Exercise 7.6.2 (Find some fractions $a/b$ such that $|\pi- a/b|

Question

Find two rational number $a/b$ satisfying
$$\left|\pi - \frac{a}{b} \right | < \frac{1}{\sqrt{5}\, b^2}.$$

Richard Ganaye · Accepted Answer

\begin{proof} 
With Sagemath:
\begin{verbatim}        
sage: dfc = continued_fraction(pi); dfc
[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...]
sage: cvg = [dfc.convergent(i) for i in range(7)]; cvg
[3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317]
sage: for f in cvg:
....:     a,b = f.numer(), f.denom()
....:     if abs(pi - f) < 1/(sqrt(5)* b^2):
....:         print(f)
....:         
3
22/7
355/113
104348/33215
\end{verbatim}
So $3/1$ and $22/7$ are suitable (but not $333/106$ or $103993/33102$).
\end{proof}

Exercise 7.6.2 (Find some fractions $a/b$ such that $|\pi- a/b| < 1/(\sqrt{5} b^2)$)

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Exercise 7.6.2 (Find some fractions $a/b$ such that $|\pi- a/b| < 1/(\sqrt{5} b^2)$)

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