How do scientists go about calculating pi to umpteen decimal places?

[RM_Mentock]

In http://www.straightdope.com/classics/a3_357.html
a Dr. Neil Basescu criticizes Cecil thusly

It’s even more absurd that the “learned treatise” you quoted said “thirty-nine places of pi suffice for computing the circumference of a circle girdling the known universe with an error no greater than the radius of a hydrogen atom.” Actually, only 35 places are required. Here’s why: a reasonable value for the radius of the universe is 2 x 10^34 angstroms. That’s just 20 billion years (the time since the big bang) times the speed of light (the upper limit on the rate of expansion). Since pi equals the circumference divided by twice the radius, the uncertainty in pi equals the uncertainty in the circumference (one half angstrom, the radius of a hydrogen atom) divided by twice the radius. That’s (1/2 / (2[2 x 10^34]) or 1/(8 x 10^34) or about 10^-35. Knowing pi to 39 decimal places would nearly suffice for computing the circumference of a circle enclosing the known universe with an error no greater than the nucleus of a hydrogen atom, and that’s a whole lot smaller than the entire atom. I’m sure you’d want to get a thing like that straight.

I’ve done the math three times, and I don’t get the same answer. Instead of 10^35, I get 10^37, which is still only 37 places–but the 3 to the left of the decimal point is another place, and surely you’d throw in just one more place just for good measure. In other words, I think Cecil was right in the first place, and so was his “learned treatise.” Can you all check me on this?

[RM_Mentock]

I meant 10 to the minus 37, of course

[dlv]

I know first hand that the Chudnovsky brothers (who computed pi to a billion zillion decimal places) are pretty weird in many other respects. Not nice people at all.

[handy]

pi [symbol], is a video that explores another way of looking at pi.

[_Df]

This may be a stupid question - so sue me - but regarding Cec’s OC:

The ancient Greeks used a simple method: You draw polygons (e.g., hexagons) around a circle with a diameter of one–one hexagon inside the circle, one out. Calculate the perimeter of the polygons (which is pretty straightforward), take an average, and you get a rough idea of pi. Use polygons with more sides and your approximation of pi gets closer and closer. The mathematician Archimedes got as far as 96 sides, calculating that pi was between 3.1408 and 3.1428.

• Archimedes did this with the Roman numbering system? (The mind absolutely boggles…)

Or, if not, how did the ancient Greeks notate their calculations?

[Arnold_Winkelried]

Archimedes did this with the Roman numbering system? (The mind absolutely boggles…)
Or, if not, how did the ancient Greeks notate their calculations

I don’t think the Greeks would have used Roman numerals, since they were doing mathematics a long time before the Romans.

see thread How did ancient Greeks do mathematics? in the General Questions section.

---

La franchise ne consiste pas à dire tout ce que l’on pense, mais à penser tout ce que l’on dit.
H. de Livry

[RM_Mentock]

Mark Brader, in alt.fan.cecil-adams, has just pointed out that the error can be isolated even further. The single sentence here:

Actually, only 35 places are required. Here’s why: a reasonable value for the radius of the universe is 2 x 10^34 angstroms. That’s just 20 billion years (the time since the big bang) times the speed of light (the upper limit on the rate of expansion).

contains the error. If you multiply the speed of light by 20 billion years, you get 2 x 10^36 angstroms. Surely, the good Dr. Basescu just used the number of angstroms in a centimeter instead of in a meter (just like I almost did five minutes ago). Anyway, the error still sits in the Return of the Straight Dope, p.359.

.

[RM_Mentock]

Update

[CalMeacham]

As I’ve noted before, an excellent read on this topic is The History of Pi by the late Petr Beckmann (That’s not a misspelling – he really did write it “Petr”). My. Beckmann’s history is lively, entertainly, crankily biased (he hated Aristotle, the Roman Empire, and the Soviet Union), and really well-informed on the subject of mathematics and proofs. He takes you from the crude beginnings (how t get pi approximately equal to 22/7 in one easy step) to how you get formulae for calculating pi to a zillion places. And, more important, why you would want to calculate pi to a zillion places when more than a handful is wasted.

[RM_Mentock]

*Originally posted by CalMeacham *
As I’ve noted before, an excellent read on this topic

Sorry, that’s off topic.