Amazing Formula

Monday 11 October 2004 at 9:47 pm | In Articles | 3 Comments

There are many interesting formulae in mathematics;

    \displaystyle \sum _{i=1}^\infty \frac{1}{n^2} = \frac {\pi^2}{6}

must be one of the most amazing of all.

The first reaction is where did that \pi come from? You can find 14 different proofs of this in a paper on Robin Chapman’s Home Page [look for Evaluating zeta(2)]

Given this result can you prove another amazing result?
If you pick two positive integers at random, the probability of them having no common divisor is \dfrac{6}{\pi^2}

\pi gets everywhere! See Wikipedia for more such as

    \displaystyle \left(\frac{1}{2}\right)!=\frac{\sqrt{\pi}}{2} or \displaystyle \frac{2}{\pi}=\frac{\sqrt{2}}{2}\frac{\sqrt{2+\sqrt{2}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2} \ldots or \displaystyle \int_{-\infty}^{\infty}e^{-x^2}\;dx=\sqrt{\pi}

3 Comments »

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  1. I don’t know if your probability makes sense…
    How do you pick two positive integers at random?
    Is it possible to pick at random from an infinite set?

    Comment by Ronald — Sunday 14 November 2004 3:48 pm #

  2. Have a look at http://www.physics.harvard.edu/probweek/sol44.pdf where they make the statement more precise:

    To be precise about what we mean by probabilities in this problem, we really should
    word the question as: Let N be a very large integer. Pick two random integers less
    than or equal to N. What is the probability that these numbers are relatively prime,
    in the limit where N goes to infinity?

    Comment by Steve — Sunday 14 November 2004 4:10 pm #

  3. That’s clever… thank you.

    Comment by Ronald — Thursday 23 December 2004 12:18 pm #

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