10 ms·

I think society would be a lot better if BASIC math and statistics would be better understood. How many times do you see a study posted here with N=23 and peop

by iopq 10y ago

I think society would be a lot better if BASIC math and statistics would be better understood. How many times do you see a study posted here with N=23 and people say "the sample size is too small" when it's clearly not? How many people ask for a card deck change to change their luck? How many times do people read a poll like 49% +/- 3% vs. 43% +/- 3% and conclude the two candidates are statistically tied? I could probably keep going with just examples from statistics/probability/combinatorics. But there are other examples of people misunderstanding math. I mean I wonder how many people even understand that 0.999... = 1?

NhanH 10y ago
To be fair, 0.9999... = 1 is not quite basic. You need to know things like infinitesimals, the distinction between value and representation of numbers etc.
- sp527 10y ago
  It's also not strictly, theoretically true. More an 'for all intents and purposes' kind of thing.
  - moron4hire 10y ago
    No, it is true in the strictest sense of the word.
    - sp527 10y ago
      Hardly. We use this reduction because it's essential from a mathematical and axiomatic perspective and because limits are a fundamental construct. But more philosophically, a natural and nonterminating simulation of 0.999... (were it possible) would never strictly equal 1. You would wait an infinite amount of time for it to do so. This comes down to how you view the problem. I would never argue with this through the lens of abstract higher mathematics. I think about it from a computability perspective, which is one way in which you can discount the observation.
      - iopq 10y ago
        It's not a reduction. If you try to find where to put 0.999... on the number line, it has to go exactly where 1 is. For one thing, 1 - 0.999... = 0.000... because you never get to have any remainder since 0.999... is infinite. Or here's another proof: x = 0.999... 10x = 9.999... 10x - x = 9.999... - 0.999... 9x = 9.000... = 9 9x = 9 x = 1
        kazinator 10y ago
        Your "proofs" simply assumes that 0.999... is a notation denoting 1, without examining the underpinnings which might legitimize that. 9.999.. - 0.999 is 9 no matter how we define .999... just as long as two or more occurrences of the 0.999... notation all denote the same entity, and we understand that the syntax 9.999... is 9 + 0.999... For example, if we define 0.999... as "rubber duck" then 9.999... stands for 9 + "rubber duck", and 9.999... - 0.999... stands for 9 + "rubber duck" - "rubber duck" = 9.
        deleted 10y ago
        [deleted]
        iopq 10y ago
        While that's true, the whole key to the proof is that 0.999... * 10 = 9.999... Because "one less 9 than infinity is still infinity" it's what really closes the loop on the proof.
      - moron4hire 10y ago
        You are discarding an essential part of the representation. You are moving the goal post to make yourself right. Just because people don't always understand infinitesimals doesn't make them right. It's the reason they are wrong. There just plain don't know what they are talking about. You can't represent 0.2 exactly using IEEE floats, either, but that doesn't mean the representation 0.2 is not exactly equal to 1/5th.
        anthk 10y ago
  - chestervonwinch 10y ago
    The sum, 0.9 * Σ (0.1)^n from n=0 to n=N-1, is 0.999... as N goes to ∞ (write out the terms). On the other hand, for any finite value of N, the sum [1] is equal to 0.9 * (1-(0.1)^N) / (1-0.1) = 1 - (0.1)^N. This value goes to 1 as N goes to ∞. More rigorously, for any positive value, ε, there is a value N, such that the value of the finite sum is within ε of 1. [1]: https://en.wikipedia.org/wiki/Geometric_series#Formula https://en.wikipedia.org/wiki/Geometric_series#Formula
- minikites 10y ago
  Most people are happy to accept 0.33333... is the same as one third and from there it's a quick hop to 3*0.3333...
  - iopq 10y ago
    Well, that's if they accept that an infinitely repeating number is actually EQUAL to the fraction. Some people just accept that a non-infinite number will never equal the fraction, but don't understand the concept of infinite repetition as an actual number (point on the number scale), not an estimate.
  - mcbits 10y ago
    It's easy to imagine that eating 1 out of 3 slices of pie is "the same" in a very specific sense as eating 333333... out of 1000000... slices of the same pie (although that would be infinity out of infinity slices, which is meaningless). What slips people up is that ignoring everything but the total pie consumed (taking the limit) is embedded in the definition of real numbers. There's an analogous story with rationals: Suppose x1 = 1, y1 = 3, x2 = 2, and y2 = 6. If we plot them, (x1, y1) and (x2, y2) are clearly different points, but x1/y1 "equals" x2/y2 because they lie on the same line through the origin. We decide that we don't need to know about those individual points.
  - umanwizard 10y ago
    If people accept 0.33... = 1/3 but not 0.99... = 1, they are being logically inconsistent. A "proof" that just plays a trick on people's logically inconsistent assumptions to derive a result isn't very satisfying. You're not really uncovering anything fundamental through that proof, just playing games. Edit: what I mean to say is there are reasonable things "..." could mean such that .99... and .33... are both not equal to 1 and 1/3, respectively. But there are none for which one pair is equal and the other not, as you rightly point out. So all your proof does is show that the other person's viewpoint is inconsistent, but it doesn't give any evidence for one of the two consistent viewpoints over the other.
  - dantillberg 10y ago
    I didn't quite accept that until I challenged a professor in a 300-level university math class. This came up in class, and I spoke up and was like, "no, that can't be true, because 0.3333 repeated N times will never equal 1/3." Up to that point, I had only a fuzzy notion in my mind of what 0.33333... even was or how it was defined. But the professor helped clear this up for me: that "infinitely-repeating decimals" was actually just shorthand for a limit definition, i.e. "0.333..." is defined to be the limit as N approaches infinity of (3/10^1)+(3/10^2)+...+(3/10^N).
- rntz 10y ago
  You don't need to understand anything about infinitesimals to understand 0.999... = 1. Perhaps you meant limits? The "standard" approach would be to point out that Σ_{i=1}^∞ 9/(10^i) = 1 (that is, the sum from i = 1 to infinity of 9/(10^i) is 1), and understanding an infinite summation requires the concept of a limit. (Of course, there are simpler proofs that use only basic algebra and intuition about decimals; a limit is just the most direct approach.) You're spot-on about needing to understand there's a distinction between a number and its decimal representation, though.
  - soVeryTired 10y ago
    Limits use a construction that's pretty similar to an infinitesimal. The epsilon-delta definition of a limit is no joke for students.
    - alejohausner 10y ago
      Oddly enough, I never understood the epsilon-delta description of limits until I read David Foster Wallace's book on infinity. All through my degree in math I was taught about things without learning the historical context that created those things.
      - geoka9 10y ago
        Getting a degree in math without understanding epsilon-delta is quite an achievement! I am serious.
      - SomeStupidPoint 10y ago
        I'm always super happy that I stumbled in to taking topology before real analysis. It meant that I understood the topological idea of limits before I had to do proofs using just the epsilon (for sequence) or epsilon-delta (for functions) definition, and so could translate the logic of showing things about neighborhoods in to the terminology of (real analysis) limits. Limits, in the abstract, are a fairly simple concept: in the case of sequences, for any neighborhood of the limit, the entire tail of the sequence (past some point) is contained in the neighborhood; in the case of functions, for any neighborhood of the limit at f(x), there's a neighborhood around x, such that every point in that neighborhood maps to the neighborhood around the limit.
- yequalsx 10y ago
  Property of real numbers: between distinct real numbers is at least one other number. Now try to find a decimal representation of a number bigger than 0.9999999... but less than 1.0. You clearly can't. They must be equal. No need for infinitesimals.
  - NhanH 10y ago
    Doesn't that property come from the fact that there is no infinitesimal in real number?
  - kazinator 10y ago
    > find a decimal representation of a number 0.99999999...1 This denotes the abstract idea that we take 0.999... (infinite number of 9's) and add another digit. This is no more or less abstract than 0.999... to begin with. 0.999... is a unicorn, and 0.999...9 is a unicorn with a pink ribbon on the tip of its horn.
    - lisper 10y ago
      > add another digit But you can't. All of the places where you might want to add a digit are already occupied by 9's.
      - kazinator 10y ago
        So you're saying a countable infinity can be occupied, so that there is no room for one more. You must then disbelieve concepts such as that the even integers can be put in 1:1 correspondence with all integers; i.e. that there are exactly as many even integers as there are integers.
        lisper 10y ago
        > So you're saying a countable infinity can be occupied, so that there is no room for one more. No. I'm saying that a countable infinity doesn't have room for one more at the end. There is no end. You can obviously stick a 1 in the middle, but then you have a number strictly less than 0.999...
- S4M 10y ago
  I think I came to the conclusion that 0.9999999... = 1 when I was 12 or 13. My reasoning was that for any n, we had 0 <= 1 - 0.999999... <= 10^(-n), so the difference had to be 0.
- JoeAltmaier 10y ago
  Its just algebra. As noted elsewhere in this thread, 10X = 9.999... so 10X - X = 9.999... - 0.999... = 9 = 9X. So X=1.
  - nilkn 10y ago
    This proof is more subtle than it appears. Here's a bogus rewrite, for instance: Let 10X = 9.9. Then 10X - X = 9.9 - 0.9 = 9 = 9X. Hence X = 1, but X is actually 0.99 in this case (not 0.9). You need 0.99 = 0.9 for this to work with the exact same structure as your version. Your proof only works because appending a 9 to an infinite expansion of 9s does not actually add a 9. But at this point you're forced to establish meaning for an infinite expansion of 9s, at which point this is really not just algebra anymore.
    - banach2 10y ago
      What? Why -0.9?
      - nilkn 10y ago
        Because that's exactly what the argument above does. It simply subtracts away the decimal part of 9.999.... This is always wrong except in the case of infinitely many repeated digits, and the proof does not explain this. More rigorously, let 9.999{n} denote an expansion with n 9s after the decimal point, where n can also be infinity. The subtlety with the argument is that it needs X to be the same as everything after the decimal point (so that the result of the subtraction is just 9). This is never true for finite values of n, and the proof does not establish that it's true for an infinite value of n -- indeed, it can't do so without supplying a meaning in the first place. Another way of phrasing it is that it assumes that if X = 0.999..., then 10X = 9.999..., where there are the "same number" of 9s after the decimal point in 10X as there are in X. This seems intuitive for an infinite repeating sequence of 9s, because "one less than infinity" is still infinity, but it's not very rigorous, and the argument as written certainly doesn't explain this.
        10y ago
rrtwo 10y ago
What is the generalised rule/case where small sample sizes are sufficient?
- edtechdev 10y ago
  Depends on what effect size or power you are looking for and/or other factors like variance in the data (standard deviation), etc. There are some different ways to calculate sample sizes here: https://en.wikipedia.org/wiki/Sample_size_determination https://en.wikipedia.org/wiki/Sample_size_determination or google: sample size calculator
- iopq 10y ago
  If the difference between samples is VERY large, you don't need a very large sample size. In other words, we're trying to find the chance that the result we got was due to chance. Let's say you have numbers like these: A: 11, 11, 12, 12, 13, 13, 13, 13, 13, 13, 14, 15 B: 90, 92, 93, 94, 94, 95, 95, 96, 97, 99, 99, 101, 101 What is the chance that those two samples come from the same distribution? On the other hand, if A averaged something like 12.5 and B averaged something like 12.4 it would require a huge sample size to prove that those two samples come from two different distributions.
  - dredmorbius 10y ago
    That sounds intuitively reasonable. Is there a cononical reference argument that you're aware of?
    - iopq 10y ago
      I'm really talking about https://en.wikipedia.org/wiki/P-value https://en.wikipedia.org/wiki/P-value Most people just look at the number of subjects in a study, but not the p-values
      - dredmorbius 10y ago
        Thanks.
MichaelBurge 10y ago
> I mean I wonder how many people even understand that 0.999... = 1? To be honest, I think it's unreasonable to expect anybody - even with a Ph.D in a field other than mathematics - to be able to even define the real numbers: My definition is probably very different from yours(I tend to say there's countably many real numbers).
- iopq 10y ago
  https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
  - MichaelBurge 10y ago
    https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_...
    - iopq 10y ago
      I didn't understand anything on that page.
      - MrManatee 10y ago
        This is not the easiest thing to explain briefly, but let's give it a shot anyway. There are several ways of defining real numbers, and one of them is the axiomatic definition. Real numbers are defined by a list of axioms they must satisfy. These would include, among others: (1) If x and y are reals, then x + y = y + x. (2) If S is a nonempty subset of reals with an upper bound, then S has a least upper bound. There is a crucial difference between these two. In (1) the variables x and y only quantify over reals, but in (2) the variable S quantifies over subsets of reals. We say that (1) is a first-order axiom and (2) is a second-order axiom. Actually, among all of the axioms of real numbers, (2) is the only one that is second-order. Therefore, it is natural to ask: can we rid of it? No, we cannot. Löwenheim-Skolem theorem says that if we only have first-order axioms, then it is impossible to distinguish between countable and uncountable sets - even if we have an infinite number of first-order axioms. In particular, this means that if we try to define real numbers using only first-order axioms, then the definition cannot even capture the basic fact that there is an uncountable number of reals. From here on, there are two roads you could take. If you're like me, then you just accept that real numbers cannot be defined using first-order axioms. By my standards, any definition that only uses first-order axioms cannot be a satisfactory definition of the real numbers. But some people don't want to accept definitions that are not based on first-order axioms. And this is not as crazy as it might sound. First-order axioms are very nice from a theoretic point of view. For example, with first-order axioms it is absolutely clear what it means to prove something based on those axioms. With second-order axioms, the situation is a lot hairier.
analog31 10y ago
Speaking of BASIC, I took a high school course in BASIC, in 1981, and it changed my life. For one thing, it changed my approach to math. I think that computation should be part of elementary math, not to produce the next generation of career programmers, but because computation is actually how a lot of math is done. And it might change the curriculum -- having students think about more complex problems that they have the tools to solve, rather than learning algorithms by rote. I speculate that people might have a better grasp of statistics if they could just play around with artificial distributions generated by a computer -- even just in a spreadsheet.
keithgabryelski 10y ago
I just wish they would teach the common curtesy of putting (page#/totalPages) on each slide of their power point presentation so I can decide whether I can make it through the deck or cut my wrists now.
kazinator 10y ago
0.999... is equal to 1 only if we assign a particular semantics to the "..." notation. Namely if "..." means "the limit of the decimal number to the left, as the repetitions of the last digit grow ever larger", then 0.999... is an alternative notation for 1 since that limit is 1. The actual number formed by repeating 9's an infinite number of times is not constructable. Whereas 1 is constructable. So they cannot be the same thing. That's because, philosophically, two objects must be identical in every property to be the same object, and constructability is a property. As we add 9's, we are getting ever closer to 1, and the concept of a limit lets us "fast forward" to that value. If we agree that this "..." denotes the limit, rather than the non-constructable number implied by the notation's "face value", then the equality holds. To actually regard 0.999... = 1 to hold without involving the limit shows an ignorance of (or denial of) the validity of induction. Because, look: Base case: 0.9 is not equal to 1. Inductive hypothesis: Adding another digit to a decimal fraction which is not equal to 1 produces a new decimal fraction which is also not equal to 1. Therefore, by induction, no matter how many 9's we add, we do not get 1. Induction is not somehow canceled by infinity; induction is how we understand that a property holds for infinity: a property such as "not equal to 1".
- soVeryTired 10y ago
  They're coming out of the woodwork... Edit: sorry, didn't mean to ad hominem, but I'm going to leave my original comment there all the same. To make my post a bit more constructive, OP, what exactly is your definition of "constructible"? Because it doesn't relate to any mathematical concept I'm familiar with. Other than maybe "finitary".
  - kazinator 10y ago
    https://en.wikipedia.org/wiki/Constructivism_(mathematics) https://en.wikipedia.org/wiki/Constructivism_(mathematics) Do you have remarks not regarding the semantics of "constructable"?
    - soVeryTired 10y ago
      But the "example from real analysis" section in your link uses exactly the same limiting technique to construct e. That section shows that 0.999... is constructible.

by iopq 10y ago

NhanH 10y ago

To be fair, 0.9999... = 1 is not quite basic. You need to know things like infinitesimals, the distinction between value and representation of numbers etc.

sp527 10y ago
It's also not strictly, theoretically true. More an 'for all intents and purposes' kind of thing.
- moron4hire 10y ago
  No, it is true in the strictest sense of the word.
  - sp527 10y ago
    Hardly. We use this reduction because it's essential from a mathematical and axiomatic perspective and because limits are a fundamental construct. But more philosophically, a natural and nonterminating simulation of 0.999... (were it possible) would never strictly equal 1. You would wait an infinite amount of time for it to do so. This comes down to how you view the problem. I would never argue with this through the lens of abstract higher mathematics. I think about it from a computability perspective, which is one way in which you can discount the observation.
    - iopq 10y ago
      It's not a reduction. If you try to find where to put 0.999... on the number line, it has to go exactly where 1 is. For one thing, 1 - 0.999... = 0.000... because you never get to have any remainder since 0.999... is infinite. Or here's another proof: x = 0.999... 10x = 9.999... 10x - x = 9.999... - 0.999... 9x = 9.000... = 9 9x = 9 x = 1
      - kazinator 10y ago
        Your "proofs" simply assumes that 0.999... is a notation denoting 1, without examining the underpinnings which might legitimize that. 9.999.. - 0.999 is 9 no matter how we define .999... just as long as two or more occurrences of the 0.999... notation all denote the same entity, and we understand that the syntax 9.999... is 9 + 0.999... For example, if we define 0.999... as "rubber duck" then 9.999... stands for 9 + "rubber duck", and 9.999... - 0.999... stands for 9 + "rubber duck" - "rubber duck" = 9.
        deleted 10y ago
        [deleted]
        iopq 10y ago
        While that's true, the whole key to the proof is that 0.999... * 10 = 9.999... Because "one less 9 than infinity is still infinity" it's what really closes the loop on the proof.
    - moron4hire 10y ago
      You are discarding an essential part of the representation. You are moving the goal post to make yourself right. Just because people don't always understand infinitesimals doesn't make them right. It's the reason they are wrong. There just plain don't know what they are talking about. You can't represent 0.2 exactly using IEEE floats, either, but that doesn't mean the representation 0.2 is not exactly equal to 1/5th.
      - anthk 10y ago
- chestervonwinch 10y ago
  The sum, 0.9 * Σ (0.1)^n from n=0 to n=N-1, is 0.999... as N goes to ∞ (write out the terms). On the other hand, for any finite value of N, the sum [1] is equal to 0.9 * (1-(0.1)^N) / (1-0.1) = 1 - (0.1)^N. This value goes to 1 as N goes to ∞. More rigorously, for any positive value, ε, there is a value N, such that the value of the finite sum is within ε of 1. [1]: https://en.wikipedia.org/wiki/Geometric_series#Formula https://en.wikipedia.org/wiki/Geometric_series#Formula
minikites 10y ago
Most people are happy to accept 0.33333... is the same as one third and from there it's a quick hop to 3*0.3333...
- iopq 10y ago
  Well, that's if they accept that an infinitely repeating number is actually EQUAL to the fraction. Some people just accept that a non-infinite number will never equal the fraction, but don't understand the concept of infinite repetition as an actual number (point on the number scale), not an estimate.
- mcbits 10y ago
  It's easy to imagine that eating 1 out of 3 slices of pie is "the same" in a very specific sense as eating 333333... out of 1000000... slices of the same pie (although that would be infinity out of infinity slices, which is meaningless). What slips people up is that ignoring everything but the total pie consumed (taking the limit) is embedded in the definition of real numbers. There's an analogous story with rationals: Suppose x1 = 1, y1 = 3, x2 = 2, and y2 = 6. If we plot them, (x1, y1) and (x2, y2) are clearly different points, but x1/y1 "equals" x2/y2 because they lie on the same line through the origin. We decide that we don't need to know about those individual points.
- umanwizard 10y ago
  If people accept 0.33... = 1/3 but not 0.99... = 1, they are being logically inconsistent. A "proof" that just plays a trick on people's logically inconsistent assumptions to derive a result isn't very satisfying. You're not really uncovering anything fundamental through that proof, just playing games. Edit: what I mean to say is there are reasonable things "..." could mean such that .99... and .33... are both not equal to 1 and 1/3, respectively. But there are none for which one pair is equal and the other not, as you rightly point out. So all your proof does is show that the other person's viewpoint is inconsistent, but it doesn't give any evidence for one of the two consistent viewpoints over the other.
- dantillberg 10y ago
  I didn't quite accept that until I challenged a professor in a 300-level university math class. This came up in class, and I spoke up and was like, "no, that can't be true, because 0.3333 repeated N times will never equal 1/3." Up to that point, I had only a fuzzy notion in my mind of what 0.33333... even was or how it was defined. But the professor helped clear this up for me: that "infinitely-repeating decimals" was actually just shorthand for a limit definition, i.e. "0.333..." is defined to be the limit as N approaches infinity of (3/10^1)+(3/10^2)+...+(3/10^N).
rntz 10y ago
You don't need to understand anything about infinitesimals to understand 0.999... = 1. Perhaps you meant limits? The "standard" approach would be to point out that Σ_{i=1}^∞ 9/(10^i) = 1 (that is, the sum from i = 1 to infinity of 9/(10^i) is 1), and understanding an infinite summation requires the concept of a limit. (Of course, there are simpler proofs that use only basic algebra and intuition about decimals; a limit is just the most direct approach.) You're spot-on about needing to understand there's a distinction between a number and its decimal representation, though.
- soVeryTired 10y ago
  Limits use a construction that's pretty similar to an infinitesimal. The epsilon-delta definition of a limit is no joke for students.
  - alejohausner 10y ago
    Oddly enough, I never understood the epsilon-delta description of limits until I read David Foster Wallace's book on infinity. All through my degree in math I was taught about things without learning the historical context that created those things.
    - geoka9 10y ago
      Getting a degree in math without understanding epsilon-delta is quite an achievement! I am serious.
    - SomeStupidPoint 10y ago
      I'm always super happy that I stumbled in to taking topology before real analysis. It meant that I understood the topological idea of limits before I had to do proofs using just the epsilon (for sequence) or epsilon-delta (for functions) definition, and so could translate the logic of showing things about neighborhoods in to the terminology of (real analysis) limits. Limits, in the abstract, are a fairly simple concept: in the case of sequences, for any neighborhood of the limit, the entire tail of the sequence (past some point) is contained in the neighborhood; in the case of functions, for any neighborhood of the limit at f(x), there's a neighborhood around x, such that every point in that neighborhood maps to the neighborhood around the limit.
yequalsx 10y ago
Property of real numbers: between distinct real numbers is at least one other number. Now try to find a decimal representation of a number bigger than 0.9999999... but less than 1.0. You clearly can't. They must be equal. No need for infinitesimals.
- NhanH 10y ago
  Doesn't that property come from the fact that there is no infinitesimal in real number?
- kazinator 10y ago
  > find a decimal representation of a number 0.99999999...1 This denotes the abstract idea that we take 0.999... (infinite number of 9's) and add another digit. This is no more or less abstract than 0.999... to begin with. 0.999... is a unicorn, and 0.999...9 is a unicorn with a pink ribbon on the tip of its horn.
  - lisper 10y ago
    > add another digit But you can't. All of the places where you might want to add a digit are already occupied by 9's.
    - kazinator 10y ago
      So you're saying a countable infinity can be occupied, so that there is no room for one more. You must then disbelieve concepts such as that the even integers can be put in 1:1 correspondence with all integers; i.e. that there are exactly as many even integers as there are integers.
      - lisper 10y ago
        > So you're saying a countable infinity can be occupied, so that there is no room for one more. No. I'm saying that a countable infinity doesn't have room for one more at the end. There is no end. You can obviously stick a 1 in the middle, but then you have a number strictly less than 0.999...
S4M 10y ago
I think I came to the conclusion that 0.9999999... = 1 when I was 12 or 13. My reasoning was that for any n, we had 0 <= 1 - 0.999999... <= 10^(-n), so the difference had to be 0.
JoeAltmaier 10y ago
Its just algebra. As noted elsewhere in this thread, 10X = 9.999... so 10X - X = 9.999... - 0.999... = 9 = 9X. So X=1.
- nilkn 10y ago
  This proof is more subtle than it appears. Here's a bogus rewrite, for instance: Let 10X = 9.9. Then 10X - X = 9.9 - 0.9 = 9 = 9X. Hence X = 1, but X is actually 0.99 in this case (not 0.9). You need 0.99 = 0.9 for this to work with the exact same structure as your version. Your proof only works because appending a 9 to an infinite expansion of 9s does not actually add a 9. But at this point you're forced to establish meaning for an infinite expansion of 9s, at which point this is really not just algebra anymore.
  - banach2 10y ago
    What? Why -0.9?
    - nilkn 10y ago
      Because that's exactly what the argument above does. It simply subtracts away the decimal part of 9.999.... This is always wrong except in the case of infinitely many repeated digits, and the proof does not explain this. More rigorously, let 9.999{n} denote an expansion with n 9s after the decimal point, where n can also be infinity. The subtlety with the argument is that it needs X to be the same as everything after the decimal point (so that the result of the subtraction is just 9). This is never true for finite values of n, and the proof does not establish that it's true for an infinite value of n -- indeed, it can't do so without supplying a meaning in the first place. Another way of phrasing it is that it assumes that if X = 0.999..., then 10X = 9.999..., where there are the "same number" of 9s after the decimal point in 10X as there are in X. This seems intuitive for an infinite repeating sequence of 9s, because "one less than infinity" is still infinity, but it's not very rigorous, and the argument as written certainly doesn't explain this.
      - 10y ago

rrtwo 10y ago

What is the generalised rule/case where small sample sizes are sufficient?

edtechdev 10y ago
Depends on what effect size or power you are looking for and/or other factors like variance in the data (standard deviation), etc. There are some different ways to calculate sample sizes here: https://en.wikipedia.org/wiki/Sample_size_determination https://en.wikipedia.org/wiki/Sample_size_determination or google: sample size calculator
iopq 10y ago
If the difference between samples is VERY large, you don't need a very large sample size. In other words, we're trying to find the chance that the result we got was due to chance. Let's say you have numbers like these: A: 11, 11, 12, 12, 13, 13, 13, 13, 13, 13, 14, 15 B: 90, 92, 93, 94, 94, 95, 95, 96, 97, 99, 99, 101, 101 What is the chance that those two samples come from the same distribution? On the other hand, if A averaged something like 12.5 and B averaged something like 12.4 it would require a huge sample size to prove that those two samples come from two different distributions.
- dredmorbius 10y ago
  That sounds intuitively reasonable. Is there a cononical reference argument that you're aware of?
  - iopq 10y ago
    I'm really talking about https://en.wikipedia.org/wiki/P-value https://en.wikipedia.org/wiki/P-value Most people just look at the number of subjects in a study, but not the p-values
    - dredmorbius 10y ago
      Thanks.

MichaelBurge 10y ago

> I mean I wonder how many people even understand that 0.999... = 1? To be honest, I think it's unreasonable to expect anybody - even with a Ph.D in a field other than mathematics - to be able to even define the real numbers: My definition is probably very different from yours(I tend to say there's countably many real numbers).

iopq 10y ago
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
- MichaelBurge 10y ago
  https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_...
  - iopq 10y ago
    I didn't understand anything on that page.
    - MrManatee 10y ago
      This is not the easiest thing to explain briefly, but let's give it a shot anyway. There are several ways of defining real numbers, and one of them is the axiomatic definition. Real numbers are defined by a list of axioms they must satisfy. These would include, among others: (1) If x and y are reals, then x + y = y + x. (2) If S is a nonempty subset of reals with an upper bound, then S has a least upper bound. There is a crucial difference between these two. In (1) the variables x and y only quantify over reals, but in (2) the variable S quantifies over subsets of reals. We say that (1) is a first-order axiom and (2) is a second-order axiom. Actually, among all of the axioms of real numbers, (2) is the only one that is second-order. Therefore, it is natural to ask: can we rid of it? No, we cannot. Löwenheim-Skolem theorem says that if we only have first-order axioms, then it is impossible to distinguish between countable and uncountable sets - even if we have an infinite number of first-order axioms. In particular, this means that if we try to define real numbers using only first-order axioms, then the definition cannot even capture the basic fact that there is an uncountable number of reals. From here on, there are two roads you could take. If you're like me, then you just accept that real numbers cannot be defined using first-order axioms. By my standards, any definition that only uses first-order axioms cannot be a satisfactory definition of the real numbers. But some people don't want to accept definitions that are not based on first-order axioms. And this is not as crazy as it might sound. First-order axioms are very nice from a theoretic point of view. For example, with first-order axioms it is absolutely clear what it means to prove something based on those axioms. With second-order axioms, the situation is a lot hairier.

analog31 10y ago

Speaking of BASIC, I took a high school course in BASIC, in 1981, and it changed my life. For one thing, it changed my approach to math. I think that computation should be part of elementary math, not to produce the next generation of career programmers, but because computation is actually how a lot of math is done. And it might change the curriculum -- having students think about more complex problems that they have the tools to solve, rather than learning algorithms by rote. I speculate that people might have a better grasp of statistics if they could just play around with artificial distributions generated by a computer -- even just in a spreadsheet.

keithgabryelski 10y ago

I just wish they would teach the common curtesy of putting (page#/totalPages) on each slide of their power point presentation so I can decide whether I can make it through the deck or cut my wrists now.

kazinator 10y ago

0.999... is equal to 1 only if we assign a particular semantics to the "..." notation. Namely if "..." means "the limit of the decimal number to the left, as the repetitions of the last digit grow ever larger", then 0.999... is an alternative notation for 1 since that limit is 1. The actual number formed by repeating 9's an infinite number of times is not constructable. Whereas 1 is constructable. So they cannot be the same thing. That's because, philosophically, two objects must be identical in every property to be the same object, and constructability is a property. As we add 9's, we are getting ever closer to 1, and the concept of a limit lets us "fast forward" to that value. If we agree that this "..." denotes the limit, rather than the non-constructable number implied by the notation's "face value", then the equality holds. To actually regard 0.999... = 1 to hold without involving the limit shows an ignorance of (or denial of) the validity of induction. Because, look: Base case: 0.9 is not equal to 1. Inductive hypothesis: Adding another digit to a decimal fraction which is not equal to 1 produces a new decimal fraction which is also not equal to 1. Therefore, by induction, no matter how many 9's we add, we do not get 1. Induction is not somehow canceled by infinity; induction is how we understand that a property holds for infinity: a property such as "not equal to 1".

soVeryTired 10y ago
They're coming out of the woodwork... Edit: sorry, didn't mean to ad hominem, but I'm going to leave my original comment there all the same. To make my post a bit more constructive, OP, what exactly is your definition of "constructible"? Because it doesn't relate to any mathematical concept I'm familiar with. Other than maybe "finitary".
- kazinator 10y ago
  https://en.wikipedia.org/wiki/Constructivism_(mathematics) https://en.wikipedia.org/wiki/Constructivism_(mathematics) Do you have remarks not regarding the semantics of "constructable"?
  - soVeryTired 10y ago
    But the "example from real analysis" section in your link uses exactly the same limiting technique to construct e. That section shows that 0.999... is constructible.