Trigonometry functions

[Unregistered]

I have always been confused as why "pi" is an irrational number. Since it is the ration of circumference/diameter, it should be a rational number (since all fractions are rational).
Just lately I have come to the conclusion that circumference (or for that matter, even diameter) can be measured EXACTLY. There is always some degree of precission that must be rounded off to. Now, to get the EXACT measurement would mean that you would have to get 'infinitely' precise. Is this why "pi" has infinitely many places and is irrational?
Now, further, how can it be calculated so accurately (are they up to a billion places yet?)? I have tried looking up the answer to this question but the answer is that trigonometry is used. Another method is that an "infinite sum" approach is taken (the arctan series). How is it known that these methods actually converge in on what the REAL value of pi is?
I gather that there are three main methods to calculate pi. So, is this the way we are sure that is the value of pi (by comparing the results of all three methods to make sure they are coming out to the exact same value for each decimal place)? What if one of the methods turns out to have a different value after the one billion billionth place? How could anyone know if that is the real value and the other method(s) are wrong?

[Unregistered]

A rational number is a ratio of INTEGERS. There is no circle that has an integer for a diameter and circumference simultaneously.

[Unregistered]

Thank you.
So, either the diameter AND/OR the circumference WILL not be able to have any EXACT measurement (except if stated in terms of pi) at the same time?
No one has answered my other question yet----
How is it known if the largest value for pi known to date (what is it up to now, one billion places?) is REALLY true? Wouldn't it have to be computed by two or more independent methods before it is really known if the method used to get the largest value really has worked for the decimal computed to?
For that matter, how is it known if all the methods that have been used so far REALLY are giving accurate results or if they are just ending up being random numbers that have nothing to do with what the REAL value of pi is? I mean no one can actually measure a circumference and a diameter that accurately to check to see if the one billionth place of pi REALLY is what has been "computed"...

[Unregistered]

In the field of "numerical analysis" one of the quesitons that is studied is the precison of numerical approximations for various procedures. In particulare if one desires an appoximation that is correct in the first x-decimal places one can set the presison at or above that level. For example if you want an approximation that is valid for the first two decimal places you use an algorythim that gives you accuracy of at least .001. Some algorythms are based on applying a function to a "seed" value (i.e. first guess) and then reapplying the function to the result and continue itterating. The algorythms that use this method typically will get better and better results for each itteration, so my statement earlier about desireing a certain precison is actually easy to meet, you just figure out how many times you need to itterate the prcedure and then do it.

Other methods are more series based where you take a finite portion of the infinite series and use that to give you an approximation. Again you can compute the accurasy of your approximation and it usually will depend on how many terms of the series you keep. So if you need a higher presicion you just keep more terms.

On another note you ask how we know that people are computing approximations to the "real value of pi". This sounds like a definition type question. I'm assuming that you are taking the "real" value of pi to be the mythical figure that is the ratio of circumphrence and diameter of a circle. As you pointed out we can't measure with infinite presision so this definition isn't the most practical. Some other definitions could be half the angle measure of a cricle (measured in radians). Your aforementioned arctan series method. Or as one book I have defines pi, it's twice the smallest positive number such that cos(x)=0. And there are many others. The equivilence of all these definitions can be shown by various methods. Some equivilences needing only a basic trig background to prove, others a first year calculus class, and still others require tools that most people don't learn untill after having a few years of college level math.

If your interested in finding out for yourself how some of these ideas work The book "Numerical Analysis Mathematics of Scientific computing" by Kincaid and Cheney is a well used text on the subject. Portions of the book should be readable by people with only a Calc II level background and a will to learn, and to understand the majority of the text the only other prerequisite is some linear algebra (or matrix theory) knowledge.

[Unregistered]

Excellent analysis.

However, wouldn't a series solution of the arccos(x) be more fruitful? After all, if a series has been proven to be convergent to a certain number of terms, then the series solution of arccos(1/2) should converge to one-third of pi to within that degree of precision.

Just curious.

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I'm no sure of the reason why people tend to use the arctan series to approximate pi instead of another one (like arccos) but time honered tradition and a nod to the direction of leibnitz has kept people using his approximation to pi (via the arctan(1) sereies). I'm too lazy to compute (or even look up) a series expansion for arccos but I do know that the arctan series is beautiful in its simplicity and this may be one reason why people tend to favor it over some others.

[Unregistered]

another reason may be that to retrieve pi from pi/4 (from the arctan approximation) you multiply by 4... (or 2 twice) a task much easier in binary than multiplying by three to get pi from pi/3.

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I'm too lazy to compute (or even look up) a series expansion for arccos but I do know that the arctan series is beautiful in its simplicity

True, if you are using a Taylor series expansion. But they use far more convergent expansions and so I don't think the simplicity has much to do with it. I agree, however, that multiplying by 4 is easier than multiplying by 3, so maybe the arctan(x) is really the best to use after all.