Is Pi r squared away?

[ultrafilter]

*Originally posted by DSeid *
Math is mainly deductive. We accept a few postulates and given those postulates we see what follows. (I state “mainly” in reference to the Monte Carlo method.)

I seriously doubt that anything other than deductive reasoning has been accepted as a method for theorem proving. If you’re talking about applications, then yes, the Monte Carlo methods are very important for finding approximate solutions. Care to elaborate?

[DSeid]

I am not a mathematician; just a pediatrician who tries to reduce his personal ignorance load when he can. As such, I am dependent on my sources.

“The History of Pi” is the source for that statement, and note that I said “math” not “theorum proving”.

In that book, the Monte Carlo method is characterized as being used to solve a wide range of problems,

…suppose we wish to find a complicated multiple integral … let the computer “shoot” n-tuplets of random numbers. These represent a coordinate in (n-dimensional) space and the coordinate either lies in the volume determined by the integral (“hit”) or it does not (“miss”). Then we let the computer shoot at the target, say, half a million times. The number of hits is then proportional to the n-tuple interval.

Beckmann also states that many theorums were used on the basis of experimental evidence long before being proved. He uses Heaviside and “operational calculus”, later explained by an integral transform and the delta function really being a distribution.

But not for proof. And that’s as good of an elaboration as I am capable of.

[ultrafilter]

*Originally posted by DSeid *
"The History of Pi" is the source for that statement, and note that I said “math” not “theorum proving”.

The two terms are more or less interchangeable. Even solving a simple equation is proving a theorem.

However, for applications, less rigor may be acceptable at times.

**In that book, the Monte Carlo method is characterized as being used to solve a wide range of problems,

…suppose we wish to find a complicated multiple integral … let the computer “shoot” n-tuplets of random numbers. These represent a coordinate in (n-dimensional) space and the coordinate either lies in the volume determined by the integral (“hit”) or it does not (“miss”). Then we let the computer shoot at the target, say, half a million times. The number of hits is then proportional to the n-tuple interval.

Beckmann also states that many theorums were used on the basis of experimental evidence long before being proved. He uses Heaviside and “operational calculus”, later explained by an integral transform and the delta function really being a distribution.**

OK, this does sound more like my understanding.

[Cyberhwk]

Actually someone corrected Cecil here by saying…

“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”

:eek::eek::eek::eek::eek:

[virtually_yours]

“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”

Hey that’s more like it!! that really answers my question.
Ok ie the cosine law ,does that have the same limtitations of pi or is that more “exact” without a bunch of decimal points?I’m asking cause I trying to work something out.
I mean specifically the way that the sine/cosines are determined.as opposed to the way pi is used

[ultrafilter]

Sines and cosines can be determined very accurately through use of their Taylor series. These series converge fairly quickly, and they are what people use to calculate the trigonometric functions.

[RM_Mentock]

*Originally posted by Cyberhwk *
Actually someone corrected Cecil here by saying…

Cecil corrected? Absurd.

I don’t know why Cecil hasn’t corrected that “correction.” Dr. Neil Basescu made a silly error of his own. Notice that he says “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).”

Here’s the math:

2 x 10^10 year times 3.2 x 10^7 sec/year times 3 x 10^8 meter/sec times 10^10 angstroms/meter = 1.92 x 10^36 angstroms, which is a hundred times the value that Basescu calculated.

See this thread for the original discussion of that issue. With reasonable assumptions, Cecil was right in the first place.

virtually yours

I’m not sure this is relevant to your inquiry now, but it seems relevant to the OP. Two hundred years ago, Gauss actually led a survey to study whether the sum of the angles of large triangles add up to 180 degrees. He triangulated some mountain tops. Had they not added up to 180, of course, then pi r squared would have been off also. He didn’t find any discrepancy. And was criticized for a waste of time and money.

[DSeid]

RM,

Of course if that study had been done accurately enough, with mountains seperated far enough, the angles would not have added up to 180 degrees. The surface of the Earth is not a Euclidian plane.

[RM_Mentock]

*Originally posted by DSeid *
**Of course if that study had been done accurately enough, with mountains separated far enough, the angles would not have added up to 180 degrees. The surface of the Earth is not a Euclidian plane. **

Sorry, couldn’t resist. Gauss was no fool, he knew that. But he was using line-of-sight measurements, which would not be influenced by the curvature of the earth’s surface. When you think about it that way, Gauss was probably the canonical no-fool.