Quality of rational approximations

In summary, 22/7, 99/70, 193/71 and 355/113 are all good approximations for π, with 355/113 being the best. This phenomenon is not limited to π, as e also has a similar pattern of continued fractions. However, it is interesting to note that algebraic numbers are not as well approximated by rationals, which may explain why π and e have better approximations with smaller denominators. This topic has been studied by mathematicians such as Dyson and is known as Roth's theorem. It is unclear if there is any specific reason for this pattern or if it is simply a coincidence.
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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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  • #3
Sure, but e has one too. What's special, if anythnig, about pi?
 
  • #4
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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1. What is the definition of "quality of rational approximations"?

The quality of rational approximations refers to how closely a rational number (a number that can be expressed as a ratio of two integers) approximates a given real number. It is a measure of how accurate the approximation is.

2. How is the quality of rational approximations measured?

The quality of rational approximations is typically measured using the error or difference between the rational approximation and the real number. The smaller the error, the higher the quality of the approximation.

3. Why is the quality of rational approximations important?

The quality of rational approximations is important because it allows us to represent real numbers in a more manageable and precise way. It is also crucial in many mathematical and scientific calculations where exact values are not necessary or practical.

4. Can the quality of rational approximations be improved?

Yes, the quality of rational approximations can be improved by using more precise and accurate methods of approximation, such as continued fractions or rational interpolation. In some cases, using a larger denominator in the rational approximation can also improve its quality.

5. How is the quality of rational approximations related to the concept of convergence?

The quality of rational approximations is closely related to the concept of convergence in mathematics. As the quality of the approximation improves, the rational numbers converge to the real number they are approximating. This means that the error between the approximation and the real number becomes smaller and smaller as the approximation approaches the real number.

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