Find Optimal Int Ratios to Approximate Pi

[Astronuc]

TL;DR Summary

Integer ratio(s) approximating pi

On July 22, V50 wrote https://www.physicsforums.com/threads/happy-pi-day.985513/post-6518554,

and it got me thinking about the ratio of two integers approximating pi. I presume someone has done it, and that there is an algorithm for an optimal calculation.

Then I found - https://en.wikipedia.org/wiki/Approximations_of_π#Practical_approximations

Depending on the purpose of a calculation, π can be approximated by using fractions for ease of calculation. The most notable such approximations are 22⁄7 (relative error of about 4·10⁻⁴) and 355⁄113 (relative error of about 8·10⁻⁸).

When reading V50's post, I thought of 7/22 rather than 22/7.

7/22 = 0.3181818... but that's ~pi/10, but one can multiply by 10 and get 70/22, 3.181818... I didn't immediately realize that ratio is 35/11, which would have lead me to 355/113, which by the way gives me an error to pi of 2.6676418940E-07 (using Excel on a Mac). Nevertheless, 355/113 gives an efficient approximation with a very small error that 22/7

Anyway, I soon realized that I had started with 7/22 and not 22/7, the latter being the classic approximation with the smallest integers.

So, I started expanding the numerator and denominator.

22/7 = 220/70 = 3.142857

2200/701 = 3.13837375 (in wrong direction)

2203/701 = 3.14265335 overshoot

2202/701 = 3.14122682 undershoot
----------------------------------
a different path

22000 / 7001 = 3.142408 overshoot but less than 2203/701

22000 / 7002 = 3.141959 better

22000 / 7003 = 3.141511 better, but undershoot

22001 / 7003 = 3.141654

220010 / 70035 = 3.141429 scaling up

220010 / 70034 = 3.141474 moving in the right direction

220010 / 70033 = 3.14151900

220010 / 77032 = 3.141564 (pi = 3.14159265358979...)

Adjusting the numerator
220012 / 77032 = 3.1415924

2200121 / 700320 = 3.1415938

2200121 / 700321 = 3.1415894

22001212 / 7003200 = 3.1415941 (I didn't think to divided by 2, which would give me 11000606/3501600,
or by 4 giving 5500303 / 1750800

22001212 / 7003201 = 3.1415937

22001212 / 7003202 = 3.1415932

22001212 / 7003203 = 3.14159278

I started using Excel rather than my calculator
220012121 / 70032040 = 3.141592348302290

220012121 / 70032040 = 3.14159234830229 err -3.0528750194E-07

220012143 / 70032040 = 3.14159266244422 err 8.8544251931E-09

220012121 / 70032033 = 3.14159266231783 err 8.7280351835E-09

220012127 / 70032035 = 3.14159265827417 err 4.6843808832E-09

I tried a few more steps

2200121267 / 700320350 = 3.14159265399042 err 4.0062708706E-10

2200121270 / 700320351 = 3.14159265378824 err 1.9844437205E-10

2200121430 / 700320402 = 3.14159265347235 err -1.1744738515E-10
1100060715 / 350160201 divide by 2
366686905 / 116720067 divide previous line by 3, or the line before by 6. I realized this while creating this post. There were other opportunities to realize this earlier.

So I was wondering if there is a program or algorithm that has been developed to identify the best set of integers based on the desired error.

Or is 355⁄113 the best (most efficient, with a reasonably small error) and we're done.

 

[fresh_42]

The best quotient is the one that takes the most digits. The shortest possible is $3$. So we have two conflicting targets: minimal error and shortest length. To answer the question, we would need a combination of them to a single-valued function which we then can optimize. As long as we have both, I see no way to get to a result.

With respect to the many formulas to calculate $\pi,$ it might be more reasonable to ask for the fastest algorithm to achieve a given error margin or to ask for the shortest formula to describe an algorithm.

 

[Gaussian97]

Summary:: Integer ratio(s) approximating pi

On July 22, V50 wrote https://www.physicsforums.com/threads/happy-pi-day.985513/post-6518554,

and it got me thinking about the ratio of two integers approximating pi. I presume someone has done it, and that there is an algorithm for an optimal calculation.

Then I found - https://en.wikipedia.org/wiki/Approximations_of_π#Practical_approximations

When reading V50's post, I thought of 7/22 rather than 22/7.

7/22 = 0.3181818... but that's ~pi/10, but one can multiply by 10 and get 70/22, 3.181818... I didn't immediately realize that ratio is 35/11, which would have lead me to 355/113, which by the way gives me an error to pi of 2.6676418940E-07 (using Excel on a Mac). Nevertheless, 355/113 gives an efficient approximation with a very small error that 22/7

Anyway, I soon realized that I had started with 7/22 and not 22/7, the latter being the classic approximation with the smallest integers.

So, I started expanding the numerator and denominator.

22/7 = 220/70 = 3.142857

2200/701 = 3.13837375 (in wrong direction)

2203/701 = 3.14265335 overshoot

2202/701 = 3.14122682 undershoot
----------------------------------
a different path

22000 / 7001 = 3.142408 overshoot but less than 2203/701

22000 / 7002 = 3.141959 better

22000 / 7003 = 3.141511 better, but undershoot

22001 / 7003 = 3.141654

220010 / 70035 = 3.141429 scaling up

220010 / 70034 = 3.141474 moving in the right direction

220010 / 70033 = 3.14151900

220010 / 77032 = 3.141564 (pi = 3.14159265358979...)

Adjusting the numerator
220012 / 77032 = 3.1415924

2200121 / 700320 = 3.1415938

2200121 / 700321 = 3.1415894

22001212 / 7003200 = 3.1415941 (I didn't think to divided by 2, which would give me 11000606/3501600,
or by 4 giving 5500303 / 1750800

22001212 / 7003201 = 3.1415937

22001212 / 7003202 = 3.1415932

22001212 / 7003203 = 3.14159278

I started using Excel rather than my calculator
220012121 / 70032040 = 3.141592348302290

220012121 / 70032040 = 3.14159234830229 err -3.0528750194E-07

220012143 / 70032040 = 3.14159266244422 err 8.8544251931E-09

220012121 / 70032033 = 3.14159266231783 err 8.7280351835E-09

220012127 / 70032035 = 3.14159265827417 err 4.6843808832E-09

I tried a few more steps

2200121267 / 700320350 = 3.14159265399042 err 4.0062708706E-10

2200121270 / 700320351 = 3.14159265378824 err 1.9844437205E-10

2200121430 / 700320402 = 3.14159265347235 err -1.1744738515E-10
1100060715 / 350160201 divide by 2
366686905 / 116720067 divide previous line by 3, or the line before by 6. I realized this while creating this post. There were other opportunities to realize this earlier.

So I was wondering if there is a program or algorithm that has been developed to identify the best set of integers based on the desired error.

Or is 355⁄113 the best (most efficient, with a reasonably small error) and we're done.

Another good approximation is 312689/99532, which has an error of 3E-11, and you can continue indefinitely. Also, 3.1415926536 has even a smaller error...
You may want to look at the concept of continued fractions.

 

[FactChecker]

IMO, the fraction approximation only has a benefit when it requires remembering fewer digits than simply memorizing the digits of pi to an equal accuracy. Very few meet that criteria. Even the benefit of 22/7 is debatable, since who does not already know 3.14?

 

[Vanadium 50]

A few points:

One is that @Astronuc has introduced the idea of a quality of approximation. I assert without proof that I would expect that on average adding a digit to the numerator improves the approximation by a factor of 10. 22/7 is off by a part in 2500, so I would expect a three digit denominator to be good to a part in 25,000. However, 355/113 is good to a part in 11 million - about 400 times better than expected.

It's a very good approximation indeed. To get a better approximation, you need to go to a 5-digit denominator: 103993/33102, although the best 5 digit approximation is 312689/99532.

Computers are so fast these days that if you want to know what the best approximation is with 6 digits in the denominator, one can brute-force test all million numbers. At this point, approximations lose their utility. Is it easier to memorize 4272943/1360120 than pi itself to 11 or 12 digits?

So, as far as useful approximations go, I think the case can be made for just two: 22/7 and 355/113.

PS My calculator tells me 4272943/1360120 is exactly equal to pi.

 

[mfb]

Continued fractions give the "best" approximation in some sense - there won't be a better fraction (closer to pi) with a smaller denominator. The larger the next integer in the denominator of the continued fraction the better the approximation relative to the size of the denominator.

From Wikipedia:$$\pi=3+\textstyle \cfrac{1}{7+\textstyle \cfrac{1}{15+\textstyle \cfrac{1}{1+\textstyle \cfrac{1}{292+\textstyle \cfrac{1}{1+\textstyle \cfrac{1}{1+\textstyle \cfrac{1}{1+\ddots}}}}}}}$$

3+1/7 = 22/7, the next term is 333/106, and the following one is 355/113. The very large 292 that would enter the following term tells us that 355/113 is an unusually good approximation (and the next two will be quite poor again relative to their size).

 

[Frabjous]

Check out
https://en.wikipedia.org/wiki/Milü
https://www.macalester.edu/aratra/edition2/chapter3/chapt3c.pdf

 

[Astronuc]

3+1/7 = 22/7, the next term is 333/106, and the following one is 355/113. The very large 292 that would enter the following term tells us that 355/113 is an unusually good approximation (and the next two will be quite poor again relative to their size).

Yeah, I overlooked that section
https://en.wikipedia.org/wiki/Pi#Approximate_value_and_digits

Fractions: Approximate fractions include (in order of increasing accuracy) 22/7, 333/106, 355/113, 52163/16604, 103993/33102, 104348/33215, and 245850922/78256779.