source file: mills2.txt Date: Mon, 6 Nov 1995 08:44:23 -0800 Subject: Re: Transcendental Numbers From: td@plan9.att.com bf250@freenet.carleton.ca (John Sankey) guesses: >Gary Morrison <71670.2576@compuserve.com asks >>" What the bloody hell is a transcendental number?! " >Another definition is that it's a number that can not be >calculated by any finite number of arithmetic operations (add, >subtract, multiply, divide), i.e. it's one with an infinite >number of terms in *any* series expansion (not just Taylor). >In other words, the decimals go on forever without repeating >with *any* pattern; this lack of pattern is not a lack of >knowledge, but a logical impossibility. Not quite. A transcendental number is one that is not algebraic. That is, it is not a solution of a polynomial equation with rational coefficients. The smallest non-trivial set closed under a `finite number of arithmetic operations (add, subtract, multiply, divide)' is the rational numbers. While the square root of 12, for example, whose `decimals go on forever without repeating in *any* pattern', is not rational. Neither is it transcendental, being a root of the equation x^12=2. A remarkable fact about transcendental numbers is that, despite their name, they are as commonplace as numbers get -- the sets of rational or algebraic numbers are vanishingly small compared to the transcendentals (first proven by Cantor.) Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Mon, 6 Nov 1995 19:18 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id JAA24935; Mon, 6 Nov 1995 09:18:32 -0800 Date: Mon, 6 Nov 1995 09:18:32 -0800 Message-Id: <9511060916.aa20625@cyber.cyber.net> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu