Quoting from Dan: "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."
No, this might be called one of the pathologies of mathematics. What seems
to be so isn't. It is certain that Pi is a "never ending series" as you put it.
Perhaps this will help. Sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... to
infinity as the fractions become smaller and smaller (1/(2^n) as n increases
without limit). The sum is 2. Can you Dan accept that a never ending sum
of smaller and smaller terms has a precise finite value? It is an integer
at that. Plus we never get to add all the terms -- there is always just
one more and it would take infinite time to add the infinity of terms.
Now look at 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ... to infinity as the
fractions become smaller and smaller. The sum is infinity, that is, it
never stops increasing. Can you Dan accept that this sum of smaller and
smaller terms has no precise value as it slowly and endlessly grows larger
(hence infinity)? And there is a neat or "pathological" property of this
infinitely large sum. Let's say we get larger than integer N after a
million billion billion terms. So now we are adding 1/(million billion
billion + 1), then 1/(million billion billion + 2), then 1/(million billion
billion + 3), then + 4 etc. As the sum crawls toward N + 1, we are less
than .000 000 000 000 000 000 000 001 away from N + 1 eventually (an
American billion being 9 zeros). The sum never "lands on" exactly N + 1 and
skips landing on all integers (and there are an infinity of those). In
fact, to get as close as one wants, say .000 (million more zeros) 01, to an
integer, another N' is needed where N' is >> N (N' is much larger than N).
The partial sum can be made as close to an integer as we like. But the
partial sum is never exactly an integer.
In other words, for all integer M, the fixed sum 1 + 1/2 + 1/3 + 1/4 + ....
+ 1/M (a fixed sum because M is the last one in a finite summation) is never
an integer even though the infinite series (as M grows to infinity) passes
through all integers.
The better mathematicians in this group (that is, all other mathematicians
:-) may give a better explanation.
At 12:06 AM 2/9/00 -0600, Dan 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
>
_________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
Mersenne Prime FAQ -- http://www.tasam.com/~lrwiman/FAQ-mers
