Approximation of pi based on sqrt(10)

[Gerson W. Barbosa]

(09-07-2017 09:18 PM)Namir Wrote:  

(09-07-2017 05:46 PM)Dieter Wrote:  I don't know what "RHS" is (assuming that you don't refer to the Retired Husband Syndrome), but I got different results for the relative error:

The √10-based approximation has a relative error of 9,11 E–9
while for 355/113 it's 8,49 E–8.

So the former yields seven correct decimal places while for the latter it's six digits: *)
p = 3,1415926822...
q = 3,1415929203...

In other words, your approximation actually is a bit more accurate than 355/113.


I don't know how you get these results, but the √10-approximation is 9–10x more accurate than 355/113. That's why it yields one more correct digit.

BTW, the continued fraction part is just a complicated way of writing 61/2949. So pi ~ √10 – 61/2949. ;-)

Dieter

--
*) German is known as a language where composite nouns are quite common. For instance, the digits right of the decimal point/comma are simply "Nachkommastellen". Maybe a native speaker can tell me if there is a comparably compact term in English ?-)

I was using the new NumWorks calculator. I am not surprised about the difference in the results you obtained.

Namir

NumWorks appears to round the results to the number of digit in the display. SCI 5 and FIX 5 are the only available formats, but I may not have explored the online simulator enough yet. Anyway, thanks to this feature I was able to find yet another \(\pi\) approximation:

\[\frac{3141593-\frac{\sqrt{3 }}{5}}{1000000}\]

Please don't use NumWorks to see how good it is. Use the WP34S instead:-)

Gerson.

PS:

On NumWorks do

\(\pi\) EXE
Ans - \(\pi\) EXE

=> 3.464102E-7

The first significant digits of \(\sqrt{12}\) are easily recognizable, aren't they?