I think my favorite counterexample to arguments like this is Gabriel's
Horn. Take the function 1/x, and revolve it around the x-axis. You now
have something that looks very similar to a trumpet's bell. Now, find the
volume of this from 0 to infinity. It has a finite volume. However, it
has an infinite surface area. (These can both be determined by
integration, however, it's been long enough since my last Calc class that
I'd probably mess up the integral for surface area, so I won't try...)
Also, spend a bit of time with the concept of limits. Limits can
approach, to an infinitesimally small difference, a finite value. So think
of the circle calculation as a limit problem, something along the lines of:
lim(y->pi)y * r^2 = area of circle.
Since we know the area of the circle to be finite, we know the limit must
be finite. However, this makes no stipulations on the properties of y (and
essentially pi, from that standpoint), so in no way is pi limited to having
an infinite number of decimals.
And then there's just all the standard research on pi. Check out the
arctan function, and some of the newer "distributed calculating digits of
pi" projects on the web for other formulas that can solve for any digit
(or up to and including that digit) of pi. Especially looking at the
summations, it's somewhat apparent that they don't resolve as simply as all
that.
My 1 3/4 cents.
George
At 12:06 AM 2/9/00 -0600, [EMAIL PROTECTED] wrote:
>Hi, I have been considering the possible role pi might play in the
>progression of mersennes. It is generally accepted that the value of pi is
>a never ending series.
>
>But when I look at the circle, the formula for the area of a circle with a
>radius of 6 inches is: A=pi*r^2 = 3.1416 * (6)^2 = 113.0976.
>
>We did not, however, use the full and correct expansion of pi in the
>calculation.
>
>Pi has been figured out to over a billion (not sure of the exact figure)
>digits with no apparent end or pattern.
>
>But when I look at a circle I see a finite area within the circle with no
>means of growing or escape. Logic seems to indicate that pi would have to
>be a finite exact value since the area in the circle is finite.
>
>So, either the figure for pi is in error (not likely) or pi has a end.
>
>The end.
>What say ye?
>Dan
>
>
>
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