I Quality of rational approximations

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22/7 is recognized as a strong approximation for π, while approximations for √2 and e require larger ratios, such as 99/70 and 193/71 respectively. The discussion raises questions about the reasons behind the effectiveness of small integer ratios for π compared to other numbers, suggesting potential geometrical or mathematical patterns, such as continued fractions. It notes that algebraic numbers generally do not have good rational approximations when keeping denominators small. The conversation also references the work of mathematician Freeman Dyson in this area, highlighting ongoing interest in the topic. The underlying reasons for these phenomena remain an intriguing subject of exploration.
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22/7 is a very good approximation for π. Sqrt(2) doesn’t do that well until 99/70 and e doesn’t do that well until 193/71. 355/113 is even better.

Is there some reason for this? Perhaps geometrical? Why do the ratios of small integers work better for π than other numbers? Or is it just coincidence? ("Gotta be something...")
 
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The pattern of continued fractions?
 
Sure, but e has one too. What's special, if anythnig, about pi?
 
If you want to keep the denominators of the rationals somewhat small, then algebraic numbers are not well approximated by rationals.
https://en.wikipedia.org/wiki/Roth's_theorem

ps. Interesting that Dyson had worked on these matters.
 
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