"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).

-- 
Henry Goodman
henry dot goodman at virgin dot net