Decimals to improper fractions program?

[Dieter]

(06-09-2018 12:55 AM)Valentin Albillo Wrote:  Rational people can "agree to disagree" in fields which are amenable to subjective opinions but Mathematics is not one of them, Mathematics is fact-based and evidence-based, so there's little or no place for subjectivism. No one can rationally "agree to disagree" whether 11111 is prime or not, you just produce its factors or lack of prime factorization and that settles the question once and for all.

Great. So let's look at the facts.

355/113 = 3,141592920353982...
while pi = 3,141592653589793...

How many digits agree? Seven. A simple matter of fact. The difference is ≈2,7 E–7, so the 7th decimal or 8th significant digit differs. That's a mathematical fact, nothing that could be subject to opinions, exactly as you said. So I think we can agree on this.

Now the subjective part comes in: you say that the 8th digit is only 2 units off. 2 out of 10. And this allows to describe the accuracy as "almost 8 digits". Here is the point where we do not agree. It's the way how we name a difference of a few units in the 8th digit. I say "that's 7 digits" while you say "but it's almost 8".

Finally you said that 355/113 is an exceptionally good approximation to pi. Agreed. It's much better than 333/106 and far more efficient than the "next best" fraction 103993/33102. It's indeed a quite accurate and effective approximation. Accurate to... ;-)

Dieter