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RE: Re: Re: bug in IntegerPart ?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg48003] RE: [mg47988] Re: [mg47970] Re: bug in IntegerPart ?
*From*: "DrBob" <drbob at bigfoot.com>
*Date*: Wed, 5 May 2004 08:11:08 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
Rational numbers are not representable as terminating decimals unless, in
reduced form, the denominator has no prime factors other than 2 and 5.
Representable and unrepresentable rationals have the same cardinality --
they're each countably infinite -- so I won't say MOST rationals are not
terminating decimals. Still, it seems that way when you consider how many
denominators have divisors other than 2 and 5.
All rationals have terminating representations in SOME base (e.g. 1/3 in
base 3), but there's a countable infinity of algebraic numbers (such as the
square root of 2) that aren't rational, and hence don't terminate in ANY
integer base system.
On the whole, algebraic numbers include all the rationals, yet algebraic
numbers are a set of measure zero in the real line.
100% of real numbers (in the measure theory or probability sense) are
transcendental, not even algebraic. None of these terminate in any base.
Spending much time on terminating decimals or BCD arithmetic seems rather
myopic at best, therefore. WRI should (and will, I think) concentrate on
math, rather than base ten arithmetic.
DrBob
www.eclecticdreams.net
-----Original Message-----
From: J. McKenzie Alexander [mailto:jalex at lse.ac.uk]
To: mathgroup at smc.vnet.net
Subject: [mg48003] [mg47988] Re: [mg47970] Re: bug in IntegerPart ?
>> The subset of rationals that can be expressed in decimal isn't
>> especially useful for exact calculation anyway.
>
> Would you be more specific, please? It sounds like a first class
> nonsense but I don't want to jump the gun.
He's referring to the fact that many rationals, such as 2/3, lack a
finite representation in decimal.
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