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Phew!, Finally I am blogging again. This weblog isnt about SAP, related technologies etc etc. I read this mathematical statement and underlying logic somehere few days back. And it took me by surprise. I have been studying Mathematics for past 12-13 years now but never thought this way. But anyways, I wanted to share this with fellow SDNers.

Ok, Consider two concentric circles. Here is a diagram to assist you:

Circles

 

For a moment, forget about the extra lines drawn. Now my question is,

Which cirlce has greater number of points? I mean, does circle with radius r0 has more points or does the circle of with radius ri has more points? By “number of points” I mean the points on the circumference.

Most of you will say: “Duh! Isnt the answer obvious?”. Bigger circle will have more number of points. But what we never realise is:

“Consider radius(line) r0. It passes through both circles. Similarly if we extend rm, it passes through both circles. So, it means for every point on outer circle, there is a corresponding point on inner circle and its one to one mapping. This implies that number of points of both circles are equal. And this defies our common logic”.

Any thoughts?

P.S: Thanks to Boris for two wonderful sessions on Community Day, Bangalore. I cherished those.

P.P.S: I have been reading a lot of “such” matematical paradoxes/puzzles/uncommon logic these days. If anyone is interested please drop me a mail. I would be more than delighted to exchange views of such topics OR I may even blog them.(But then I do not want to overload SDN with matematics).

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8 Comments

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  1. Anonymous
    Puru,
    Cool blog!
    But, what happens to the concept of Circumference now? Doesnot a bigger circumference(=2*pi*r which the outer circle has) mean more number of points on it?
    Cheers!
    Bidwan
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  2. Steve Rumsby
    So no matter how long or short your line segment, you can fit infinitely many points. In that sense, every line segment has the same number of points.

    This is similar to the question of how many “numbers” there are of different types. First think about natural numbers (positive whole numbers) and integers (positive & negative whole numbers). You may or may not be surprised that there are the same number of each.

    Now think about fractions (rational numbers, in math-speak). It is somewhat more surprising that there are the same number of those as there are integers or natural numbers.

    These sets of numbers all have an infinite number of members, obviously (i.e. you can keep counting and never reach the end). They are called “countably infinite”, since you can count the members.

    Now think about the “real numbers” – numbers with decimal fractions. The set of real numbers is “uncountably infinite”, meaning there are more real numbers than integers! Strange, but true…

    An exercise for the reader. I said above that there are an infinite number of points in a line segment. Is that a countably infinite, or uncountably infinite number?

    We now return you to the less confusing world of SAP!!

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    1. Achim Hauck
      I’d guess “countable infinite”, because you can approauch the amount by fractions again:

      half the line (and count two times)
      half the first half again (and count 4 times)
      etc…

      am I right?

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      1. Steve Rumsby
        No, I’m afraid not!

        Think of it this way. The reason there are more real numbers than rationals (fractions) is that there are gaps between the fractions. No matter how often you split the gap between two fractions to create another fraction, you can’t make them meet. This is why there are real numbers that you can’t represent as fractions – irrational numbers. The most famous example being pi. The irrational numbers fit in the gaps between the fractions, if you like.

        If the number of points on a line was countable, there’s be gaps in the line.

        Another strange, related fact. The number of irrational numbers, that is the real numbers that fit “in the gaps” between the fractions, is uncountable infinite. That is, if you take the uncountably infinite reals and remove the countably infinite fractions you are still left with uncountably many irrationals.

        Infinity is a strange thing…

        Steve.

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        1. Puru Govind Post author
          “Infinity is a strange thing”
          Just to demonstrate:
          A = 1 + 1/2 + 1/2 + 1/2 to infinty
          B = 1+ 1/2 + 1/3 + 1/4 to infinity

          Which is bigger?

          There are two ways to see this problem:
          Way 1:
          1 and 1/2 is common in both the series… After that each term in B in less than corresponding term in A. => A is bigger.

          Way 2:
          1 is common to both series. We can also regroup the terms in B. So 1/3 +1/4 > 1/2. Similarly we can group other terms..

          So is A>B or B>A? 🙂

          Btw I have reading all these in Book “A certain Ambiguity” by Gauravi Suri. Get a copy of this book if this problem itches you 😀

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  3. Anoop Dosi
    Dude,
    Good thougtht… but as I said, in practical world, we need to define weight with each point. And than what will happen there could be case when same dot in inner circumference could correspond with 2 or more dots in outer circimference…
    For example when this two circle will go on some printer for printing, printer will have some weight for each pixel. like 1000 points/cm… something of that sort…. so when printer with print these circle, it will use different no of points…

    Anoop

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  4. Pratap Sone
    Good one Puru.

    But considering the concept of number of points on a line ( finite or infinite ) there will be infinite number of points on a line. And if we imagine the circumference to be a straight line, then there are infinite number of points. and you can never compare two infinite number.

    Even if you try to draw the lines to bisect a curve of the circumference you will go bisecting the parts and never end up finishing your job, Infinity!!!

    Though your thought looks logically good, but I doubt we can prove this point of “equal number of points”.

    Whats your opinion.

    Best Regards,
    Pratap Sone

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