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Everything contained in the ratio of Pi

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BoogieMonster
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« Reply #15 on: April 09, 2013, 16:42:08 PM »

Since some of these operations leave behind sub-sequences of the original number’s expansion, it should also be clear that an irrational number “contains” other irrational numbers and is in its own turn “contained” in others.

I meant just dropping the decimal point, without further "operations". But yeah, on second pondering I seem to recall that an infinite sequence is a subset of an infinite sequence, so there is no problem of "having space", and since the irrational number is both infinite and non repeating, it means that EVERY infinite sequence of digits would be contained in it, including a sequence of digits that make up another irrational number sans the decimal. By that logic every irrational number would necessarily contain every other. Cool! Thanks.
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brianvds
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« Reply #16 on: April 09, 2013, 18:14:52 PM »

And to paraphrase some or other physicist, anyone who does not find irrational numbers disturbing has not understood them. :-)
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Mefiante
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« Reply #17 on: April 09, 2013, 19:10:34 PM »

I wonder if that means that there is in principle a systematic operation by which one could extract, say, "Macbeth" from pi...
I’m sorry, I don’t follow.  Is this an obscure joke I’m not getting?  I see nothing in what I wrote that would suggest or imply such a thing.  Within the prescribed circumstances, a systematic extraction of digits will typically extract random (or, more accurately, pseudorandom) digit sequences.   Of course, you could always use some existing pattern’s encoding, e.g. the text of a Shakespearean play, as a basis for defining how digits are to be extracted.



I meant just dropping the decimal point, without further "operations".
In that case, you’d just end up with an integer, albeit one that is infinitely long.  The set of all integers is a proper subset of the rational numbers and so cannot be irrational.

'Luthon64
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