A special fraction that is as close to Pi as possible

The fraction 937246158 / 298341567 is a close approximation to the value of Pi. 937246158 / 298341567 = 3.14152053

What is interesting about both the numerator and the denominator is that each of these nine-digit numbers both consist of all of the digits 1 through 9, with no repeats.

The fraction 573819264 / 182653479 also contains these properties and is even closer to Pi, albeit very slightly, than the first fraction. 573819264 / 182653479 = 3.14157314

Question: What nine-digit numerator and nine-digit denominator, each containing the digits 1-9, is CLOSEST to the value of Pi?

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P.S. While looking for something else, I came across this question and I thought it was very interesting. I had never heard the question before, so hopefully one or two others here have not heard it either.

The website I visited did not know or provide the answer, so I spent a few hours earlier this morning computing it on my own. I had a lot of fun in doing so.

However, as I should have expected, after arriving at my answer I soon realized many other websites also contained this question and many of them often posted the correct answer...meaning I had not discovered anything new. Arrgh! Darn!

That's one of the big disadvantages with math. When you learn a new concept, or discover something interesting, you soon realize you did not learn anything that hasn't been known long before. In fact, there's a good chance people like Euler and Pascal probably knew it when they were still in their diapers.

Since the answer to the above question is so readily available, here's a bonus question that might not be so easy to find: What is the SECOND closest nine-digit numerator / nine-digit denominator that comes as close to Pi as possible?

some other interesting fact, the smallest fraction with an impressing 6 decimal accuracy:

355/113 = 3.14159292035...

Quote: EdCollins

Since the answer to the above question is so readily available, here's a bonus question that might not be so easy to find: What is the SECOND closest nine-digit numerator / nine-digit denominator that comes as close to Pi as possible?

Here are the first 20 :)

467895213/148935672=3.1415926535048
429751836/136794258=3.14159265369165
813569274/258967143=3.14159265370588
928146375/295438167=3.14159265346376
546213789/173865249=3.14159265374531
937186245/298315647=3.14159265336826
621583974/197856324=3.14159265386938
896452713/285349761=3.1415926540762
613987452/195438276=3.14159265301747
784321596/249657318=3.14159265301408
529436817/168524973=3.14159265285864
481573629/153289647=3.14159265432975
842935167/268314597=3.14159265438697
529136487/168429375=3.14159265270681
549231876/174825936=3.14159265247692
683719254/217634598=3.14159265246971
915628734/291453678=3.14159265473397
613825497/195386724=3.1415926549851
621574983/197853462=3.14159265507318
579234168/184375962=3.14159265512063

22/7 is close enough in my book

Quote: andysif

some other interesting fact, the smallest fraction with an impressing 6 decimal accuracy:

355/113 = 3.14159292035...

That uses 6 total digits in the fraction and approximates pi to 3.1415929. The real pi is 3.1415926. So it uses 6 digits to come up with 7 correct digits.

22/7 uses 3 and is right to 3.

Can anyone get improve upon 355/113, where the degree of success is the number of total digits saved, in this case 1 (using 6 to get pi right to 7 digits).

Ref: pi=3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679

I'm not going to risk severe brain trauma by attempting to come up with better fractions. I just thought I would mention that the very first time that I saw the 355/113 approximation for Pi was in the instruction manual for an HP35 calculator I purchased in 1973. As an exercise in learning to use the RPN calculation technique, they suggested calculating the percentage error of this ratio as an approximation to Pi. The manual noted that an easy way to remember this particular fraction is to write the first three odd numbers twice each, then separate them in the middle with a long division symbol like:


Off-topic but related to my own post: I think the HP35 was the first commonly-available portable electronic calculator that could perform calculations with transcendental functions like logarithms, trig functions, etc. It provided a lot more significant digits than a slide rule. I bought mine after they came out with the HP45 and the price on the HP35 dropped from $400 to $300. About a year later, TI came out with their comparable calculators for around $150. Within a couple more years, slide rules were novelty "antiques". I still have one or two in my desk drawer.

This is getting off topic, but my father gave me an HP15C as a high school graduation present, and I've been using daily, almost hourly, ever since. How far HP has fallen, in my opinion, to the computer that caught fire on me a few years ago.

Quote: andysif

some other interesting fact, the smallest fraction with an impressing 6 decimal accuracy:

355/113 = 3.14159292035...

Archimedes concluded that PI was smaller than 3+1/7 and bigger than 3+10/71.
As 3+1/7 = 3+16/112 at first glance it seems fairly unremarkable that 3+16/113 =355/113 is a better approximation. But that improvement took humanity almost 800 years to make (and the western world over 1800 years to accomplish).

It is not clear to me if the Chinese in the 5th century knew that 355/113 was also an upper bound.

If you go back and read about how these calculations were done, even getting pi = 22/7 was a remarkable accomplishment.

I do think that Archimedes may have been one of the greatest minds of all of history. I am big fan of his. His murder was an unbelievable tragedy.

All I knowl about pi is the mnemonic device for the first nine places...may I have a large container of orange juice

Quote: Wizard

This is getting off topic, but my father gave me an HP15C as a high school graduation present, and I've been using daily, almost hourly, ever since. How far HP has fallen, in my opinion, to the computer that caught fire on me a few years ago.

Ah... HP calculators. I remember saving $200 so I could buy a used HP48SX in high school. It was worth every penny and put my TI83 to shame, which I gave to my sister. HP calculators were awesome.