/sci/ - Science & Math

[Anonymous No.7001165]

what are, currently, the largest two WHOLE numbers that can be divided by each other to give, up to a point (decimal), an equal value of pi? note I know it will always just be an approximation.


x>221/71
x<22/7
x>???
x<???

question:
Does man know an accurate formula method of finding larger and larger whole numbers that divide more accurately "into" pi?
or are such whole numbers only 'found' by analysis, through other formulas ie. BBP

[Anonymous No.7001179]

>>7001165
A History of Pi has them

You can also derive them from making inscribe and circumscribed n-gons and increasing n

[Anonymous No.7001189]

>>7001165
Look up "continued fraction". It gives you a nice algorithm to determine rationals that are good approximations of a certain number.

For π it is [3; 7, 15, 1, 292, 1, 1, 1,...]. That is equal to the approximations:

3, 22/7, 333/106, 355/113, 97591/31065, and so forth.

[Anonymous No.7001198]

>>7001189
Also, just to make it complete, a continued fraction is defined as
[display]a_0 + \frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+...}}}}[/display]
and is usually written down as
<span class="math">[a_0; a_1, a_2,a_3,...][/spoiler].

[Anonymous No.7001200]

>>7001198
Whoops.
<span class="math">a_0 + \frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+...}}}}[/spoiler]
What was the command for the larger latex math again?

[Anonymous No.7001204]

>>7001200
Oh fuck this.
<span class="math">a_0 + \frac{1}{a_1+\frac{1}{a_2+\frac{1}{ a_3+...}}}[/spoiler]

[Anonymous No.7001217]

>>7001189
>97591/31065
Oh, and that is wrong, sorry. It's 103993/33102 actually.

[Anonymous No.7001218]

>>7001189
will it help with upper and lower boundaries?

I guess I can try to guess them as long as I have a linear structure of the upper boundary by adding n

>>7001179
I'm imagining what that does for upper and lower, if the inscribed means inside and circum means outside. What is this called and who discovered it first?

[Anonymous No.7001222]

>>7001218
> What is this called and who discovered it first?
Huygens?
It's been many years since I read the book

[Anonymous No.7001227]

>>7001165
Take x digits of pi, and remove the decimal. Now divide that by 10^(x-1).

The numbers can be infinitely large.

[Anonymous No.7001236]

>>7001198
>>7001204
>>7001217

thanks, I don't have latex atm, I'd like to play with continued fractions more but I just have excel.

So I guess to answer my own question, we don't have a formula per se that dictates whole numbers of pi, we instead have continued fractions that can sometimes lead to whole numbers at an interval

is that interval expressed?

I mean, after a certain number of fractions in these sequences, we can find a whole number
can we predict the number of sequences ahead before another fraction can be expressed as a larger (and therefore more accurate) whole numbered fraction?

here's a prediction I know is wrong
2222-1/700+1
2221/701 = pi?

holy shit why do I see phi?

[Anonymous No.7001270]

>>7001165
31415926535897932384626433832795028841971693993751058209749445923078164/10000000000000000000000000000000000000000000000000000000000000000000000

[Anonymous No.7001275]

>>7001227
This is a pretty inefficient way of representing the number.

[Anonymous No.7001280]

>>7001275
then ask the question correctly. that's the answer to op's question. you want something else? ask for it

[Anonymous No.7001281]

>>7001280
I'm not OP, so I'm not asking for anything.

[Anonymous No.7001285]

>>7001281
then go away, your nitpicking isnt helping anyone. if you have something to say, go ahead.

[Anonymous No.7001288]

>>7001285
Trying to improve things by a great margin is nitpicking? Do you even do math?

[Anonymous No.7001290]

>>7001288
see >>7001280, set standards for "improve", ask again

[Anonymous No.7001297]

>>7001218
Those troll comics are so fucking stupid it's not even funny

[Anonymous No.7001366]

thanks, I don't have latex atm, I'd like to play with continued fractions more but I just have excel.

Why exactly is 221/71 important?

I could just say that pi is greater than 110/35

is it because 22*10+1 and 7*10-1 have the same feature, multiply by 10 and subtract by a tenth of ten?

[Anonymous No.7001371]

>>7001227

/thread

[Anonymous No.7001374]

>>7001366
<span class="math"> 221/71-\pi < 110/35 - \pi [/spoiler]

[Anonymous No.7001377]

>>7001290
>>7001280
>>7001227

thanks for the help so far, I don't have latex atm, I'd like to play with continued fractions more but I just have excel.

Why exactly is 221/71 important?

I could just say that pi is greater than 110/35

is it because 22*10+1 and 7*10-1 have the same feature, multiply by 10 and subtract by a tenth of ten?

>>7001297
true, it's just what came up on the google search in Aristotle, which is sad.

>>7001288
you aren't trying to improve things by nitpicking, you try to improve things by showing how something doesn't work and asking a question within some logical limits to determine why it's not efficient

we haven't deciphered pi very well so it's only natural we have a large number of poor ways to find it out, even the complex accurate ways of finding pi were just based on analysing what was working and what wasn't working, finding the degree of inefficiency, and realigning it.

for example
if X is true to a degree, and false to this degree
and Y is fase at this degree and true to this degree
how can X and Y both have similar truths, and similar falses, and is the pattern determinable?

[Anonymous No.7001380]

>>7001374
oh.
>>7001290
>>7001280
>>7001227

thanks for the help so far, I don't have latex atm, I'd like to play with continued fractions more but I just have excel.

>>7001297
true, it's just what came up on the google search in Aristotle, which is sad.

>>7001288
you aren't trying to improve things by nitpicking, you try to improve things by showing how something doesn't work and asking a question within some logical limits to determine why it's not efficient

we haven't deciphered pi very well so it's only natural we have a large number of poor ways to find it out, even the complex accurate ways of finding pi were just based on analysing what was working and what wasn't working, finding the degree of inefficiency, and realigning it.

for example
if X is true to a degree, and false to this degree
and Y is fase at this degree and true to this degree
how can X and Y both have similar truths, and similar falses, and is the pattern determinable?

[Anonymous No.7002941]

>>7001165
355/113 is the simplest and most efficient approximation for pi. It gets right 6 decimal digits, to do better than that you have to get to 5/6 digits divisions.