| Subject: Re: Astropulse and BOINC |
| From: f/f george |
| Date: 31/05/2004, 14:36 |
On Mon, 31 May 2004 10:14:56 +0100, "Henry Goodman"
<henry.goodman@virgin.net> wrote:
"Odysseus" <odysseus1479-at@yahoo-dot.ca> wrote in message
news:40BA7B8A.89FB7514@yahoo-dot.ca...
Henry Goodman wrote:
"baskitcaise" <baskitcaise@hotmail.com> wrote in message
news:2huff2Fh6uduU1@uni-berlin.de...
Henry Goodman adjusted his tin foil beanie and asbestos
underwear to
write:
Um, Pi is transcendental. That means it never finishes or
repeats.
Not quite. See below.
Has that been proven yet?
Yes. I think in the 18th century. (Hardy's "Pure Mathematics "
says
that pi was proved to be transcendental by Lambert in1761)
No; that was Ferdinand Lindemann, in 1882. Johann Lambert's proof
was
of the *irrationality* of pi -- but you're right to mention his
result in this context because it indeed established that pi "never
finishes or repeats". The transcendental numbers are a subset of the
irrationals; what distinguishes them is that they are not roots of
any algebraic equation. The square root of 2 is an example of an
irrational number that is *not* transcendental; while its decimal
representation never repeats, it *can* be (easily) constructed with
a
compass and straight-edge. OTOH 'squaring the circle' is impossible
because pi is transcendental.
See <http://mathworld.wolfram.com/Pi.html> and
<http://mathworld.wolfram.com/TranscendentalNumber.html>.
Very interesting. Your sources certainly say that Lambert only proved
pi to be irrational. When I read pure maths at Cambridge some 50 years
ago Hardy was considered to be the main text book (I notice it is
still available on Amazon). On rereading page 70 of Hardy I agree that
he says Lambert proved only that it was irrational (by means of
continued fractions).
Um look here:
http://mathworld.wolfram.com/Pi.html
In part it says:
Pi is known to be irrational (Lambert 1761; Legendre 1794; Hermite
1873; Nagell 1951; Niven 1956; Struik 1969; Königsberger 1990;
Schröder 1993; Stevens 1999; Borwein and Bailey 2003, pp. 139-140). In
1794, Legendre also proved that Pisquared is irrational (Wells 1986,
p. 76). Pi is also transcendental (Lindemann 1882). An immediate
consequence of Lindemann's proof of the transcendence of also proved
that the geometric problem of antiquity known as circle squaring is
impossible. A simplified, but still difficult, version of Lindemann's
proof is given by Klein (1955).
The actual web page has symbols and links.