Constant Failure

[pokerfan]

From physicsworld.com

Have we defined our fundamental constants with maximum efficiency? Robert P Crease invites your comments



In Proposition 3 of On the Measurement of the Circle, Archimedes asserts, based on calculations involving regular polygons circumscribed around and inscribed in a circle, that “the ratio of the circumference of any circle to its diameter is less than 3 1/7 but greater than 3 10/71”. He thereby strongly reinforced, if he did not actually create, the tradition of considering that ratio, two millennia later referred to as π, to be fundamental.

Was Archimedes wrong?

When I was in elementary school, my teachers told me that π was the most important number in mathematics, which scared me into cramming as much of it as I could into my brain. I got to all of 35 places. This many places actually turned out to be harmful to my science career, for whenever I used all of them in calculations in high-school physics, I was penalized for something my teacher called “misplaced precision”.

http://physicsworld.com/cws/article/print/32679">More

[Muttocracy]

1. The problem would not be in using extra decimals in pi, but in the final answer.

Just because you use 30-some digits for pi doesn't mean your final answer is extra-precise - it's constrained by the number of significant digits in the other factors in your calculation.

If you do a sloppy job measuring the diameter of a circle to the nearest inch, your calculation of the circumference can't suddenly be down to the millimeter just because you used a lot of digits of pi.

[pokerfan]

2. Measure with a micrometer, mark with chalk, cut with an ax

Precision and accuracy are different things, of course.

Pi to 39 decimal places allows one to compute the circumference of the entire universe to the accuracy of less than the diameter of a hydrogen atom.

[pnorman]

3. Pi-lovers anyone?

Here's an algorithm I had found in a computer magazine a long time back, for the computation of Pi:

p=2: k=1
start:
s=sqr(s+2)
p=2*p/s
print p
k=k+1
goto start

Nothing more exacting than square root calculations!

I have yBasic in my cellphone, and it converges to the correct value (3.14159267) in 15 steps. I had another PDA that had double-precision basic, and it calculated correctly to about 16 decimal places. My gut feeling is that the algorithm would work at any precision available, but there's nothing on the web to help me. Without exception, all the "classic" formulas are incredibly complex, and usually slow in converging. Can anyone here give me some ideas?

pnorman

[pokerfan]

4. Look up the Gauss–Legendre algorithm

Converges extremely quickly.

[pnorman]

5. Thanks! That's getting me much closer to what I was looking for.

Here's one hit: http://nostalgia.wikipedia.org/wiki/Gauss-Legendre_algorithm I'll do some more Googling from there.

What bothered me was that, after seeing all those incredibly complex formulas on the web (and in my books), that EXTREMELY "simple" algorithm couldn't possibly work out to higher precisions. But I'm a lot more confident now!

pnorman
PS: That yBasic (Hotpaw Basic) may be of interest to any Palm OS fans on DU. Check it out here:
http://www.hotpaw.com/rhn/hotpaw/

[Dr. Strange]

6. I've often wondered the same thing about pi.

The correct constant really should be 2pi.
I attended a talk given by a professor back in my grad school days, in which he argued that an alien race, even if it did not have any concept of geometry, would still discover pi. (He didn't mention that it would be 2pi, but it seems more likely than the aliens coming up with pi.)

[caraher]

7. I'll have to think about this one...

Reminds me of a fellow student in grad school (physics) who was fond of referring to factors of "pi-bar" (which is, of course, one-half; if "h bar" is h/(2 pi), then "pi bar" must be pi/(2 pi)...)

Now how would 2 pi be written in base 12? ;)

[BlooInBloo]

8. Nothing like people who know nothing about numerical analysis talking about numerical analysis.