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Pi = 2

  1. Feb 22, 2005 #1
    This was brought to my attention today, and I haven't had much time to think about it; I think it has something to do with fractals.

    If you have half a circle with diameter of 2, the circumference will be [tex]\pi[/tex].

    If you create two circles, each with diameter of 1, the combined length of the circumference is also [tex]\pi[/tex], and the sum of their diameters will remain at 2.

    If you continue in this fashion, the sum of the circumferences will remain at [tex]\pi[/tex] until the semicircles become points, at which point the sum of the circumferences remains at [tex]\pi[/tex], but the is now the line segment which was the diameter that should actually measure 2 (because the semicircles become points).

    Can anyone explain?
     
  2. jcsd
  3. Feb 22, 2005 #2
    How are you so sure that this is true?
    Diameter of 2, so Pi*d = 2Pi, how's that make Pi 2?
     
  4. Feb 22, 2005 #3
    If you have two circles with a diameter of 1, the sum of their circumferences is not [tex]\pi[/tex] , it is [tex]2\pi[/tex]

    [tex] C = 2\pi{r} [/tex]

    r is 1/2. so the Circumference of each circle is [tex]\pi[/tex]... add both of these together and you get [tex]2\pi[/tex]
     
  5. Feb 22, 2005 #4
    Your original post contains some trivial errors that make it difficult to read.

    This is one of the better limit paradoxes I have seen. It just goes to illustrate that a naive approach to limits is doomed to failure.

    Do you know the definition of a limit? If you do, apply it this problem (hint: as the limit of the number of circles goes to infinity, does the sum of the circumferences approach a limit?)

    You will find that there is a limit to the sum of the circumferences: pi, not 2.
    So in the limit of infinitely many circles, the sum of the diameters is still pi.

    The mistake occurs in the phrase "the semi circles become points", this is not true using the definition of a limit. Good one though, I will remember it to show people the perils of having a slippery concept of limits.
     
  6. Feb 23, 2005 #5
    There is a vaguely similar problem involving limits to infinity and fractals, known as the Koch Snowflake. In this situation the perimeter of a shape goes to infinity, whereas the area converges to some finite value. If you're struggling with limits of this kind then check it out. http://mathworld.wolfram.com/KochSnowflake.html
     
  7. Feb 23, 2005 #6

    matt grime

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    And another one is the step approximation to te diagonal.

    Suppose you want to go from (0,0) to (1,1) in the plane. You can take the diagonal, length sqrt(2). Or you can do it in a series of steps: go 1/n along, then 1/n up, then 1/n along for some integer n and so on. As n tends to infinity the steps approach the diagonal, but the length of the steps is always 2, hence sqrt(2)=2.
     
  8. Feb 23, 2005 #7

    DaveC426913

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    I'm afraid I don't see the paradox.

    A half circle has a diameter of one, and a circumference of [tex]pi[/tex].
    An infinite number of half circles have a summed diameter of one, and a summed circumference of [tex]pi[/tex].
    No paradox so far.

    Do you mean that the infinitely small circumferences become a line segment, thus must have a summed length of 1?

    No. See Matt's response about the diagonal paradox. It's the same thing, and it's conceptually much simpler.
     
  9. Feb 23, 2005 #8

    dextercioby

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    THERE IS A PARADOX.Your whole post contains a paradox (more of an oximoron) repeated twice...Be my guest and identify it... :tongue:

    HINT:it's about language...


    Daniel.
     
  10. Feb 23, 2005 #9
    as the diameters of the semicircles become zero, what does the semicircle become if it's not a point?
     
  11. Feb 23, 2005 #10

    NateTG

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    They stay (semi-)circles. They're just smaller and smaller.
     
  12. Feb 23, 2005 #11
    Interesting. Is there a way to prove it?
     
  13. Feb 23, 2005 #12

    NateTG

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    Sort of.

    If you go to the limit (more on this below) then structure you describe has some odd properties, and it may make as much sense to talk about it as a 'line' as as 'circles' since neither of those is really accurate - it would be more accurate to call the structure as a whole a fractal.

    However, no matter how big an n you choose, if you have n half circles lined up along your line segment - they will still be half circles. That is to say, while you're approaching (rather than at) the limit, the structure will always be a line of half circles. In this case, the jump from the approach to the limit is a quantum jump.

    The reason mathematicians refer to something as a limit as foo approaches bar is because that this limit is fundementally different than evaluating if foo is equal to bar. Going to the limit (as this example illustrates nicely) can have unexpected results. It's also frequently the case that it's unclear whether the limit exists at all.

    Let's say we have a circle, with diameter d. We know that the ratio of the circumference to the diameter is π. Now, if we take your approach to letting the circle go to a point, we get the final result that π should be equal to the circumference of a point (0 or undefined) divided by the diameter of a point (also 0 or undefined) and therefore π is undefined - rather than 3.14...
     
  14. Feb 25, 2005 #13

    both of these statements are wrong.

    first of all when you say half circles i think you mean circles with half the diameter of the previous mentioned one (1)

    an infinite number of circles with a finite nonzero diameter will obviously have a summed circumference of infinity as well as a summed diamter of infinity.

    the second statement is unclear. how many circles are you talking about?

    say you have a circle with diameter of 1 and circumference of pi. next you have 2 circles but the diamber of each is 1/2. then you have 3 circles with a diameter of 1/3 and so on.

    so the sum of the circumferences of n circles is n*(pi*d) = n * (pi * (1/n)) = pi

    there for if you follow this trend as n approaches infinity, the sum of the circumferences will tend to pi

    you can see from this that following this trend, the diameter of the nth circle will be (1/n)

    there fore the sum of the diameters of n circles is n*(d) = n*(1/n) = 1

    there as n approaches infinity the sum of the diameters tends to 1

    this i what i think you were trying to say
     
  15. Feb 28, 2005 #14
    How is this a paradox?

    By moving 1/n up and 1/n along, you are creating very small right triangles, each with a hypotenuse of sqrt(2/n^2), so the resultant (which is what you are actually looking for) would be N * sqrt(2/N^2). If the number of moves were 10, you'd have a total length of 10( sqrt(2/10^2) ) giving 10sqrt(2)/10 = sqrt(2). The same if it were 1000 moves. 1000srt(2)/1000 = sqrt(2).

    Paradox? Where? How does this infer sqrt(2) = 2?

    :smile: Dave
     
  16. Feb 28, 2005 #15

    AKG

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    Here's an incredibly crude sketch of what's supposed to be going on. On the first line, you have two bumps, which are in total, the circumference of a circle of diameter 1cm. The next one shows 4 bumps, totaling the circumferences of two circles, each of diameter 1/2 cm. For the first line, the curve made up of two bumps will be pi cm long, and the horizontal line is obviously 2cm long. In the second line, the curve of 4 bumps is again pi cm long, and the horizontal is again 2cm long. As we make more and more circles (or bumps), their radii get smaller (to fit on the 2cm segment), and we can see that the bumpy curve appears to be approaching the straight line segment. If the lenght of the curve follows the sequence {pi, pi, pi, ...} and the line segment follows the sequence {2, 2, 2, ...} then the limit of the sequence of length of curves is pi, and the limit of the length of segments is 2. But since the bumpy curve appears to approach the line segment in the limit, their lengths "should" approach one another, so pi must be 2. Of course, this is not really a good argument, so it doesn't give us good reason to believe pi = 2, it is based on vague notions of curves appearing to approach other curves.
     

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  17. Feb 10, 2009 #16

    Sure, I can explain. If by "continue in this fashion" you mean "keep dividing the diameter by 2 and doubling the amount of circles", the circles would never "become points" because a point has no diameter and no matter how many times you divide, you will never reach zero.

    That's enough to derail this thought experiment, but let's go on.

    Assume the circles did "become points". A point has no dimensions. No dimensions means no radius, which means no circumference. "Circumference" is a term we use when we're talking about circles. A point is just as much a circle as it is a square as it is a line as it is a Labrador retriever.

    So, there's another nail in the coffin, but let's assume that was okay too.

    Assume that from this logic, we could come to the conclusion that pi = 2.
    But wait, you started the whole thing off by saying that "If you have half a circle with diameter of 2, the circumference will be [tex]\pi[/tex]." You used C=PiD to find the circumference of a circle with diameter 2. You said the circumference came out Pi. Which means that, for the first component of your thought experiment, you assumed Pi = Pi, not 2. You used Pi = Pi as a logical step toward concluding that Pi = 2. That's a big no no.

    So, I hope that clears it all up...
     
  18. Feb 10, 2009 #17
    True, the math is wrong, but the principle isn't affected by that. See? Back up. He also said that ONE circle with diameter 2 would have circumference Pi. That's also wrong. It would have circumference 2Pi, just like the added circumferences of the two R=1/2 circles.

    His point is that, if you keep halving the radius while doubling the number of circles, the total circumference of all the circles will never change.

    The only thing this mathematical correction does is make it come out as
    2Pi = 2 instead of Pi = 2.

    So, you're right, the math is wrong, and it does change the outcome, but it doesn't make it anymore true. The flaw is in his logic, not his calculations.
     
  19. Feb 10, 2009 #18
    Nope. Keep in mind that tending to infinity is not the same thing as being at infinity (because the latter isn't even a "thing"). Therefore the steps "approaching the diagonal" isn't the same thing as the steps being the diagonal. The steps are still steps, albeit very small and very many. So, yes, your total distance traveled is 2, but you haven't traveled on the diagonal at all. Radical 2 is still just radical 2.
     
  20. Feb 10, 2009 #19
    Sure, he said "an infinite number" and "infinitely small".

    Haha, those aren't just oxymorons. They're downright grammatically incorrect!

    Most people will agree that infinity is not a number, but not many people truly grasp what that means. It means there is no such thing as "an infinite number", "infinitely many", "at infinity", or anything of the sort.

    I don't mean to dig on you, Dave. It actually is a really difficult thing to grasp.

    For example, how many points are there on a line segment that is 1 unit long?

    How can you answer this, except by saying "infinitely many"?

    The fact is, there isn't an "infinite number" of points, nor is there a finite number of points.
    There isn't a number of points.

    It can't be expressed as a number of any kind. The points don't exist in "quantity".
    They can only be described as a range.

    In fact, it isn't even correct to ask the question "how many points are there on a line segment that is 1 unit long?"
    Because asking "how many" suggests that I want you to give me a number as your answer.

    I might as well ask you "how many degrees Celsius is purple?"

    It doesn't make sense.
     
  21. Feb 10, 2009 #20

    Office_Shredder

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    So you're not a fan of proof by contradiction?

    Besides, he never stated what pi is... it would be like assuming every circle has a diameter to circumference ratio x, which he doesn't know, and then proving that x=2. There's no logical flaw in this part
     
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