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I just stumbled upon

$ \pi \approx \sqrt{ \frac{9}{5} } + \frac{9}{5} = 3.141640786 $

which is $\delta = 0.0000481330$ different from $\pi$. Although this is a rather crude approximation I wonder if it has been every used in past times (historically). Note that the above might also be related to the golden ratio $\Phi = \frac{\sqrt 5 + 1}{2} $ somehow (the $\sqrt5$ is common in both).

$ \Phi = \frac{5}{6} \left( \sqrt{ \frac{9}{5} } + \frac{9}{5} \right) - 1 $

or

$ \Phi \approx \frac{5}{6} \pi - 1 $

I would like to know if someone (known) has used this, or something similar, in their work. Is it at all familiar to any of you?

Related Question (link).

  • 0
    The _closest_ approximation for $\pi$ I have found is $\pi \approx \frac{9}{5}\sqrt{3}$, which comes from the first term of a series: http://math.stackexchange.com/a/1682189/1347912016-03-05

3 Answers 3

1

This is not a complete answer, but it may be useful.

The largest root of the simple polynomial $x^2-3x+1$

is $\Phi^2=\frac{3+\sqrt{5}}{2}=\left(\frac{1+\sqrt{5}}{2}\right)^2=\Phi+1$

Modifying the coefficients of the polynomial using $5$ and $6$ it becomes

$5^2x^2-5\times6\times 3 x+6^2$

and its largest root is this approximation to $\pi$.

$\pi\approx \frac{6}{5}\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)=\frac{9}{5}+\sqrt{\frac{9}{5}}$

This procedure seems related to the one for another approximation by Ramanujan.

  • 0
    @ja72 Corrected, thank you!2016-04-01
27

I have not seen it before. Note that $\pi = \sqrt{a} + a$ where $a = (1+2\,\pi -\sqrt {1+4\,\pi })/2$, and what you're saying is that a rational approximation of $a$ is $9/5$. In fact, we have a continued fraction $ a = 1 + \dfrac{1}{1 + \dfrac{1}{3+ \dfrac{1}{1+\dfrac{1}{1139 + \ldots}}}}$ and $1+1/(1+1/(3+1/1)) = 9/5$. The fact that the first omitted element, $1139$, is so large makes this a very good approximation: the error in approximating $a$ by $9/5$ is only about $3.5 \times 10^{-5}$. Four elements later comes $7574$, so an even better approximation is $1+1/(1+1/(3+1/(1+1/(1139+1/(1+1/(15+1/1)))))) = 174530/96963$ with error about $1.4 \times 10^{-14}$.

EDIT: Perhaps even more remarkable are $ \eqalign{\pi - \sqrt{1 + \dfrac{47}{35} \pi} &\approx \dfrac{6}{7}\cr \pi - \sqrt{\dfrac{3}{5} + \dfrac{5}{2} \pi } &\approx \dfrac{216}{923}\cr}$

corresponding to the continued fractions

$ \eqalign{\pi - \sqrt{1 + \dfrac{47}{35} \pi} &= \dfrac{1}{1+ \dfrac{1}{6 + \dfrac{1}{126402+ \ldots}}}\cr \pi - \sqrt{\dfrac{3}{5} + \dfrac{5}{2} \pi} &= \dfrac{1}{4+\dfrac{1}{3+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{19+\dfrac{1}{133286+\ldots}}}}}}}\cr}$

21

Ramanujan found this approximation, among many others, according to Wolfram MathWorld equation 21 in linked page.

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    Perfect! Exactly what I was looking for.2012-05-18