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    Re: Interesting mathematical statements

    Quote Originally Posted by braintic View Post
    There is a lot of highly theoretical mathematics here.
    Does anyone have more real-world mathematical statements?
    (The birthday problem was a good one)
    The Kakeya Needle Problem: What is the smallest area of a parking lot in which you can have a needle of length 1 turn around 180 degrees and return to its starting position, pointing in the other direction?

    Answer: The area can be made arbitrarily small through a series of divisions and transformations of the shape required for the needle to turn around. Hence, no smallest area exists.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: Interesting mathematical statements

    Quote Originally Posted by Paradoxica View Post
    The Kakeya Needle Problem: What is the smallest area of a parking lot in which you can have a needle of length 1 turn around 180 degrees and return to its starting position, pointing in the other direction?

    Answer: The area can be made arbitrarily small through a series of divisions and transformations of the shape required for the needle to turn around. Hence, no smallest area exists.
    This isn't very "practical" though, since the car / needle needs to be made arbitrarily thin if you want the area to be arbitrarily small. For anyone interested, there is a Numberphile video on the Kakeya Needle Problem: www.youtube.com/watch?v=j-dce6QmVAQ

    Last edited by InteGrand; 31 Dec 2015 at 9:10 PM. Reason: Embedded video
    kawaiipotato likes this.

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    Re: Interesting mathematical statements

    Quote Originally Posted by InteGrand View Post
    This isn't very "practical" though, since the car / needle needs to be made arbitrarily thin if you want the area to be arbitrarily small. For anyone interested, there is a Numberphile video on the Kakeya Needle Problem: www.youtube.com/watch?v=j-dce6QmVAQ
    Well the problem never stated any width. It is a mathematical solution, after all.





    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: Interesting mathematical statements

    Another good one is that there are always two opposite points on Earth with exactly the same temperature. This is proved quickly and accessibly in this video by Dr James Grime:

    https://youtube.com/watch?v=5Px6fajpSio

    Last edited by InteGrand; 1 Jan 2016 at 1:15 AM.

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    Re: Interesting mathematical statements

    Quote Originally Posted by Paradoxica View Post
    Well the problem never stated any width. It is a mathematical solution, after all.





    Haha yeah, it is an interesting result, I just meant that it's not really real-world as braintic wanted.

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    Re: Interesting mathematical statements

    Quote Originally Posted by Paradoxica View Post
    Well the problem never stated any width. It is a mathematical solution, after all.





    It can't be that EACH of the individual faculties has a bias towards women, just that most of them do. At least one faculty must have male bias if the overall bias is male.

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    Re: Interesting mathematical statements

    This is the nature of weighted averages.

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    Re: Interesting mathematical statements

    Quote Originally Posted by InteGrand View Post
    Another good one is that there are always two opposite points on Earh with exactly the same temperature. This is proved quickly and accessibly in this video by Dr James Grime:

    https://youtube.com/watch?v=5Px6fajpSio

    I love intermediate value theorem consequences like this. You can get quite a lot out of such a seemingly common sense result.

    Another one is the one that colloquially states that you can always rotate a square four legged table on wobbly ground into a position in which it balances.

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    Re: Interesting mathematical statements

    Quote Originally Posted by glittergal96 View Post
    I love intermediate value theorem consequences like this. You can get quite a lot out of such a seemingly common sense result.

    Another one is the one that colloquially states that you can always rotate a square four legged table on wobbly ground into a position in which it balances.
    I'd just rather have the triangular legged table in the first place. Stable on almost any readily available terrain.

    Last edited by Paradoxica; 29 Dec 2015 at 11:35 PM.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: Interesting mathematical statements

    Quote Originally Posted by glittergal96 View Post
    I love intermediate value theorem consequences like this. You can get quite a lot out of such a seemingly common sense result.

    Another one is the one that colloquially states that you can always rotate a square four legged table on wobbly ground into a position in which it balances.
    Haha yeah that table one is really nice. Here's a math / physics paper about it: http://arxiv.org/pdf/math-ph/0510065.pdf

    Here's the Numbephile video about it:


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    Re: Interesting mathematical statements

    I don't have the mathematical vocabulary or ability to precisely describe it but I find Benford's Law fascinating and very strange

    https://en.m.wikipedia.org/wiki/Benford%27s_law


    ~ First in Drama 2015 ~

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    Re: Interesting mathematical statements

    Quote Originally Posted by glittergal96 View Post
    It can't be that EACH of the individual faculties has a bias towards women, just that most of them do. At least one faculty must have male bias if the overall bias is male.
    Imagine that
    (a) In 2013/14, Steve Smith has a batting average of 30 in 5 innings, while David Warner has a batting average of 31 in 20 innings
    (b) In 2014/15, Steve Smith has a batting average of 40 in 50 innings, while David Warner has a batting average of 41 in 15 innings

    In both years, Warner's average was higher than Smith's.
    Yet when you work out the combined average for the two seasons, Smith's average beats Warner's 39.1 to 35.3


    So it doesn't have to be simply a majority. It can be all.
    Last edited by braintic; 30 Dec 2015 at 12:05 AM.

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    Re: Interesting mathematical statements

    I love the "wobbly circle" and "dragoncurve" videos on Numberphile.

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    Re: Interesting mathematical statements

    Quote Originally Posted by InteGrand View Post
    Haha yeah, it is an interesting result, I just meant that it's not really real-world as braintic wanted.
    No, that is good enough for me. Statistical paradoxes are definitely real-world.

    I thought someone might give a chaos theory example. That seems to be a topic that is never given the attention it deserves.

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    Re: Interesting mathematical statements

    Quote Originally Posted by braintic View Post
    No, that is good enough for me. Statistical paradoxes are definitely real-world.

    I thought someone might give a chaos theory example. That seems to be a topic that is never given the attention it deserves.
    I was referring to the Kakeya Needle one in those posts.

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    Re: Interesting mathematical statements

    Quote Originally Posted by InteGrand View Post
    I was referring to the Kakeya Needle one in those posts.
    Oops - sorry. Nevertheless, I wonder if someone can come up with an example of the "regression to the mean" fallacy.

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    Re: Interesting mathematical statements

    Quote Originally Posted by braintic View Post
    Imagine that
    (a) In 2013/14, Steve Smith has a batting average of 30 in 5 innings, while David Warner has a batting average of 31 in 20 innings
    (b) In 2014/15, Steve Smith has a batting average of 40 in 50 innings, while David Warner has a batting average of 41 in 15 innings

    In both years, Warner's average was higher than Smith's.
    Yet when you work out the combined average for the two seasons, Smith's average beats Warner's 39.1 to 35.3


    So it doesn't have to be simply a majority. It can be all.
    I think glittergal96 might have thought that the bias thing with men and women was referring to the percentage out of all those accepted who are male. The study actually considered the acceptance rates of women and men applicants separately (like your cricket example basically).

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    Re: Interesting mathematical statements

    By the way, my example with averages partially shows why, when your maths teacher is calculating your yearly mark based on say 4 tests, the individual test marks should NOT be artificially scaled to the same mean before adding.

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    Re: Interesting mathematical statements

    Quote Originally Posted by braintic View Post
    By the way, my example with averages partially shows why, when your maths teacher is calculating your yearly mark based on say 4 tests, the individual test marks should NOT be artificially scaled to the same mean before adding.
    What about when English teachers calculate your yearly mark? :P

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    Re: Interesting mathematical statements

    Quote Originally Posted by braintic View Post
    Imagine that
    (a) In 2013/14, Steve Smith has a batting average of 30 in 5 innings, while David Warner has a batting average of 31 in 20 innings
    (b) In 2014/15, Steve Smith has a batting average of 40 in 50 innings, while David Warner has a batting average of 41 in 15 innings

    In both years, Warner's average was higher than Smith's.
    Yet when you work out the combined average for the two seasons, Smith's average beats Warner's 39.1 to 35.3


    So it doesn't have to be simply a majority. It can be all.
    Actually, let me explain why people have trouble seeing how this works.

    People try to compare 30 to 31 and 40 to 41.
    Instead, they should be comparing diagonally: 30 to 41 in Warner's favour, and 40 to 31 in Smith's favour.
    The differences are 11 and 9 - roughly the same.
    But the weightings of the two pairs are 20 to 70.
    So the slightly lower difference which is in Smith's favour has a much higher weighting.

    So the key is the large discrepancy between the number of innings each year, and the turnaround in those numbers between the two years.

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    Re: Interesting mathematical statements

    Quote Originally Posted by InteGrand View Post
    What about when English teachers calculate your yearly mark? :P
    They call in the maths staff

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    Re: Interesting mathematical statements

    Quote Originally Posted by braintic View Post
    Oops - sorry. Nevertheless, I wonder if someone can come up with an example of the "regression to the mean" fallacy.
    braintic is having trouble with his mobile data connection. He tries walking around the room until the signal strength improves enough for him to be able to view the latest BOS forum posts on his phone. Incidentally, braintic subconsciously picks up on background details, and his subconscious observation is that he is near a pineapple which is there for no apparent reason. A short while later, Integrand enters the room and removes the pineapple for some nefarious purpose. Later braintic is in a pyrotechnics lab, which happens to be the room where Integrand placed the pineapple. It turns out Integrand is making a pineapple bomb. braintic tries to view the latest posts, but is unable to load the page, and walks around the room to get a better signal. The page loads as soon as braintic walks by the pineapple. braintic realises that the pineapple was near his phone when he was able to load the previous time. braintic concludes the pineapple caused his signal to improve, and kills Integrand to prevent him from destroying the pineapple.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: Interesting mathematical statements

    Quote Originally Posted by braintic View Post
    No, that is good enough for me. Statistical paradoxes are definitely real-world.

    I thought someone might give a chaos theory example. That seems to be a topic that is never given the attention it deserves.
    Two celestial bodies orbiting each other, assuming there exists no other sources of gravity in their entire universe, are comprised by a perfectly deterministic system where the exact location of each body can be predicted with 100% accuracy infinitely far into the future.

    Introduce a third body into the simulation, and it is analytically impossible to determine with any amount of accuracy where any of the bodies will be located infinitely far into the future. This has to do with the resulting calculations giving rise to an integral that cannot be expressed in terms of elementary functions, and can only be resolved numerically.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: Interesting mathematical statements

    Quote Originally Posted by braintic View Post
    Imagine that
    (a) In 2013/14, Steve Smith has a batting average of 30 in 5 innings, while David Warner has a batting average of 31 in 20 innings
    (b) In 2014/15, Steve Smith has a batting average of 40 in 50 innings, while David Warner has a batting average of 41 in 15 innings

    In both years, Warner's average was higher than Smith's.
    Yet when you work out the combined average for the two seasons, Smith's average beats Warner's 39.1 to 35.3


    So it doesn't have to be simply a majority. It can be all.
    Yeah, sorry about that. Misinterpreted the original post and what kind of bias was entailed. (overall students vs acceptance rates)

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    Re: Interesting mathematical statements

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