Approximation of pi based on sqrt(10)

[Dieter]

(09-07-2017 02:44 PM)Namir Wrote:  let the RHS be p, I got

(p - pi)/pi * 1E9 = 110.26

Using the approximation of q=355/113 I get

(p - pi)/pi * 1E9 = 84.91

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.

(09-07-2017 02:44 PM)Namir Wrote:  So my approximation, based on sqrt(10), is a bit less accurate than 355/113 but by not a whopping difference (ratio of errors is about 1.3)!

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 ?-)