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Pi - too many digits

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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

 
Peterzx
1332996.  Sun Oct 13, 2019 10:49 am Reply with quote

This great nmemonic give you Pi to 15 significant figures:

"How I need a drink, alcoholic of course, after the heavy chapters involving quantum mechanics"

 
Jenny
1333097.  Mon Oct 14, 2019 3:31 pm Reply with quote

I like that!

 

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