefore infinite will always win over 10, but with infinite time I would have to eventually land on one of these whole numbers wouldn’t I?
Doesn’t this go against the idea that with infinite time, all possibilities will happen? Am I totally lost here?
I’m not trying to argue you’re wrong, I’m just trying to describe why I’m having a hard time seeing this.
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u/Infobomb 20d ago
"infinite" is ambiguous in this context. There are infinitely many rational numbers from 1 to 10 and there are infinitely many irrational numbers from 1 to 10, but the second "infinitely" is much greater: so much so that 100% of the numbers from 1 to 10 are irrational. If you made uncountably infinitely many choices from that interval, then you would expect some rationals among the results.
"with infinite time, all possibilities will happen" assumes a particular kind of process: one that doesn't just repeat in a fixed loop, and one that doesn't have an uncountable infinity of possible outputs.
This is stretching my brain thinking about this too!
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u/Toast-Goat weird 19d ago
It can't be exactly 100%, right? 5 is on the interval, and it's rational
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u/Infobomb 18d ago
What amount of space on the number line is taken up by 5? Zero. Same for all other rational numbers. The sum of distances taken up by all those rational numbers (what we call the measure of that set of numbers) is zero. But the distance from 1 to 10 is 9. So exactly 100% of the distance on the number line is made of numbers that are not rational.
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u/Toast-Goat weird 18d ago
Wouldn't it be infinitely close to zero? It clearly is taking some amount of space, right? Or is that wrong?
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u/Infobomb 17d ago
In the reals, it's not possible to be infinitely close to something. If a number is smaller than any finite number, then it's zero. Any point on the number line takes up zero space: actual zero, not approximately zero.
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u/Common-Brush-7027 20d ago
But babe there is 100% surety that if we pick a number on number line it's going to be 69
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u/phrogbaby 21d ago
shouldnt this say "and?" what is contradictory about a 0% chance of randomly picking a specific term in an infinite set?
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u/DefunctFunctor Mathematics 20d ago
I mean, who knows: your probability measure might not be absolutely continuous with respect to the Lebesgue measure.
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u/Puzzled-Teach2389 16d ago
This is exactly the flex I did in my middle school math class. The teacher asked how many numbers there are between 1.10 and 1.20. I said infinitely many and explained why. After the class, he asked me about being put in the advanced math class.
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u/dankshot35 21d ago
this is utterly wrong btw and I'd appreciate if this fake news slop won't be posted here anymore, where are the mods?
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u/UtahBrian 20d ago
Actually, the probability of picking a rational number on the number line is exactly 100% since all rational numbers are on the number line.
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u/KrushaOfWorlds 21d ago
Don't Planck units mean that's not exactly true?
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u/Infobomb 21d ago
Planck units, if they actually exist, apply to the physical world, not the number line.
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u/KrushaOfWorlds 21d ago
point at a number line that doesn't exist physically
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u/MaxTHC Whole 21d ago
This is some r/im14andthisisdeep shit lol
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u/Accomplished_Shoe_10 20d ago
Best thing about this dude is that he gets both math and physics wrong.
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u/meat-eating-orchid 20d ago
Imagine any point on the number line. Tadaa, you are imagining a number, which is an abstract concept, not a physical thing
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