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> There is no proof that will ever satisfy a person dead-set against this. Indeed. I've torn my hair out trying to convince smart people with PhDs in hard scie

by gspr 7y ago

> There is no proof that will ever satisfy a person dead-set against this. Indeed. I've torn my hair out trying to convince smart people with PhDs in hard sciences and had to give up in frustration. I usually find that the most success can be had by kicking the ball to them immediately and having them define what they actually mean when they say "0.999…". If we're going to debate whether that thing equals another thing, we better make sure we know what we're talking about. Inevitably, this either causes the dead-set person to give up, or give a myriad of definitions that are either meaningless, ill-defined, or causes them to realize that they don't actually know what "0.999…" means (or what they want it to mean). It is hard to have the patience to chase down the consequences of their ill-fated definitions, though.

2OEH8eoCRo0 7y ago
Ask for a number between .9 repeated and 1
- chris_wot 7y ago
  So does this mean that an infinitely small number is zero? As in 1/∞ ?
  - thaumasiotes 7y ago
    You can read about this in any work on nonstandard analysis. ("Nonstandard" is just the name, much like "imaginary" numbers.) An infinitely small number is zero when projected onto the real number line. If you introduce an infinitesimal quantity to the reals, then for every number there is a unique real number to which that first number is infinitely close (that is, the difference between them is infinitesimal). You can use that real number as a (good) approximation of all the nonstandard numbers in its halo. (As long as you're comparing it to other real numbers.)
  - selectodude 7y ago
    In calculus, yes.
  - GolDDranks 7y ago
    In real numbers, there doesn't exist such a thing as "infinitely small number" that is apart from zero. Yes, there exists infinitely many numbers between any minisculely small number and zero, but the way they are defined, every single number you can grasp, is finitely small. The "infinitely" small gap is inaccessible. In some other number systems it isn't, but in the standard reals it is. That means that the "infinitely small" doesn't exist; "smallest apart from zero" doesn't exist either.
  - brlewis 7y ago
    There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero.
    - gspr 7y ago
      > There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero. Zero. Just zero. The difference is zero. 0. Because 0.999… = 1.
      - brlewis 7y ago
        You are stating that 0.999... = 1 proves that 1 - 0.999... equals zero. I am stating that 1 - 0.999... = 0.000... proves that 0.999... = 1. I think people intuitively see that infinitely zero equals zero.
        gspr 7y ago
        > 1 - 0.999... = 0.000... proves that 0.999... = 1. If people accept the former, and that the RHS of the former is in fact 0, they've already also accepted that 0.999…=1. I don't see what the discussion is at that point
        brlewis 7y ago
        If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious. I don't have a direct computation for making the latter obvious, just indirect ones like 1 - 0.999... and 3 x 0.333...
        gspr 7y ago
        > If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious. How can they compute 1-0.999… when they clearly have no idea what 0.999… is?
  - gspr 7y ago
    > So does this mean that an infinitely small number is zero? What does "infinitely small" mean? > As in 1/∞ ? What notion of division are we talking about here? The division most people expect is that of real numbers. ∞ is not a real number, so you'll have to specify what you mean.
- traderjane 7y ago
  It's the smallest number bigger than 0.
  - naringas 7y ago
    strictly bigger than 0
    - gspr 7y ago
      Doesn't exist.
  - TuringTest 7y ago
    bigger or equal
    - gspr 7y ago
      0.
  - thinkharderdev 7y ago
    That doesn't exist. An open interval doesn't have a smallest number.
    - da_chicken 7y ago
      Of course it does. It's called the infinitesimal. It's common definition for real number is 1 / infinity: https://en.wikipedia.org/wiki/Infinitesimal https://en.wikipedia.org/wiki/Infinitesimal If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist. It's not a value you can meaningfully write out, but you can't write out pi, e, phi, root 2, 1 / 3 in base 10, root -1, etc. "I can't write it down" isn't a particularly unique property for numbers.
      - pwdisswordfish2 7y ago
        There is no smallest positive infinitesimal either. At least in theories that manage to define those rigorously. And it’s mostly a formal trick anyway; standard epsilon-delta calculus avoids them entirely. Had you actually meaningfully studied this subject, or did you just link to a Wikipedia article you half-heartedly skimmed one day?
- brutt 7y ago
  0.00...1
  - brlewis 7y ago
    No, there's no 1. 2OEH8eoCRo0 is exactly right. Subtract 0.999... from 1 and you get 0.000...
    - brutt 7y ago
      0.999... + 0.000...1 = 1 0.000...1 = 1/∞ 0.999... = 1 - 1/∞
      - brlewis 7y ago
        You're repeating the same wrong thing you said earlier. It's 0.999... and not 0.999...0 In the same way, it's 0.000... and not 0.000...1.
        brutt 7y ago
        0.999... = 0.999...9 0.999...9 + 0.000...1 = 1 0.999...0 + 0.000..1 = 0.999..1 0.000...1 = 1/∞ 0.999...9 = 1 - 1/∞ 0.999...0 = 1 - 1/∞ - 9/∞ = 1 - 10/∞ If x/∞ = 0, then 0.999...x = 1. If x/∞ ≠ 0, then 0.999...x ≠ 1.
        brlewis 7y ago
        Ok, now you're saying that infinite decimals have final digits.
        brutt 7y ago
        If Universe is infinite, then if we compare you to size of Universe, you are infinitely small, so you don't exists at all. Why I should waste my time? If Universe is finite, then finite number of elements can make only finite number of combinations, thus this discussion is repeated infinite number of times again. Why I should waste my time again?
- throwaway287391 7y ago
  "There isn't one, 1 is the very next number right after 0.999... Checkmate atheists." (In all seriousness I don't think it's a very convincing argument for someone who doesn't buy the proofs -- it requires you to believe and have internalized the idea that there are an infinite number of reals between any two distinct reals, and therefore that any pair of reals with nothing between are the same number. Those seem like bigger logical leaps to me than the simple proofs for someone who hasn't thought about this stuff.)
- burnte 7y ago
  Ask for a letter between G and H.
  - gspr 7y ago
    You're missing the point. This would be an analogy fit for talking with someone who's looking for an integer between 1 and 2.
- burmer 7y ago
  How about, ask for an integer between 1 and 2. Can't think of one? Guess they're the same number then.
  - brlewis 7y ago
    Apples and oranges. For any two different real numbers, there's a number between them. Integers work differently.
    - andrewprock 7y ago
      This is a bold assertion, and one that is not obviously true, especially in cases like 0.999... and 1.0
      - Pudini 7y ago
        Obviously saying it is not obviously true is false if 0.999... == 1.0
        perl4ever 7y ago
        Can something be true and not obvious?
        Pudini 7y ago
        Obviously not. Which is the point in using the word obvious, obviously. Namely, using it to feel superior or to not provide a better argument.
        perl4ever 7y ago
        When I use the word, the point is to call out the fact that I think it's obvious, so if others don't, they can explain why. Not to forestall any discussion. Anyone who uses the word differently is doing it wrong.
      - 7y ago
- naikrovek 7y ago
  I don't understand how they're the same number. I will never accept that they are the same. The difference between 0.9 repeating infinitely and 1 is infinitely small, but it isn't zero.
  - gowld 7y ago
    What is an "infinitely small" number? Is 9999..... the same as infinity? What is 1.0 - 0.99999.... = ? What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer?
    - jdashg 7y ago
      Those are great questions that not every system is required to address in the same way. (In a similar vein, +0 != -0 in Java) This is breakdown in notation and/or convention. There is no ground truth, just what's true within the system.
    - naikrovek 7y ago
      > What is an "infinitely small" number? What is an infinitely large number? > What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer? By that logic, 0.99 repeating isn't a number at all, and therefore can't be equivalent to 1, because you can't subtract it from 1. So my understanding that they are different is correct.
      - gspr 7y ago
        > > What is an "infinitely small" number? > What is an infinitely large number? Neither is a well-defined concept within the standard reals, and completely unnecessary for understanding that 0.999…=1. > > What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer? > By that logic, 0.99 repeating isn't a number at all, and therefore can't be equivalent to 1, because you can't subtract it from 1. So my understanding that they are different is correct. 0.99… is a real number. The sequence (a_n)_{n positive integer} with a_n = 9/10^1 + 9/10^2 + … + 9/10^n has a limit (do you want me to prove that?). 0.99… is defined as that limit. That limit is 1. Therefore 0.99… = 1. I think you're struggling to grasp the definition here. The defintion of 0.ddd…, where d is an integer between 0 and 9, is the limit of the above sequence with 9 replaced by d. That limit always exists, and the definition is therefore OK. In the case of d=9, the limit is 1.
crispyambulance 7y ago
> It is hard to have the patience to chase down the consequences of their ill-fated definitions, though. Of course it's hard because in day to day life, even for the vast majority of STEM practitioners, the nuance of the proof that 0.9999... is 1 is not of much utility. Whenever one sees a 0.999[... to however many digits] one can safely assume it's less than one or perhaps more realistically "almost 1". To say 0.999... with the very specific detail that the 9's go on forever is actually a strange thing to say and outside of most people's experience. There are simple enough proofs of this that normal folks who paid attention in high school can follow, but I think it has to be framed more as a clever brain-teaser than as a proof.
- gspr 7y ago
  > Of course it's hard because in day to day life, even for the vast majority of STEM practitioners, the nuance of the proof that 0.9999... is 1 is not of much utility. Oh absolutely. I'm not expecting STE(no M this time!) practitioners to necessarily be aware of why 0.999…=1 in their daily lives, but I do expect them to have encountered enough situations in their field of expertise where scraping the surface using shallow intuition and gut feeling lead them wildly astray. I'm therefore surprised that they're willing to deny this basic fact to the face of mathematicians. The ones I've interacted with also don't happen to be the types that'll start arguing Anatomy 101 facts with a heart surgeon at a bar, but somehow arguing over basic calculus with mathematicians is fine.
  - gowld 7y ago
    > start arguing Anatomy 101 facts with a heart surgeon at a bar Well, there's this: wikipedia.org/wiki/Vaccine_hesitancy wikipedia.org/wiki/Homeopathy

by gspr 7y ago

2OEH8eoCRo0 7y ago

Ask for a number between .9 repeated and 1

chris_wot 7y ago
So does this mean that an infinitely small number is zero? As in 1/∞ ?
- thaumasiotes 7y ago
  You can read about this in any work on nonstandard analysis. ("Nonstandard" is just the name, much like "imaginary" numbers.) An infinitely small number is zero when projected onto the real number line. If you introduce an infinitesimal quantity to the reals, then for every number there is a unique real number to which that first number is infinitely close (that is, the difference between them is infinitesimal). You can use that real number as a (good) approximation of all the nonstandard numbers in its halo. (As long as you're comparing it to other real numbers.)
- selectodude 7y ago
  In calculus, yes.
- GolDDranks 7y ago
  In real numbers, there doesn't exist such a thing as "infinitely small number" that is apart from zero. Yes, there exists infinitely many numbers between any minisculely small number and zero, but the way they are defined, every single number you can grasp, is finitely small. The "infinitely" small gap is inaccessible. In some other number systems it isn't, but in the standard reals it is. That means that the "infinitely small" doesn't exist; "smallest apart from zero" doesn't exist either.
- brlewis 7y ago
  There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero.
  - gspr 7y ago
    > There is no infinitely small number between 0.999... and 1. The difference is 0.000... Not infinitely small, but infinitely zero. Zero. Just zero. The difference is zero. 0. Because 0.999… = 1.
    - brlewis 7y ago
      You are stating that 0.999... = 1 proves that 1 - 0.999... equals zero. I am stating that 1 - 0.999... = 0.000... proves that 0.999... = 1. I think people intuitively see that infinitely zero equals zero.
      - gspr 7y ago
        > 1 - 0.999... = 0.000... proves that 0.999... = 1. If people accept the former, and that the RHS of the former is in fact 0, they've already also accepted that 0.999…=1. I don't see what the discussion is at that point
        brlewis 7y ago
        If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious. I don't have a direct computation for making the latter obvious, just indirect ones like 1 - 0.999... and 3 x 0.333...
        gspr 7y ago
        > If people don't accept the former, they can take out a pencil and paper to compute it themselves. After a few digits it will become obvious. How can they compute 1-0.999… when they clearly have no idea what 0.999… is?
- gspr 7y ago
  > So does this mean that an infinitely small number is zero? What does "infinitely small" mean? > As in 1/∞ ? What notion of division are we talking about here? The division most people expect is that of real numbers. ∞ is not a real number, so you'll have to specify what you mean.
traderjane 7y ago
It's the smallest number bigger than 0.
- naringas 7y ago
  strictly bigger than 0
  - gspr 7y ago
    Doesn't exist.
- TuringTest 7y ago
  bigger or equal
  - gspr 7y ago
    0.
- thinkharderdev 7y ago
  That doesn't exist. An open interval doesn't have a smallest number.
  - da_chicken 7y ago
    Of course it does. It's called the infinitesimal. It's common definition for real number is 1 / infinity: https://en.wikipedia.org/wiki/Infinitesimal https://en.wikipedia.org/wiki/Infinitesimal If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist. It's not a value you can meaningfully write out, but you can't write out pi, e, phi, root 2, 1 / 3 in base 10, root -1, etc. "I can't write it down" isn't a particularly unique property for numbers.
    - pwdisswordfish2 7y ago
      There is no smallest positive infinitesimal either. At least in theories that manage to define those rigorously. And it’s mostly a formal trick anyway; standard epsilon-delta calculus avoids them entirely. Had you actually meaningfully studied this subject, or did you just link to a Wikipedia article you half-heartedly skimmed one day?
brutt 7y ago
0.00...1
- brlewis 7y ago
  No, there's no 1. 2OEH8eoCRo0 is exactly right. Subtract 0.999... from 1 and you get 0.000...
  - brutt 7y ago
    0.999... + 0.000...1 = 1 0.000...1 = 1/∞ 0.999... = 1 - 1/∞
    - brlewis 7y ago
      You're repeating the same wrong thing you said earlier. It's 0.999... and not 0.999...0 In the same way, it's 0.000... and not 0.000...1.
      - brutt 7y ago
        0.999... = 0.999...9 0.999...9 + 0.000...1 = 1 0.999...0 + 0.000..1 = 0.999..1 0.000...1 = 1/∞ 0.999...9 = 1 - 1/∞ 0.999...0 = 1 - 1/∞ - 9/∞ = 1 - 10/∞ If x/∞ = 0, then 0.999...x = 1. If x/∞ ≠ 0, then 0.999...x ≠ 1.
        brlewis 7y ago
        Ok, now you're saying that infinite decimals have final digits.
        brutt 7y ago
        If Universe is infinite, then if we compare you to size of Universe, you are infinitely small, so you don't exists at all. Why I should waste my time? If Universe is finite, then finite number of elements can make only finite number of combinations, thus this discussion is repeated infinite number of times again. Why I should waste my time again?
throwaway287391 7y ago
"There isn't one, 1 is the very next number right after 0.999... Checkmate atheists." (In all seriousness I don't think it's a very convincing argument for someone who doesn't buy the proofs -- it requires you to believe and have internalized the idea that there are an infinite number of reals between any two distinct reals, and therefore that any pair of reals with nothing between are the same number. Those seem like bigger logical leaps to me than the simple proofs for someone who hasn't thought about this stuff.)
burnte 7y ago
Ask for a letter between G and H.
- gspr 7y ago
  You're missing the point. This would be an analogy fit for talking with someone who's looking for an integer between 1 and 2.
burmer 7y ago
How about, ask for an integer between 1 and 2. Can't think of one? Guess they're the same number then.
- brlewis 7y ago
  Apples and oranges. For any two different real numbers, there's a number between them. Integers work differently.
  - andrewprock 7y ago
    This is a bold assertion, and one that is not obviously true, especially in cases like 0.999... and 1.0
    - Pudini 7y ago
      Obviously saying it is not obviously true is false if 0.999... == 1.0
      - perl4ever 7y ago
        Can something be true and not obvious?
        Pudini 7y ago
        Obviously not. Which is the point in using the word obvious, obviously. Namely, using it to feel superior or to not provide a better argument.
        perl4ever 7y ago
        When I use the word, the point is to call out the fact that I think it's obvious, so if others don't, they can explain why. Not to forestall any discussion. Anyone who uses the word differently is doing it wrong.
    - 7y ago
naikrovek 7y ago
I don't understand how they're the same number. I will never accept that they are the same. The difference between 0.9 repeating infinitely and 1 is infinitely small, but it isn't zero.
- gowld 7y ago
  What is an "infinitely small" number? Is 9999..... the same as infinity? What is 1.0 - 0.99999.... = ? What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer?
  - jdashg 7y ago
    Those are great questions that not every system is required to address in the same way. (In a similar vein, +0 != -0 in Java) This is breakdown in notation and/or convention. There is no ground truth, just what's true within the system.
  - naikrovek 7y ago
    > What is an "infinitely small" number? What is an infinitely large number? > What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer? By that logic, 0.99 repeating isn't a number at all, and therefore can't be equivalent to 1, because you can't subtract it from 1. So my understanding that they are different is correct.
    - gspr 7y ago
      > > What is an "infinitely small" number? > What is an infinitely large number? Neither is a well-defined concept within the standard reals, and completely unnecessary for understanding that 0.999…=1. > > What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer? > By that logic, 0.99 repeating isn't a number at all, and therefore can't be equivalent to 1, because you can't subtract it from 1. So my understanding that they are different is correct. 0.99… is a real number. The sequence (a_n)_{n positive integer} with a_n = 9/10^1 + 9/10^2 + … + 9/10^n has a limit (do you want me to prove that?). 0.99… is defined as that limit. That limit is 1. Therefore 0.99… = 1. I think you're struggling to grasp the definition here. The defintion of 0.ddd…, where d is an integer between 0 and 9, is the limit of the above sequence with 9 replaced by d. That limit always exists, and the definition is therefore OK. In the case of d=9, the limit is 1.

crispyambulance 7y ago

> It is hard to have the patience to chase down the consequences of their ill-fated definitions, though. Of course it's hard because in day to day life, even for the vast majority of STEM practitioners, the nuance of the proof that 0.9999... is 1 is not of much utility. Whenever one sees a 0.999[... to however many digits] one can safely assume it's less than one or perhaps more realistically "almost 1". To say 0.999... with the very specific detail that the 9's go on forever is actually a strange thing to say and outside of most people's experience. There are simple enough proofs of this that normal folks who paid attention in high school can follow, but I think it has to be framed more as a clever brain-teaser than as a proof.

gspr 7y ago
> Of course it's hard because in day to day life, even for the vast majority of STEM practitioners, the nuance of the proof that 0.9999... is 1 is not of much utility. Oh absolutely. I'm not expecting STE(no M this time!) practitioners to necessarily be aware of why 0.999…=1 in their daily lives, but I do expect them to have encountered enough situations in their field of expertise where scraping the surface using shallow intuition and gut feeling lead them wildly astray. I'm therefore surprised that they're willing to deny this basic fact to the face of mathematicians. The ones I've interacted with also don't happen to be the types that'll start arguing Anatomy 101 facts with a heart surgeon at a bar, but somehow arguing over basic calculus with mathematicians is fine.
- gowld 7y ago
  > start arguing Anatomy 101 facts with a heart surgeon at a bar Well, there's this: wikipedia.org/wiki/Vaccine_hesitancy wikipedia.org/wiki/Homeopathy