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Math Help - inverting the Pairing function

  1. #1
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    inverting the Pairing function

    Pairing function - Wikipedia, the free encyclopedia



    (1) i cant really understand why he is using the triangle number anyway

    (2) why is t <= z ?
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  2. #2
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    Hello dayscott

    Welcome to Math Help Forum!
    Quote Originally Posted by dayscott View Post
    Pairing function - Wikipedia, the free encyclopedia



    (1) i cant really understand why he is using the triangle number anyway

    (2) why is t <= z ?
    Why is this true (1): z = t+y ? Because
    t = \frac{w(w+1)}{2}
    = \frac{(x+y)(x+y+1)}{2}, since w = x+y
    \Rightarrow z =\frac{(x+y)(x+y+1)}{2}+y = t+y
    Why is this true (2): t \le z ? Because
    t=z-y, and y is a natural number.
    As far as your question about triangle numbers is concerned, it is simply that the expression \tfrac12w(w+1) happens to be the sum of the first w natural numbers; which is the w^{th} triangle number.

    Grandad
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  3. #3
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    thx a lot !

    another question - the red arrow marks the gap in my thinking ^^:
    Last edited by dayscott; November 16th 2009 at 04:20 AM.
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  4. #4
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    no one ? : /
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  5. #5
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    Hello dayscott

    I didn't post a reply earlier, because I can't see it either!

    The LHS of the inequality is straightforward enough:
    w=\frac{\sqrt{8t+1}-1}{2}, which is strictly increasing,

    and t\le z

    \Rightarrow w\le\frac{\sqrt{8z+1}-1}{2}
    but I can't see where the right-hand part \frac{\sqrt{8z+1}-1}{2}<w+1 comes from.

    Perhaps someone else may be able to help?

    Grandad
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  6. #6
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    i posted in the wikipedia diskussion of the pairing function article - maybe the author will answer there : )
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  7. #7
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    we got an answer.. Talk:Pairing function - Wikipedia, the free encyclopedia

    ...which i still don't get - i can't even phrase a good question right now.
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