**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