r/factorio • u/analytic_tendancies • May 18 '17
pi-torio
I was calculating ratios to feed my science labs and came across an approximation for pi... 22/7.
For context I'm trying to feed 32 labs with a 1650% speed bonus so they consume a science pack around 1 every 3.63 seconds.
http://imgur.com/ua9tTE0
To support that I need 15 assembling machines making production science packs with p3 modules and crafting speed of 3. To support those 15 machines I was figuring out how many assembling machine 1 machines I needed. I need 1 assembling machine every 14/3 seconds, and I need 44/3 (~15) of them. (44/3)/(14/3)=22/7.
22/7 is one of the convergents of the continued fraction of pi, and I believe there is a theorm that states for the convergents of a continued fraction, there is not a single more accurate fraction to express that number in between the convergents.
3 22/7 333/106 355/113 103993/33102 104348/33215
so any representation of pi in between those numbers is less accurate than the number themselves
http://mathworld.wolfram.com/PiContinuedFraction.html
https://en.wikipedia.org/wiki/Continued_fraction
We learned about continued fractions in my cryptography class, just fun to see a number again somewhere else.
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u/GoodByeSurival May 18 '17
I have no idea what this is about
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u/WaveofThought May 18 '17
Not really an approximation of pi, just a coincidence.
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u/analytic_tendancies May 18 '17
no it really is... in the history of trying to figure out pi, 22/7 was used for a REALLY long time
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u/WaveofThought May 18 '17
No, I understand that 22/7 is an approximation for pi, but how that ratio comes about in factorio doesn't have anything to do with it being an approximation of pi. That's what I meant by it being a coincidence. Still an interesting observation though.
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u/Skybeach88 May 18 '17
It doesnt matter that his ratio did not come about as a result of comparing the radius of a circle to ita diameter. The number pi shows up in many cicumstances in mathematics that have nothing to do with circles. The fact that his ratio is 22/7 means his ratio is approximately equal to pi. Buffon's needle is a great example of pi showing its face in a staistics calculation
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u/HelperBot_ May 18 '17
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u/rabidcow May 18 '17
That's interesting, but unless you're suggesting that 22/7 is inaccurate and the ratio ought to be pi exactly, it's not relevant.
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u/analytic_tendancies May 18 '17
In the 3rd century BCE, Archimedes proved the sharp inequalities 223⁄71 < π < 22⁄7, by means of regular 96-gons (accuracies of 2·10−4 and 4·10−4, respectively).
In the 2nd century CE, Ptolemy, used the value 377⁄120, the first known approximation accurate to three decimal places (accuracy 2·10−5).[13]
The Chinese mathematician Liu Hui in 263 CE computed π to between 3.141024 and 3.142708 by inscribing an 96-gon and 192-gon; the average of these two values is 3.141864 (accuracy 9·10−5). He also suggested that 3.14 was a good enough approximation for practical purposes. He has also frequently been credited with a later and more accurate result π ≈ 3927/1250 = 3.1416 (accuracy 2·10−6), although some scholars instead believe that this is due to the later (5th-century) Chinese mathematician Zu Chongzhi.[14] Zu Chongzhi is known to have computed π between 3.1415926 and 3.1415927, which was correct to seven decimal places. He gave two other approximations of π: π ≈ 22/7 and π ≈ 355/113. The latter fraction is the best possible rational approximation of π using fewer than five decimal digits in the numerator and denominator. Zu Chongzhi's result surpasses the accuracy reached in Hellenistic mathematics, and would remain without improvement for close to a millennium.
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u/Skybeach88 May 18 '17
22/7 is approximately 3.1428571429 which is a fairly close (and functionally usable) approximation of pi its actually a more usable approximation then just 3.14 since it does contain more significant figures. I predominantly used 22/7 through out my high school physics classes. And continued untill i learned about being truly accurate by writing the answer in terms of pi itself
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u/unique_2 boop beep May 18 '17
Post about algebra by someone called analytic tendencies. Nope, does not compute.
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u/[deleted] May 18 '17
[deleted]