Subject:
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Re: 22/7 & infinities (was: Re: The nature of the JC god, good or evil?)
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Newsgroups:
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lugnet.off-topic.geek
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Date:
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Mon, 23 Aug 1999 03:54:09 GMT
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Viewed:
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1369 times
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In lugnet.off-topic.geek, "Simon Robinson" <simon@simonrobinson.com> writes:
> > Note that there are some irrational numbers which can't be expressed
> > algorithmically in terms of rational numbers. I forget if this subset is a
> > larger or a small infinity than the ones which can be, but it's a really
> > fascinating subset of irrationals which gets into random number territory.
>
> You mean the transcendental numbers, presumably? I seem to recall
> reading that practically every number along the real number line
> is transcendental, with just odd blips where you encounter an isolated
> rational or irrational number. But I may be mistaken on that - and that
> alone doesn't prove the transcendentals constitute a bigger infinity
> than the irrationals.
Well, since transcendentals are a subset of the irrationals, so there's no
way that transcendentals can constitute a bigger infinity than the
irrationals. All transcendentals are irrational (but the converse isn't
true). pi is transcendental but sqrt(2) is not -- but both are irrational.
I'd forgotten about transcendentals -- and thanks for bringing it up -- but
I was talking about numbers which can't be expressed _algorithmically_ (as
opposed to those which can't be expressed _algebraically_). sqrt(2) can be
expressed both algebraically and algorithmically. Pi cannot be expressed
algebraically but it can still be expressed algorithmically.
So I mean the numbers which are so obscure and random in their infinite
expansions that they're not even definable (i.e., computable) by a
finite-length[1] computer program
--Todd
[1] That is, counting the total number of instructions in the program
itself, not the total number of instructions executed by the executor of the
program. A program to compute pi exactly, for example, requires that an
infinite number of instructions be executed, but the program itself can
still be very short.
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