[net.math] Transcendental Pi

robertj@garfield.UUCP (Robert Janes) (12/18/84)

n upon the irrationality of Pi brings to
	mind a further fact relevant to this number.Not only is Pi
	irrational but furthermore it is transcendental.That is to
	say that there does not exist a polynomial which has only
	integeral co-efficents having Pi as a root.(It can also be 
	said that Pi is not an algebraic number I believe).

	Another number of this nature is e (euler's number)as well
	as the number 0.110001000..1000..010.. which has 0's in every
	place except the k!th place where k is an integer.I would 
	appreciate two things:

		1.Refences to the proof of the proof of the transcendentality
		  of Pi.

		2.References to any other interesting transcendental
		  numbers.(I'm told that the cardinality of the transcendentals
		  is the same as that of the reals so please do not send
		  any old example,just interesting ones(what ever that may
		  be)).

	Please send by mail!

	I'm also told that the fact that Pi is transcendental makes the old
	problem of constructing a square with the same area as a given 
	circle using only a straightedge and ruler impossible.An explan-
	ation of why this is so would be welcome.

	I stand to be corrected on any of the above statements.

Thanks:Robert Janes
       Memorial University of Newfoundland.

mccaugh@uiucdcs.UUCP (12/30/84)

  To: robertj
  Re: Impossibility of squaring the circle:

  I am certainly not the best source for responding to this problem, but on
  seeing that no response had not yet been made, here goes:
  i] The construction constraint pertains to straightedge and compass, NOT
     straightedge and ruler (in fact it is fairly trivial to trisect an
     arbitrary angle given a ruler, but impossible without);
 ii] The "squaring" problem--if soluble--would amount to the ability to con-
     struct a square with area equal to that of a circle with radius 1, i.e.,
     a square satisfying: side**2 = pi, and so a line-segment (the side of
     the square) could be constructed of length side = pi**0.5, which is
     also transcendental, since pi is, but:
iii] the only constructible line-segments must have as lengths algebraic
     numbers; in fact the constraints are far stricter--e.g., cube-root of 2
     is not constructible (otherwise it is easy to show that an arbitrary
     angle can be trisected).

 Towards the end of his famous text: "Algebra", the noted authority Jacobson
 (whom some say is to Algebra what Feller was to Probability) discusses the
 subject of constructibility (but not in depth) just before his launch into
 Galois Theory...Altgeld Hall's Math Library here at Urbana has several mono-
 graphs on the subject, including one on all the famous unsolvable problems
 (which includes an interesting--if laborious--complete construction of the
 17-gon using Gauss's original construction and verification). I will be most
 happy to elaborate upon the "squaring" problem (and related problems) if
 sufficient interest accrues.

 --uiucmsl!mccaugh