Undergrad Quality of rational approximations

  • Thread starter Thread starter Vanadium 50
  • Start date Start date
  • Tags Tags
    Quality Rational
Click For Summary
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.
Vanadium 50
Staff Emeritus
Science Advisor
Education Advisor
Gold Member
Dearly Missed
Messages
35,003
Reaction score
21,716
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...")
 
Mathematics news on Phys.org
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.
 
Reply
  • Like
Likes Vanadium 50, FactChecker, PeroK and 1 other person

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

High School Find Optimal Int Ratios to Approximate Pi
  • · Replies 7 ·
Replies
7
Views
2K
Approximating Arctangent and Natural Log
  • · Replies 4 ·
Replies
4
Views
4K
Perfect Espresso Water Chemistry
  • · Replies 7 ·
Replies
7
Views
4K
Undergrad Deur Gravitational self-interaction Doesn't Explain Galaxy Rotation Curves
  • · Replies 72 ·
3
Replies
72
Views
10K
Do You Have What it Takes to Be a True Mathematician?
  • · Replies 21 ·
Replies
21
Views
4K
Studying Skipping Chapters in Stewart’s Calculus? (Pearson's Edexcel IAL Background)
  • · Replies 2 ·
Replies
2
Views
637
How can I avoid large exponents in calculations with small answers?
  • · Replies 5 ·
Replies
5
Views
3K
Schools PGRE - A message to undergrads from physics grad school.
  • · Replies 25 ·
Replies
25
Views
8K
Mathematica This Week's Finds in Mathematical Physics (Week 218)
  • · Replies 1 ·
Replies
1
Views
2K
Challenge Math Challenge - December 2018
  • · Replies 67 ·
3
Replies
67
Views
15K