(If you think this discussion is off topic, jump directly to the end and read the PS for some metric stuff.)
On Jun 14 Stephen Davis wrote asking if anyone could confirm his friend's claim that Leibniz had "solved Pi using fractions in the 1670's", specifically:
Pi = 4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15 + 1/17 - 1/19 etc...)
According to a great book I have by Isaac Asimov, this is correct, with the possible exception of some minor quibbling about the meaning of the word "solved". (More about Asimov's great book in a PS, below.*)
According to Dr. Asimov, Gottfried Wilhelm von Leibniz developed this equation (involving an infinite series of terms) in 1673. However, I would point out that the equation involves an INFINITE series of fractions to be added and subtracted. Since no one can actually perform the necessary infinity of operations necessary, it is debatable whether or not this equation actually "solves" Pi; it certainly does not give Pi as a specific number that can be written. However, that doesn't make it less useful or important.
One can get a good approximation of Pi by calculating just the first few terms. By using more and more terms one can get better and better approximations to Pi. But one can do the same thing with decimals. One can write Pi as 3.14 and that's pretty close, or one can use more and more decimal places and write 3.1415926 and that is much closer BUT IT'S STILL NOT EXACT.
So if Stephen's friend was arguing that common fraction are better than decimal fractions (and hence decimally based metric measures are not as good as systems based on common fractions), he is wrong. Both the decimal representation and the common fraction representation require an impossible infinity of terms in order to be exact and both can be used with a few terms to get an approximate result. Indeed, one can get a more accurate value of Pi using just ONE decimal place than you can get with the first TEN additions and subtraction in Leinbiz's series above. (The decimal form of Pi, to ONE decimal place, is: 3.1 while the common fraction series using TEN fractions in Leibniz's series, gives a result of 3.04 .) Stephen's friend claims that the series is "amazingly accurate", but he gets such accuracy only by using 2^24 fractions (that's almost 17 MILLION terms!). The calculation had to be done by computer, of course. The only thing I find amazing is how many terms are needed before getting any accurate value at all.
Interestingly, a completely different approach to determining Pi, ALSO using an INFINITE number of steps, was developed in ancient Greece by Archimedes (287 to 212 BCE). Hi method involved finding the perimeter of a regular polygon inscribed inside a circle with a diameter of one unit (first a triangle, then a square, then a pentagon, then a hexagon, etc.). He showed that the more sides the polygon had, the closer to Pi would be the result. For an infinite number of sides the perimeter of the polygon would be the same as the circumference of a circle and would therefore equal Pi. But to find it, one would need to find the perimeter of an infinite-sided polygon. Archimedes did it for a polygon of 96 sides and got an answer of 3.1408 . Stephen's friend needed almost 17 million terms to get an accurate result out of Leibniz's equation.
It has since been proven (in 1761 by Johann Lambert, that Pi is an irrational number, one which cannot be represented by an exact decimal except by the impossible process of writing an infinitely long decimal fraction. Similarly, Pi cannot be represented by common fractions without using an infinite series of them, nor can it be represented geometrically except by an infinite sided polygon. ANY effort to represent Pi exactly requires an infinite number of steps. An exact, written value of Pi is impossible.
Regards, Bill Hooper
*PS
Asimov's book, "Asimov on Numbers" (copyright 1977, publ. by Pocket Books) contains the essay "Pieces of Pi" which was originally published in a magazine in May of 1960. The book is old (and in some ways dated) but still marvelous. I'd recommend it to any of you. Two other essays in the book are of special interest to readers of this list: "Forget it!" and "Prefixing It Up". Both extoll the virtues of the metric system. The things he recommends we "forget" in the essay "Forget It!" are all of the Olde English units of measure. And the prefixes referred to in "Prefixing It Up" are -- you guessed it -- the prefixes of the metric system.
