# Approximation for $\pi$

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?

• Another interesting discussion (related) here math.stackexchange.com/questions/108510/… Commented May 18, 2012 at 21:01
• @Artin this first link does not apply because it is an identity formula and not an approximation, and the second link is already included in the original posting. Commented May 18, 2012 at 21:03
• What are you expecting, validation of this approximation and some sort of variation of this approximation? Commented May 18, 2012 at 21:16
• I look for someone to say, I have seen this and it was used by x, or this is related to y approximation. Commented May 18, 2012 at 21:21
• 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: math.stackexchange.com/a/1682189/134791 Commented Mar 5, 2016 at 15:21

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}

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

• Perfect! Exactly what I was looking for. Commented May 18, 2012 at 21:29

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.

• Where did the negative sign of $-3x$ go? Commented Apr 1, 2016 at 15:32
• @ja72 Corrected, thank you! Commented Apr 1, 2016 at 15:53