[net.math] PI is irrational

jlg@lanl.ARPA (12/13/84)

> Sounds neat, but I'm afraid its not possible; at least not unless PI is
> rational.  I seem to remember that the non-rationality of PI has never
> been proven, but the smart money is on that side.

The irrationality of PI was proven in the 19th century (I think Gauss did
it).  Anyway, there is no question - PI is irrational.

augustss@chalmers.UUCP (Lennart Augustsson) (12/19/84)

[empty]

   PI was proved irrational by Johann Heinrich Lambert in 1767,
and more rigorously by Adrien-Marie Legendre in 1794.

   PI was proved transcendental (i.e. non-algebraic, i.e. not the
root of an algebraic equation with a finite number of terms, whose
coefficients are rational) by F. Lindemann in 1882.

-- 

		Lennart Augustsson
		{seismo,philabs,decvax}!mcvax!enea!chalmers!augustss

keithd@cadovax.UUCP (Keith Doyle) (12/20/84)

[]
>The irrationality of PI was proven in the 19th century (I think Gauss did
>it).  Anyway, there is no question - PI is irrational.

It's been too long, I'm afraid I don't even remember exactly what this means.
I'd venture a guess: does it mean that PI is not computed vi an infinite
series (and is that just an approximation?) if not, then how is it computed?
I remember a page in the Time-Life science library where PI is reproduced
in a hundred or so significant digits.  At this point, I don't even know
how to do it the hard(?) way.

Keith Doyle
{ucbvax,ihnp4,decvax}!trwrb!cadovax!keithd

PEREIRA@sri-iu.UUCP (PEREIRA) (12/23/84)

For those interested in learning about the transcendentality of pi, the
squaring of the circle and other results of field theory, I suggest ``Field
Theory and ist Classical Problems'', by Charles Robert Hadlock , The
Mathematical Association of America, 1978, ISBN 0-88385-020-6. I find this a
very readable introduction with minimal prerequisites (basic calculus and
linear algebra).

-- Fernando Pereira

holland@uiucdcs.UUCP (01/04/85)

PI can be computed by as an infinite series.  (There is a relatively
simple one, which converges slowly.)

In fact, any real number can be computed as an infinite series. Let
d(i) be the ith digit after the decimal point and let d(0) be the
integer part (to the left of the decimal point).  Then

number  =  d(0)/10^0 + d(1)/10^1 + d(2)/10^2 + d(3)/10^3 + ...