Re: "Numeric overflow"?
Re: "Numeric overflow"?
- Subject: Re: "Numeric overflow"?
- From: Michael Sullivan BCE <email@hidden>
- Date: Mon, 19 Sep 2005 16:22:33 -0400
On Sep 15, 2005, at 11:15, Doug McNutt wrote:
At 20:45 -0400 9/14/05, deivy petrescu wrote:
So, when one says Pi is irrational, one says that Pi is certainly
*not* equal to 22/7.
As long as we're being pedantic, Pi is indeed not rational but it is
also not irrational in the technical jargon of arithmetic - and
perhaps of AppleScript but one never knows. . ..
An irrational number must be representable as a root of an algebraic
polynomial. Pi is not - the circle cannot be squared. Numbers which
are exactly representable only as an infinite series are said to be
transcendental.
To be even more pedantic, that's not correct either. You are
describing algebraic numbers, which is the set of roots of polynomials
of finite degree (with integral coefficients and powers).
Irrational numbers are defined as being NOT rational (of the form p/q
with p,q integers), and include all transcendental numbers (NOT
algebraic) as well as many algebraic numbers. IIRC, the algebraics
turn out to be countable. (you can do an n-dimensional ordering similar
to triangulating rationals by modeling them as ordered ntuples -- the
coefficients of the polynomial of which they are a root followed by the
root number. As long as the polynomials can't have infinite degree,
they are countable)
Anyway, transcendental numbers are a subset of irrationals. They are
not disjoint. I'm not aware of a one-word term for irrationals which
aren't transcendentals, but irrational isn't it.
see <http://mathworld.wolfram.com/IrrationalNumber.html>
Michael
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