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I just knew that the first two were going to be the Monty Hall problem and 0.99999...==1. I'm actually surprised that so many people don't accept that last one;

by gordaco 14y ago

I just knew that the first two were going to be the Monty Hall problem and 0.99999...==1. I'm actually surprised that so many people don't accept that last one; I understood it immediately when I learned it (I was 12). The third blows your mind until you realize that, simply put, infinity is not a number and common arithmetic doesn't apply. The sixth is not really math, although it perplexes a lot of people. I remember that a few years ago my sister didn't get that, if searching for "termX" wouldn't yield any results, searching for "termX and termY" also wouldn't give anything, no matter how much "termY" would. I'd never seen the 8th, but it's beautiful, and very simple too! Does it really confuse people? Maybe I'm too mathy, but it's obvious to me. The 9th one is a classic, but I feel it's less "controversial" because if you know (or have been told) that the series diverge, chances are that you have a relatively solid base of mathematical knowledge. I encountered the 11th in a list of geometry problems recently; I had to prove it. The first time I saw the problem, just before starting a geometry course, I didn't have a clue. I shitted my pants thinking that I was going to fail the course if my exam was like that! Then, after a few weeks of study, I came back and immediately saw the proof, without thinking much (you just have to use the intercept theorem: each side of the parallelogram is parallel to one original polygon's diagonal). The 7th and the last one don't seem too different than the Monty Hall... our brains are not very well wired for probability estimation and you can see that in many situations. The rest are nice pieces of pop math (if you feel that "pop math" is not much of an oxymoron, anyway).

alok-g 14y ago
>> 0.99999...==1 I am one of the people who don't accept the above as truth. Whether it is truth or not depends on the formal meaning of the ellipsis ... which no one provides to me (I have asked several people about this). Without a formal definition, the statement is meaningless to me. Here is a way to present it formally. 0.9999... is sum of a geometric series a + a * d + a * d^2 ... a * d^n ... with a = 0.9 and d = 0.1. In the limit n -> infinity, the above sum is 1. This is however only true in the limit. It is not clear/obvious to me that ... really means "in the limit".
- lutusp 14y ago
  > Whether it is truth or not depends on the formal meaning of the ellipsis ... which no one provides to me (I have asked several people about this). Mathematical definitions aren't like dictionary definitions. A closed ellipsis: 1...100 Is accepted as meaning a list of all the intermediate values. An open-ended ellipsis is accepted as meaning an infinite repetition of what precedes it. So this ellipsis: 0.9999... Means an infinite decimal sequence of 9's. And .999... really is equal to 1. http://en.wikipedia.org/wiki/0.999.. http://en.wikipedia.org/wiki/0.999... A quote: "In mathematics, the repeating decimal 0.999... (sometimes written with more or fewer 9s before the final ellipsis, or as 0.9, , 0.(9)) denotes a real number that can be shown to be the number one. In other words, the symbols "0.999..." and "1" represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigor, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience." Remember that mathematics, like science, avoids ambiguity where possible. And mathematical notation strives for less ambiguity than mathematics itself.
  - alok-g 14y ago
    Other than pointing to the Wikipedia article (thanks for the same), you have not said anything that I did not. The repetition of the decimals before the ellipsis is the same as the geometric series representation I mentioned. The Wikipedia article also mentions the limit in one of the proofs (though I am yet to read the other proofs there, which may be illuminating! :-)
    - lutusp 14y ago
      Mathematics isn't politics or law. The point is to come to a consensus on the meaning of symbols and conventions. The consensus is that the ellipsis in 0.999... extends out to infinity. In mathematics, there is every incentive to agree on axioms like the meaning of the ellipsis. In science, there is every incentive to agree on the meaning of evidence and how it relates to theory. Only in psychology do they vote on the meaning of their "scientific" findings (all the new conditions that made their way into the new DSM got there by votes, not evidence) -- but psychology isn't a science, it's a pseudoscience. Meanwhile, based on the agreed-upon axioms, 0.999... is equal to 1.

by gordaco 14y ago

alok-g 14y ago

>> 0.99999...==1 I am one of the people who don't accept the above as truth. Whether it is truth or not depends on the formal meaning of the ellipsis ... which no one provides to me (I have asked several people about this). Without a formal definition, the statement is meaningless to me. Here is a way to present it formally. 0.9999... is sum of a geometric series a + a * d + a * d^2 ... a * d^n ... with a = 0.9 and d = 0.1. In the limit n -> infinity, the above sum is 1. This is however only true in the limit. It is not clear/obvious to me that ... really means "in the limit".

lutusp 14y ago
> Whether it is truth or not depends on the formal meaning of the ellipsis ... which no one provides to me (I have asked several people about this). Mathematical definitions aren't like dictionary definitions. A closed ellipsis: 1...100 Is accepted as meaning a list of all the intermediate values. An open-ended ellipsis is accepted as meaning an infinite repetition of what precedes it. So this ellipsis: 0.9999... Means an infinite decimal sequence of 9's. And .999... really is equal to 1. http://en.wikipedia.org/wiki/0.999.. http://en.wikipedia.org/wiki/0.999... A quote: "In mathematics, the repeating decimal 0.999... (sometimes written with more or fewer 9s before the final ellipsis, or as 0.9, , 0.(9)) denotes a real number that can be shown to be the number one. In other words, the symbols "0.999..." and "1" represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigor, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience." Remember that mathematics, like science, avoids ambiguity where possible. And mathematical notation strives for less ambiguity than mathematics itself.
- alok-g 14y ago
  Other than pointing to the Wikipedia article (thanks for the same), you have not said anything that I did not. The repetition of the decimals before the ellipsis is the same as the geometric series representation I mentioned. The Wikipedia article also mentions the limit in one of the proofs (though I am yet to read the other proofs there, which may be illuminating! :-)
  - lutusp 14y ago
    Mathematics isn't politics or law. The point is to come to a consensus on the meaning of symbols and conventions. The consensus is that the ellipsis in 0.999... extends out to infinity. In mathematics, there is every incentive to agree on axioms like the meaning of the ellipsis. In science, there is every incentive to agree on the meaning of evidence and how it relates to theory. Only in psychology do they vote on the meaning of their "scientific" findings (all the new conditions that made their way into the new DSM got there by votes, not evidence) -- but psychology isn't a science, it's a pseudoscience. Meanwhile, based on the agreed-upon axioms, 0.999... is equal to 1.