View previous topic | View next topic

Pi - too many digits

Page 1 of 2
Goto page 1, 2  Next

dstarfire
1257812.  Wed Oct 25, 2017 12:02 pm Reply with quote

Edited to: on second that, let's not go down another tangent.


Last edited by dstarfire on Wed Oct 25, 2017 12:07 pm; edited 1 time in total

 
GuyBarry
1257813.  Wed Oct 25, 2017 12:02 pm Reply with quote

Alexander Howard wrote:
There's an infamous story about an American "mathematician", Edwin J. Goodwin, who devised a system where π would equal 4. How mad it got after that I do not know, nor do you really need to know any more.


Not true. He was trying to copyright his method for "squaring the circle", which had already been proved impossible, and tried to get it passed into Indiana state law. His method assumed that pi had the value 3.2:

http://mentalfloss.com/article/30214/new-math-time-indiana-tried-change-pi-32


Last edited by GuyBarry on Wed Oct 25, 2017 12:16 pm; edited 1 time in total

 
GuyBarry
1257814.  Wed Oct 25, 2017 12:14 pm Reply with quote

dstarfire wrote:
GuyBarry wrote:
The universe doesn't have a circumference, since it doesn't have a boundary. I think you mean the circumference of the observable universe. Still a lot less than 22 trillion though :-)


How do we know there isn't a boundary? Has somebody managed to observe, even indirectly, that area? (Unlikely since it's getting father away from us at superluminal speeds.)


Well no, because it's not observable from the Earth. Any part of the universe that's further away than the distance that light could have travelled to the Earth since the Big Bang is, by the laws of physics, unobservable from the Earth. That doesn't mean that it doesn't exist.

Quote:
Also, you don't need a barrier to have an edge.
1. According to the big bang theory, all matter in the universe occupied a single point at the beginning of time.
2. The universe is expanding at a given rate.
3. The universe has a definite age.
4. Ergo there exists a common point beyond which it's impossible for matter to have travelled naturally.


And you think that the point it all expanded from is the Earth?

Do you seriously believe that the universe is a perfect sphere with its centre at the Earth? That would take the anthropocentric assumption to new limits.

Quote:
Finally: What, are you chairman of the pedantic society?


Well, you made the original point. On a forum like this, you need to be able to back up your assertions.

 
dstarfire
1257883.  Wed Oct 25, 2017 8:51 pm Reply with quote

D*mnit, didn't delete fast enough. Oh well, if we're going to go down this tangent, might as well do it properly.
GuyBarry wrote:
And you think that the point it all expanded from is the Earth?

Do you seriously believe that the universe is a perfect sphere with its centre at the Earth?


I don't recall saying the earth is the center, and I certainly didn't mean to imply that. I know that there is NO CENTER, and that everywhere is the center. Expanding space is somewhat counter-intuitive like that.

Quote:
Quote:
Finally: What, are you chairman of the pedantic society?


Well, you made the original point. On a forum like this, you need to be able to back up your assertions.


It's the start of a joke from the "Groovy" episode. The punchline is "vice-chairman, actually".

 
Paul-R
1259757.  Sun Nov 05, 2017 4:16 pm Reply with quote

Alfred E Neuman wrote:
GuyBarry wrote:
Yes, but it's not much use if you're trying to calculate quickly in your head!

True. But there’s not that much of that being done these days.

I still do occasionally (using 3.14). Also sometimes on paper (using anything up to 3.14159 depending on how pretentious I feel). I don't carry a calculator, so sometimes you need to do a quick measure. Also it keep the brain from atrophying.

 
Alfred E Neuman
1259758.  Sun Nov 05, 2017 4:18 pm Reply with quote

Paul-R wrote:
Also it keep the brain from atrophying.


You don't say.

 
Paul-R
1259763.  Sun Nov 05, 2017 4:38 pm Reply with quote

Alfred E Neuman wrote:
Paul-R wrote:
Also it keep the brain from atrophying.

You don't say.

Think what it would be like if I didn't do this!

 
Rob Andrews
1261287.  Sun Nov 12, 2017 4:43 pm Reply with quote

For a working paper approximation of Pi we (the students on the Mathematics degree course at the University of Sheffield ca. 1993 / 1994) were given the ratio 355/113. This handy approximation is correct to 6 decimal places.

 
GuyBarry
1261344.  Mon Nov 13, 2017 6:39 am Reply with quote

Indeed - it's known as Milü, and is a remarkably close approximation for a fraction with such a small denominator. To get a closer rational approximation you need five digits in the denominator - 52163/16604, which is hardly closer than 355/113. To get seven decimal places, you need 86953/27678.

There's an interesting article here about its unusualness.

 
GuyBarry
1261366.  Mon Nov 13, 2017 9:51 am Reply with quote

Footnote: the renowned Indian mathematician Srinivasa Ramanujan discovered that the fourth root of 2143/22 approximates pi to an astonishing eight decimal places (3.14159265258...). Not a lot of practical use unfortunately!

 
Rob Andrews
1261549.  Tue Nov 14, 2017 1:59 pm Reply with quote

GuyBarry wrote:
Indeed - it's known as Milü, and is a remarkably close approximation for a fraction with such a small denominator. To get a closer rational approximation you need five digits in the denominator - 52163/16604, which is hardly closer than 355/113. To get seven decimal places, you need 86953/27678.

There's an interesting article here about its unusualness.


Thanks. I'd never heard of Milü (I never got to the end of the Mathematics degree!)
I wonder if there is a similar approximation for e?

 
GuyBarry
1261557.  Tue Nov 14, 2017 2:38 pm Reply with quote

2721/1001 gives you six decimal places.

271801/99990 gives you nine - it evaluates to the repeating decimal 2.7182818281828..., and by sheer fluke the decimal expansion of e repeats four digits in the first nine places (2.7182818284590...).

I suspect that rational approximations to e aren't studied so much because, unlike pi, e isn't a ratio, but a base for logarithms.

Do you know anything about continued fractions? There's something very unusual about the value of pi - its continued fraction representation has a very high term early on, namely [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2...] Whereas the continued fraction representation for e is [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1,...]. This has a clear pattern to it, unlike the one for pi, but it means that no high terms occur early in the expansion.

 
spectacles
1267145.  Sun Dec 17, 2017 3:14 pm Reply with quote

bit of a side note from approximations but a few mathematicians, called tauists, think "pi is wrong" (as a concept) and would prefer to use tau, twice pi, as it's a more natural unit that represents the whole 360 degrees of a circle. They've even written a lengthy manifesto and prefer to refer to the 14th March (3/14 in America), agreed on by congress to be Pi Day, as Half Tau Day. However, what they really want is the recognition of the 28th June as Tau Day.

Sources:
https://www.scientificamerican.com/article/let-s-use-tau-it-s-easier-than-pi/
https://tauday.com/tau-manifesto

 
GuyBarry
1267152.  Sun Dec 17, 2017 4:10 pm Reply with quote

spectacles wrote:
bit of a side note from approximations but a few mathematicians, called tauists, think "pi is wrong" (as a concept) and would prefer to use tau, twice pi, as it's a more natural unit that represents the whole 360 degrees of a circle.


I don't think I would say that π is "wrong", but I agree that 2π does seem to crop up more often than π in mathematical formulae. About the only common formula where π appears without the factor 2 is the one for the area of a circle (πr^2).

It would probably have been better historically if the ratio of a circle's circumference to its radius had been taken as the fundamental one; but I can't see centuries of mathematical writing being overturned. It's just one of those little inconsistencies we have to put up with.

 
Baryonyx
1267240.  Mon Dec 18, 2017 9:39 am Reply with quote

Numberphile love this shit:

https://www.youtube.com/watch?v=83ofi_L6eAo

 

Page 1 of 2
Goto page 1, 2  Next

All times are GMT - 5 Hours


Display posts from previous:   

Search Search Forums

Powered by phpBB © 2001, 2002 phpBB Group