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Re: Finitely definable reals.
Posted:
Jan 15, 2013 9:32 AM
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On Tuesday, January 15, 2013 4:55:29 AM UTC-8, WM wrote:
> On 14 Jan., 23:46, Virgil <vir...@ligriv.com> wrote:
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> > > In my opinion everything that in mathematics can be used to express
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> > > 1/3 as a decimal fraction is "all its finite digits". That means, only
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> > > in the infinite we obtain 1/3, but a better phrase describing "the
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> > > infinite" is simply "never".
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> > Thus WM throws out every limit process, including all of canlculus.
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> That seems only so to unknowledgeable persons. Limit processes existed
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> long before Cantor.
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> > > We will never obtain 1/3 as a decimal.
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> > Certainly WM won't, but calculus will.
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> Why then does calculus never publish his/her/its? the achievement?
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> REgards, WM
Look, every decimal expansion of a rational number has a terminal repeating
sequence. This isn't calculus but the study of limits. It's high school
stuff, pre-calculus. It's how we turn a decimal expansion back into a
the ratio of two integers. The key is the length of the terminal repeating
sequence. For instance if it is 4 digits long then one can multiply
by 10,000 then subtract the original, thereby reducing the repeating
terminal sequence to zeros.
For instance 0.1212[1234] (the terminal sequence is 1234 and it continues
without bounds) gets computed as (1212.1234[1234] - .1212[1234])/9999
or 1212.0022/9999 or 12120022/99990000 then factorization can happen and
the fraction reduced to mutual primes. Only the infinte expansion can
be treated this way because the repeating sequence has to line up with
itself.
.[9] = (9.[9] - .[9])/9 = 1
.1[0] = (1.0[0] - .1[0])/9 = .9/9 = 9/90 = 1/10
(with repeating terminal sequence [0] one can just skip the subtraction)
Every finite expansion of a rational whose terminal repeating sequence
isn't [0] will not equal the rational. Only the infinite expansion will
do where the termial repeating sequense is identified as above.
You know this but are just playing games. You also know that some reals
are irrational and do not have a termial repeating sequence. Just as
with the rationals only the infinite expansion will equal itself--all elese
are approximations. For most human purposes sufficiently close approximations
are good enough even if they don't equal the number being approximated.
Most mathmaticians will leave the irrationals alone and eliminate them if
they can through the factor reduction process ( 10e/7e = 10/7) and
thereby retain accuracy.
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