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Pi or 2 Pi: That Is the Question
livescience.com — In February this year there appeared in Physics World an article entitled Constant Failure by Robert P Crease of Stony Brook University, in which he showed in how many formulae of physics and mathematics 2 π turns up, rather than π. This article struck a chord with me, since even after many years I remember the feeling of "cognitive dissonance"
mpind176- 531 diggs
- digg it
- humanerror, on 07/23/2008, -3/+24You can't change it now! I already spent months memorizing 3.14159..uh, two.. six... uhhmm okay change it.
- AdmiralAcbar, on 07/23/2008, -6/+13.141592653589793238462643383.... *****. I had the next 7 or 8 memorized a few years back. Ah well...
- RepoOne, on 07/23/2008, -4/+13.141592653589793238462643383279502884197169399 is what I have memorized.
- jlubawy, on 07/23/2008, -3/+11wow that guys that posted the million digits of pi in the comments is a douche
- blackanode, on 07/23/2008, -0/+3Really kind makes me think when I saved that it was just under a mb in rtf format.
- Icetype, on 07/23/2008, -0/+3Where was that?
- GassyTurd, on 07/23/2008, -10/+3This morning I tea bagged a rattle snake. It took me 3.14 hours.
- ShoesChrist, on 07/23/2008, -0/+2I'd like to read it, but it seems to freeze when I open it o.O
- thebeebs, on 07/23/2008, -0/+1Same problem here
- geobay, on 07/23/2008, -0/+3Most likely your browser doesn't like pi to a million places...
- thebeebs, on 07/23/2008, -0/+1Same problem here
- bcstereotype, on 07/23/2008, -0/+4Fascinating stuff. Dugg.
- cardshark69, on 07/23/2008, -2/+14∫(2πr)dr =πr^2 (where c is known to be 0 for all you pedants)
- humanerror, on 07/23/2008, -2/+7Get that moon language out of here, vile wizard!
- IKORKYI, on 07/23/2008, -1/+12πdπ/dr = 0
- thebeebs, on 07/23/2008, -10/+2The guy that posted a million Digits broke my browser! I had to use Firefox to open it in the End:
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In February this year there appeared in Physics World an article entitled Constant Failure by Robert P Crease of Stony Brook University, in which he showed in how many formulae of physics and mathematics 2π turns up, rather than π. This article struck a chord with me, since even after many years I remember the feeling of "cognitive dissonance" when being taught that the formula was 2πR rather than πD.
I felt it a bit much, though, suggesting that Archimedes might have been mistaken in choosing to calculate the ratio of circumference to diameter rather than to radius. In those days, the fundamental dichotomy seems to have been between the geometers who thought of circumference, diameter and their ratio, and the astronomers who used the radius in their calculation of chord tables.
Hipparchus used a radius of 3438 which is the nearest integer to the number of minutes in 1 radian, but Ptolemy preferred 3600 as this is easier to calculate within the sexagesimal system. The work of these astronomers, further developed by Hindu and Arabic mathematicians, gives us our trigonometry of today.
In particular, Aryabhata published in 499 A.D. the Aryabhatiya in which he invented the sine function (radius!) as more convenient than the chord, but nevertheless computed the most accurate value of π (diameter!) known in ancient times. However al-Kashi, who was very much an astronomer and trigonometer, set a new record in precision in his Treatise on the Circumference in July 1424, a work in which he calculated 2π to nine sexagesimal places and translated this into sixteen decimal places.
The Greek geometers did not think of their ratio as a number. To them, number, magnitude and ratio were three distinct concepts. Then who first did? As it might say at the beginning of a tale from The Thousand and One Nights, "there were three brothers from Baghdad", namely the Banu Musa in the 9th Century, who are first recorded as having described this ratio as a number.
The first person to use π to represent the ratio of the circumference to the diameter (3.14159...) was the Welshman William Jones in 1706. But the radius fought back, with the word 'radian' first appeared in print in 1873, in examination questions set by James Thomson (brother of Lord Kelvin) at Queen's College, Belfast.
He used the term as early as 1871, while in 1869 Thomas Muir, then of St. Andrew's University in Scotland, hesitated between 'rad', 'radial' and 'radian', adopting 'radian' after consultation with James Thomson. (A Welshman, an Irishman, and a Scotsman – is it a Celtic conspiracy?)
Even the difference between the two versions of Planck's constant ℎ and ℏ (aka the Dirac constant) depends on whether one is thinking physically in terms of frequency ν or mathematically in terms of angular velocity ω. Physics is not Applied Mathematics! - Seth024, on 07/23/2008, -11/+3Someone posted a million digits of pi, which causes the website to freeze when opening it. Digg effect will start in a few minutes so I just copied the article here.
Also check out http://www.math.utah.edu/~palais/pi.html It's an article from Bob Palais about why we should use a symbol for 2*pi in mathematical formulas instead of pi.
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In February this year there appeared in Physics World an article entitled Constant Failure by Robert P Crease of Stony Brook University, in which he showed in how many formulae of physics and mathematics 2π turns up, rather than π. This article struck a chord with me, since even after many years I remember the feeling of "cognitive dissonance" when being taught that the formula was 2πR rather than πD.
I felt it a bit much, though, suggesting that Archimedes might have been mistaken in choosing to calculate the ratio of circumference to diameter rather than to radius. In those days, the fundamental dichotomy seems to have been between the geometers who thought of circumference, diameter and their ratio, and the astronomers who used the radius in their calculation of chord tables.
Hipparchus used a radius of 3438 which is the nearest integer to the number of minutes in 1 radian, but Ptolemy preferred 3600 as this is easier to calculate within the sexagesimal system. The work of these astronomers, further developed by Hindu and Arabic mathematicians, gives us our trigonometry of today.
In particular, Aryabhata published in 499 A.D. the Aryabhatiya in which he invented the sine function (radius!) as more convenient than the chord, but nevertheless computed the most accurate value of π (diameter!) known in ancient times. However al-Kashi, who was very much an astronomer and trigonometer, set a new record in precision in his Treatise on the Circumference in July 1424, a work in which he calculated 2π to nine sexagesimal places and translated this into sixteen decimal places.
The Greek geometers did not think of their ratio as a number. To them, number, magnitude and ratio were three distinct concepts. Then who first did? As it might say at the beginning of a tale from The Thousand and One Nights, "there were three brothers from Baghdad", namely the Banu Musa in the 9th Century, who are first recorded as having described this ratio as a number.
The first person to use π to represent the ratio of the circumference to the diameter (3.14159...) was the Welshman William Jones in 1706. But the radius fought back, with the word 'radian' first appeared in print in 1873, in examination questions set by James Thomson (brother of Lord Kelvin) at Queen's College, Belfast.
He used the term as early as 1871, while in 1869 Thomas Muir, then of St. Andrew's University in Scotland, hesitated between 'rad', 'radial' and 'radian', adopting 'radian' after consultation with James Thomson. (A Welshman, an Irishman, and a Scotsman – is it a Celtic conspiracy?)
Even the difference between the two versions of Planck's constant ℎ and ℏ (aka the Dirac constant) depends on whether one is thinking physically in terms of frequency ν or mathematically in terms of angular velocity ω. Physics is not Applied Mathematics! - LucasVB, on 07/23/2008, -1/+9See also: Robert Palais' "Pi is Wrong!" http://www.math.utah.edu/~palais/pi.pdf
- Nitrodist88, on 07/23/2008, -0/+3Was about to post the same thing. Congrats to you, sir.
- arjie, on 07/23/2008, -1/+2Interesting, but cosmetic mainly, non?
- Nicoon, on 07/23/2008, -0/+1Was just about to post the same thing. I guess you beat me to it, Lucas ;)
- WeeBull, on 07/23/2008, -0/+10Doesn't help that the article writer doesn't actually understand what's being discussed.
The discussion is because lot's of formulae use 2pi, or multiples and powers of 2pi, shouldn't we have a constant defined with that value. e.g. pipi (for want of a better name) = 2pi. It would remove a lot of constants from our maths and science formulae.
Instead he goes on about how he should have been taught things like pi*D instead 2pi*r. That doesn't change the constant. What's being asked is "should the formula for the circumference of a circle be pipi*r?". The down side is that it makes the area of a circle (pipi*r*r)/2.- rdeml, on 07/23/2008, -1/+1wouldn't pipi mean pi * pi?
My guess is that they based it on the radius of a circle rather than the diameter because it's easy to draw a circle with a radius. Put a stick in the ground with a string and walk around it.
- rdeml, on 07/23/2008, -1/+1wouldn't pipi mean pi * pi?
- fas2, on 07/23/2008, -1/+16But then the world's most beautiful formula will not be perfect anymore: exp(i pi) + 1 = 0. It contains 5 fundamental constants (e, i, pi, 1, 0) and the two axiomatic field operators over the complex numbers: addition and multiplication.
- dakotatypes, on 07/23/2008, -0/+4It could be argued that it is even more beautiful in that case. Imagine there is a symbol for 2pi. I would suggest a unit circle with a radius from (0,0) to (1,0). Since this character doesn't exist in ascii here I will use @. Euler's formula becomes
e^(i@/2) + 1 = 0
or e^{frac{twopi}{2} i} + 1 = 0 in LaTeX
Therefore you now have *six* fundamental constants (e, i, twopi, 0, 1, 2) and the field operations, plus the implied existence of inverses.- fas2, on 07/23/2008, -0/+7I see what you mean, but the beauty of the formula lies in the fact that the two operations (*, +) are axiomatic field operators and 0 and 1 are the axiomatic neutral elements of those two operations. In this regard (and in every other actually), 2 is no fundamental constant. It uglifies the otherwise perfect formula :)
- TheKeithD, on 07/23/2008, -1/+5Too complicated, send it back.
- futur4m4, on 07/23/2008, -0/+2Dugg for saying "uglifies" lol
- dakotatypes, on 08/20/2008, -0/+1I'm a little late in this reply, but I've got to take up for my homie.
By your reasoning, e and pi shouldn't be there either, they don't have anything to do with the fundamental properties of a field.
"In this regard (and in every other actually), 2 is no fundamental constant."
Two is
* the first prime and the only even one (hence making it very odd! oh ho ho bad math joke!)
* the least nonnegative integer whose square root is irrational
* the least number of characters necessary for a place value system
* the cardinality of the trivial field (oh snap!)
* the reason the fastest integer search is called 'binary search'
* just awesome.
- mbw234, on 07/23/2008, -0/+6Well, if A = 2pi, then e^{i A} = 1, which can be written as e^{i A} - 1 = 0.
So, if pi was really equal to to 2pi we could still write Euler's formula to have the constants 0, 1, pi, e, i. It's even a little better now as it has -1.- fas2, on 07/23/2008, -0/+1How is -1 better than +1?
- dakotatypes, on 07/23/2008, -0/+4It could be argued that it is even more beautiful in that case. Imagine there is a symbol for 2pi. I would suggest a unit circle with a radius from (0,0) to (1,0). Since this character doesn't exist in ascii here I will use @. Euler's formula becomes
- Scrunion, on 07/23/2008, -5/+2mmm... irrational numbers.. π e √2
btw - firefox rules - ie is junk if it can't display large portions of irration numbers- insanebrain, on 07/23/2008, -0/+1that' because the new IE can't think outside of the box ;)
- dupeduperson, on 07/23/2008, -0/+6My favorite appearance of pi is in the "random walk". Basically if you walk a random distance in a random direction your average displacement (after so many iterations) has a pi in it.
- Origin415, on 07/23/2008, -0/+5Or the probability two numbers are relatively prime... (6/pi^2)
Or the sum of the series' 1/n^k for k>1...
etc
- Origin415, on 07/23/2008, -0/+5Or the probability two numbers are relatively prime... (6/pi^2)
- loquax, on 07/23/2008, -0/+6One of the things lacking in this discussion is why the culture of the Greeks probably favored the diameter over the radius. For the Greek mind, the division of things into halves-(think Zeno's paradox of the arrow that never hits) was a constant fascination.
- Duositex, on 07/23/2008, -4/+2This topic has come up before. It was stupid then, and it's stupid now. Making the formulas more readable is silly. The number is what it is. Multiplying it by two is sloppy and unscientific.
- JakeyG14, on 07/23/2008, -1/+4It's Pi. 2 Pi may come up more in physics but at the end of the day it's still a multiple of a fundamental, non-terminating decimal number. The last sentence in the article reiterates more elegantly what I'm trying to say; if Pi is like h (Planck's constant), then 2 Pi is like ℏ, the same number as h but just multiplied by something else (the reciprocal of 2 Pi haha).
- rohan1234, on 07/23/2008, -1/+1At the end of the day, pi is still a multiple of 2pi, a fundamental, non-terminating decimal number.
- JakeyG14, on 07/23/2008, -0/+1Dude, well it's not an integer multiple (which is what most people assume when you say "multiple"), also look up "common factor".
- rohan1234, on 07/23/2008, -1/+1At the end of the day, pi is still a multiple of 2pi, a fundamental, non-terminating decimal number.
- Napiertt, on 07/23/2008, -0/+4After the 500,000 digit in pi, the following sequence occurs: 10101020202012345678910. Did anyone else catch that? Clearly a hidden code left by the designers on the universe. Wow!
- sxreader, on 07/23/2008, -0/+5pi contains pretty much any sequence of number, somewhere, so why are you surprised?
- Napiertt, on 07/23/2008, -0/+0Suckers! I wanted to see who'd be the first to bite! :)
- stklaw, on 07/23/2008, -0/+3Makes me wonder what you do with your free time.
- Fitness03, on 07/23/2008, -0/+2So how come when I copy your little number there and paste it into the find bar in firefox and search for it on the article linked in this page does it not seem to appear, considering there is a comment that has pi with 1,000,000 decimals or something ridiculous?
- thylacine222, on 07/23/2008, -0/+1Searchstart: 12 The string 123456789 did not occur in the first 200000000 digits of pi after position 0.
(Sorry! Don't give up, Pi contains lots of other cool strings.)
according to http://www.angio.net/pi/piquery- Napiertt, on 07/23/2008, -0/+0Cool search page
- snurfle, on 07/23/2008, -0/+4If pi is truly infinite and non-repeating, then somewhere in there will be a sequence that exactly matches my OGG copy of "Stairway to Heaven."
Call the RIAA right away before this news leaks... - RogueMountie, on 07/23/2008, -0/+1If you think that's something, check out pi in base 13. It's full of all kinds of hidden messages. Left by SATAN!
- sxreader, on 07/23/2008, -0/+5pi contains pretty much any sequence of number, somewhere, so why are you surprised?
- sxreader, on 07/23/2008, -5/+0What pisses me off mostly is the fact that americans/brits pronounce pi as pie and not pee, which is the correct modern Greek pronunciation and also the pronunciation in most other countries.
- aggies11, on 07/23/2008, -0/+7That's because "pie" is delicious, while "pee" is just gross.
- innocentsinner, on 07/23/2008, -0/+2Why? P.
- zxjams, on 07/23/2008, -0/+2Why even bother? When is the last time you heard someone pronounce β as "veeta", or μ as "mee"? Even δ is pronounced like TH in our word "the" in modern Greek (most of the time).
To get technical about it, Brits pronounce θ correctly.
- Icetype, on 07/23/2008, -0/+2Lets use a new character for 2pi.. how about: 元
- Nicoon, on 07/23/2008, -0/+1There's already a symbol for 2pi. It's basically like the pi symbol but with three legs instead of two.
- roddack, on 07/23/2008, -1/+2I vote CAKE!
- cinch123, on 07/23/2008, -0/+1Between this and the article about datacenter heat, there are some good quality articles on the front page today.
- 23bpg, on 07/23/2008, -0/+0Link to referenced article: http://physicsworld.com/cws/article/print/32679
- futur4m4, on 07/23/2008, -0/+4π is exactly three!
- Professor Frink - sammydeath, on 07/25/2008, -0/+1i thought u was talking bout the film pi, if there was a second moive that would be sweet!
http://www.imdb.com/title/tt0138704/
im not even a geek and thought it was amazing
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