5 ms·

This reminds me of this reddit question https://www.reddit.com/r/askmath/comments/1eakt5c/if_you_pick_a_real_number_from_0_to_1_what_is_the/ https://www.reddit.

by jfoutz 2y ago

This reminds me of this reddit question https://www.reddit.com/r/askmath/comments/1eakt5c/if_you_pick_a_real_number_from_0_to_1_what_is_the/ https://www.reddit.com/r/askmath/comments/1eakt5c/if_you_pic... "If you pick a real number from 0 to 1, what is the probability that it is rational?" and the first response, " With a uniform random process over an interval (a, b), you can find the probability of picking an element from a set X by integrating 1/(b - a) over the set X. Since the (lebesgue) integral of the rational numbers is 0, the probability of picking a rational number is 0. " And that kinda broke my brain for an afternoon. So, when a computer represents a number, it's always rational. You can have a lot of memory, but always finite. Now, you can go further, and represent things symbolically, like sage math or a bunch of other systems. But (I believe) at the end of the day Godel always wins. There are always true statements that can't be proved. If I remember right, computable stuff is a subset (maybe improper?) of provable stuff. Looks interesting. Thank you for posting it. I'll try to read it later. Although, I'm a math tourist. I probably won't be able to give you any grand insight.

deleted 2y ago
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rjurney 2y ago
I got this one without realizing the complex basis for it... there's an infinite number of real numbers between 0 and 1, compared with a limited proportion of rational numbers.
- jerf 2y ago
  There's an infinite number of both. A way to intuit why you have a zero probability of choosing a rational is to consider randomly picking digits one at a time. Suppose you start with a "3". What is necessary to get one third as your random choice? You must now uniformly randomly select an infinite number of 3s. You can think of that as just a contradiction without much loss here, in that selecting an infinite number of 3s is infinite proof that your process must not be random. (Putting rigor on that might be a challenge but I think it's a fine intuition.) This argument similarly holds for any rational, because no matter how large any given rational's repeat may be, it is always finite, and in order for that rational to be your choice you have still fully determined the result of a putatively uniformly random process for an infinite number of exact picks, and if it is truly random, the probability of that process producing the exact necessary repeat infinitely is always 0.
  - kaashif 2y ago
    That argument works for any number. The probability of picking any particular number is zero, there's nothing special about rationals from that perspective. This has no bearing on the measure of the set as a whole.
    - jerf 2y ago
      Which is why I very, very explicitly presented it as a way to gain some intuition and not as a rigorous mathematical argument. That wasn't an accident.
      - kaashif 2y ago
        The probability of picking a single number can't be used to determine the probability of picking a set. You can't build intuition this way. This is almost exactly the same thing as saying integrating x^3 from minus infinity to plus infinity is zero because each zero width line has zero area, so adding them all up is zero too. Gives the right answer by coincidence, but also works to show incorrect results.
  - roywiggins 2y ago
    That argument proves too much: you similarly will never pick Pi, or any other irrational. And yet, the probability you pick some irrational is 1. In fact, the probability you pick an uncomputable number is 1, too.
    - jerf 2y ago
      I agree with you, except your claim that it proves too much. You will indeed never pick pi or any other specific number you can name. I don't know if you were trying to say my argument leads to those results you consider absurd, but they are in fact the correct results. One way of interpreting this result is to say that randomly selecting a real number isn't physically meaningful and this does not fit into our human brains well. You can do math on the "random number", but you can't ever actually have one in hand (as a sibling comment to yours points out correctly, it is guaranteed to be uncomputable as well), not even mathematically, so you should expect counterintuitive results. Which is generally true of the real numbers when you get close enough to them anyhow. They are popular for a reason but they have rather more spiky bits and strange behaviors than one might expect from a number called "real". I think both of you may be thinking of "random number" as something like what you get out of a RNG or something, but random reals are much stranger than random ints from a finite range.
- Quekid5 2y ago
  EDIT: Honestly, just read the wiki article on Cantor's Diagonal Argument[0]. [0] https://en.wikipedia.org/wiki/Cantor's_diagonal_argument https://en.wikipedia.org/wiki/Cantor's_diagonal_argument
yarg 2y ago
You can provide definitions for irrational numbers, but they'll be expressed in terms of infinite serieses (what the hell is the plural form of series?). So, if ever you try to expand them, you'll need to truncate at some point. There will however, be situations where you can perform operations directly on the serieses, and in those cases (but not the ones requiring expansion and truncation) you'll be able to perform the operations losslessly. Of course, no one would ever bother with that - even NASA truncates pi to 3.141592653589793 (https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/ https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimal...). (And anyway, the distance from any irrational number to the 'nearest' rational is always infinitessimal, so it's generally a moot point unless you're doing calculus.)
- omnicognate 2y ago
  > what the hell is the plural form of series? Series. It's from the Latin 5th declension (res, dies, species, etc.) for which the nominative singular and plural are the same, which carries over into English.
  - yarg 2y ago
    Well, that's annoying if you need precision of meaning in a discussion.
    - adrian_b 2y ago
      In Latin that was seldom a problem like in English, because these nouns were seldom used in the nominative case, i.e. as agents of verbs. They were used much more frequently in other cases, like accusative (singular seriem, plural seriees). In general the fact that English and many other languages borrow the Latin nouns in their nominative form, because of the rather useless tradition that this is their dictionary primary form, can be considered as a serious mistake, which frequently makes awkward the use of such words. The termination "-s" does not have anything to do with the meaning of the word "series". It is just one of the most frequently used markers for the agent of a verb. Even if it is written without delimiting spaces after "serie", it does not belong to this word any more than for example the word "at" in "at home" would be considered to belong to "home". Like "home" can be encountered in various contexts, e.g. "at home", "in the home", "to the home", "for the home" and so on, "serie-" can be found in various contexts, e.g. "serie-es", "serie-m", "serie-ebus", "serie-ii" and so on. What corresponds in Latin to an English word is the Latin word stem, without any of the terminations that may be attached to it. The Latin terminations correspond in English to either explicit prepositions or to implicit role markers provided by the fixed order of the English words. When a Latin word is borrowed into English, it should have been borrowed as its bare stem, not together with its nominative termination. Moreover, for the nouns that denoted inanimate things, the nominative was rarely used and the accusative case was the main form. Even for animate but non-human beings the most important case was still the accusative. This is the reason why in the Romance languages most of their nouns correspond to the accusative forms of the Latin nouns and not with their nominative forms. If the Latin nouns were borrowed as bare stems, there would never be conflicts between their singular nominative marker "-s" and the English plural marker "-s".
llm_trw 2y ago
That's a bit like using a steam hammer to crack a nut. A much simpler way of thinking about it is: What is the chance that if I pick an infinite number of random natural numbers between 0 and 9 that all of them are 0? You're missing the repeating fractions but it gets the flavor of what's actually happening across much better than a Lebesgue integral.
- Xcelerate 2y ago
  Hmm... if I pick an infinite number of random natural numbers between 0 and 9 and all of them are 0, can you prove they aren’t random? https://www.americanscientist.org/sites/americanscientist.org/files/20144141249210337-2014-05TechnologueFp170.jpg https://www.americanscientist.org/sites/americanscientist.or... (I kid, I kid)
nine_k 2y ago
Sets of measure 0 that are not empty or are even infinite are indeed mind-bending when first encountered. But a set of cardinality 1 may be fun, too. Take a real-valued stochastic processbS, and an arbitrary time t. The process generates a value, v = S(t). Since the width of the range from v to v is zero, the probability that S generates this number is also zero. We can replace it with a physical stochastic process. Say, take the voltage in your electric socket. It fluctuates randomly around the nominal value. At every moment there is some voltage in the socket, and the probability to encounter this voltage is zero. It looks like the impossible is actually happening at every moment! This is, of course, thevprobability being a chance to encounter a given value. If we pick a value first, and then start sampling our stochastic process, there's zero probability that we're going to encounter our (infinitely precise) number, no matter how long we'd try. (A physical voltmeter has a finite resolution, say, 0.10 V, so it's going to show you a predetermined voltage pretty often. But it would be a rounded approximation of the unpredictable real value.)

by jfoutz 2y ago

deleted 2y ago

[deleted]

rjurney 2y ago

I got this one without realizing the complex basis for it... there's an infinite number of real numbers between 0 and 1, compared with a limited proportion of rational numbers.

jerf 2y ago
There's an infinite number of both. A way to intuit why you have a zero probability of choosing a rational is to consider randomly picking digits one at a time. Suppose you start with a "3". What is necessary to get one third as your random choice? You must now uniformly randomly select an infinite number of 3s. You can think of that as just a contradiction without much loss here, in that selecting an infinite number of 3s is infinite proof that your process must not be random. (Putting rigor on that might be a challenge but I think it's a fine intuition.) This argument similarly holds for any rational, because no matter how large any given rational's repeat may be, it is always finite, and in order for that rational to be your choice you have still fully determined the result of a putatively uniformly random process for an infinite number of exact picks, and if it is truly random, the probability of that process producing the exact necessary repeat infinitely is always 0.
- kaashif 2y ago
  That argument works for any number. The probability of picking any particular number is zero, there's nothing special about rationals from that perspective. This has no bearing on the measure of the set as a whole.
  - jerf 2y ago
    Which is why I very, very explicitly presented it as a way to gain some intuition and not as a rigorous mathematical argument. That wasn't an accident.
    - kaashif 2y ago
      The probability of picking a single number can't be used to determine the probability of picking a set. You can't build intuition this way. This is almost exactly the same thing as saying integrating x^3 from minus infinity to plus infinity is zero because each zero width line has zero area, so adding them all up is zero too. Gives the right answer by coincidence, but also works to show incorrect results.
- roywiggins 2y ago
  That argument proves too much: you similarly will never pick Pi, or any other irrational. And yet, the probability you pick some irrational is 1. In fact, the probability you pick an uncomputable number is 1, too.
  - jerf 2y ago
    I agree with you, except your claim that it proves too much. You will indeed never pick pi or any other specific number you can name. I don't know if you were trying to say my argument leads to those results you consider absurd, but they are in fact the correct results. One way of interpreting this result is to say that randomly selecting a real number isn't physically meaningful and this does not fit into our human brains well. You can do math on the "random number", but you can't ever actually have one in hand (as a sibling comment to yours points out correctly, it is guaranteed to be uncomputable as well), not even mathematically, so you should expect counterintuitive results. Which is generally true of the real numbers when you get close enough to them anyhow. They are popular for a reason but they have rather more spiky bits and strange behaviors than one might expect from a number called "real". I think both of you may be thinking of "random number" as something like what you get out of a RNG or something, but random reals are much stranger than random ints from a finite range.
Quekid5 2y ago
EDIT: Honestly, just read the wiki article on Cantor's Diagonal Argument[0]. [0] https://en.wikipedia.org/wiki/Cantor's_diagonal_argument https://en.wikipedia.org/wiki/Cantor's_diagonal_argument

yarg 2y ago

You can provide definitions for irrational numbers, but they'll be expressed in terms of infinite serieses (what the hell is the plural form of series?). So, if ever you try to expand them, you'll need to truncate at some point. There will however, be situations where you can perform operations directly on the serieses, and in those cases (but not the ones requiring expansion and truncation) you'll be able to perform the operations losslessly. Of course, no one would ever bother with that - even NASA truncates pi to 3.141592653589793 (https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimals-of-pi-do-we-really-need/ https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimal...). (And anyway, the distance from any irrational number to the 'nearest' rational is always infinitessimal, so it's generally a moot point unless you're doing calculus.)

omnicognate 2y ago
> what the hell is the plural form of series? Series. It's from the Latin 5th declension (res, dies, species, etc.) for which the nominative singular and plural are the same, which carries over into English.
- yarg 2y ago
  Well, that's annoying if you need precision of meaning in a discussion.
  - adrian_b 2y ago
    In Latin that was seldom a problem like in English, because these nouns were seldom used in the nominative case, i.e. as agents of verbs. They were used much more frequently in other cases, like accusative (singular seriem, plural seriees). In general the fact that English and many other languages borrow the Latin nouns in their nominative form, because of the rather useless tradition that this is their dictionary primary form, can be considered as a serious mistake, which frequently makes awkward the use of such words. The termination "-s" does not have anything to do with the meaning of the word "series". It is just one of the most frequently used markers for the agent of a verb. Even if it is written without delimiting spaces after "serie", it does not belong to this word any more than for example the word "at" in "at home" would be considered to belong to "home". Like "home" can be encountered in various contexts, e.g. "at home", "in the home", "to the home", "for the home" and so on, "serie-" can be found in various contexts, e.g. "serie-es", "serie-m", "serie-ebus", "serie-ii" and so on. What corresponds in Latin to an English word is the Latin word stem, without any of the terminations that may be attached to it. The Latin terminations correspond in English to either explicit prepositions or to implicit role markers provided by the fixed order of the English words. When a Latin word is borrowed into English, it should have been borrowed as its bare stem, not together with its nominative termination. Moreover, for the nouns that denoted inanimate things, the nominative was rarely used and the accusative case was the main form. Even for animate but non-human beings the most important case was still the accusative. This is the reason why in the Romance languages most of their nouns correspond to the accusative forms of the Latin nouns and not with their nominative forms. If the Latin nouns were borrowed as bare stems, there would never be conflicts between their singular nominative marker "-s" and the English plural marker "-s".

llm_trw 2y ago

That's a bit like using a steam hammer to crack a nut. A much simpler way of thinking about it is: What is the chance that if I pick an infinite number of random natural numbers between 0 and 9 that all of them are 0? You're missing the repeating fractions but it gets the flavor of what's actually happening across much better than a Lebesgue integral.

Xcelerate 2y ago
Hmm... if I pick an infinite number of random natural numbers between 0 and 9 and all of them are 0, can you prove they aren’t random? https://www.americanscientist.org/sites/americanscientist.org/files/20144141249210337-2014-05TechnologueFp170.jpg https://www.americanscientist.org/sites/americanscientist.or... (I kid, I kid)

nine_k 2y ago

Sets of measure 0 that are not empty or are even infinite are indeed mind-bending when first encountered. But a set of cardinality 1 may be fun, too. Take a real-valued stochastic processbS, and an arbitrary time t. The process generates a value, v = S(t). Since the width of the range from v to v is zero, the probability that S generates this number is also zero. We can replace it with a physical stochastic process. Say, take the voltage in your electric socket. It fluctuates randomly around the nominal value. At every moment there is some voltage in the socket, and the probability to encounter this voltage is zero. It looks like the impossible is actually happening at every moment! This is, of course, thevprobability being a chance to encounter a given value. If we pick a value first, and then start sampling our stochastic process, there's zero probability that we're going to encounter our (infinitely precise) number, no matter how long we'd try. (A physical voltmeter has a finite resolution, say, 0.10 V, so it's going to show you a predetermined voltage pretty often. But it would be a rounded approximation of the unpredictable real value.)