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From: kunzmilan on 9 Aug 2008 04:33
Rational numbers have either a finite number of valid digits,
as 1/2 = 0.5, 50/100 = 0.50,
or if their length is infinite, as 1/3 = 0.33333...,
1/7 = 0.142857142857..., they contain a repeating sequence.
Any change in such infinite sequences, as 0.33133... gives
an irrational number. It is not possible to name
two finite natural numbers giving the changed sequence,
since normal infinite natural numbers can not have infinite number of
digits.
kunzmilan
From: julio on 9 Aug 2008 04:53
On 9 Aug, 09:33, kunzmilan <kunzmi...(a)atlas.cz> wrote:
> Rational numbers have either a finite number of valid digits,
> as 1/2 = 0.5, 50/100 = 0.50,
> or if their length is infinite, as 1/3 = 0.33333...,
> 1/7 = 0.142857142857..., they contain a repeating sequence.

The "period" in their decimal expansion.

> Any change in such infinite sequences, as 0.33133... gives
> an irrational number.

No, any such change would just give another rational number with the
same period. An irrational number is a number with no period, or
otherwise with an infinite period (the two statements are equivalent).

If you are thinking the diagonalization procedure, I think it leaves
any sequence unchanged as to its rationality or otherwise. This anyway
implies some reasoning about infinite sequences.

> It is not possible to name
> two finite natural numbers giving the changed sequence,
> since normal infinite natural numbers can not have infinite number of
> digits.

This one I cannot really parse, but I guess I might have already
answered.

-LV

> kunzmilan
From: smn on 9 Aug 2008 04:59
On Aug 9, 1:33 am, kunzmilan <kunzmi...(a)atlas.cz> wrote:
> Rational numbers have either a finite number of valid digits,
> as 1/2 = 0.5, 50/100 = 0.50,
> or if their length is infinite, as 1/3 = 0.33333...,
> 1/7 = 0.142857142857..., they contain a repeating sequence.
Any change in such infinite sequences, as 0.33133... gives an
irrational

NO .33133... =.331+(.333.../1000) is still rational.You have to be
a;;owed to start the repeated finite sequences of digits after a
finite number of arbitrarily chosen digits to characterize the
rationals.
Also .5=.5000... =.499999... so you needent mention "finite number of
valid digits" smn
> an irrational number. It is not possible to name
> two finite natural numbers giving the changed sequence,
> since normal infinite natural numbers can not have infinite number of
> digits.


From: translogi on 9 Aug 2008 06:32
On Aug 9, 9:59 am, smn <smnewber...(a)comcast.net> wrote:
> On Aug 9, 1:33 am, kunzmilan <kunzmi...(a)atlas.cz> wrote:> Rational numbers have either a finite number of valid digits,
> > as 1/2 = 0.5, 50/100 = 0.50,
> > or if their length is infinite, as 1/3 = 0.33333...,
> > 1/7 = 0.142857142857..., they contain a repeating sequence.
>
>  Any change in such infinite sequences, as 0.33133... gives an
> irrational
>
> NO .33133... =.331+(.333.../1000) is still rational.You have to be
> a;;owed to start the repeated finite sequences of digits after a
> finite number of arbitrarily chosen digits to characterize the
> rationals.
> Also .5=.5000... =.499999... so you needent mention "finite number of
> valid digits" smn
>
>
>
> > an irrational number. It is not possible to name
> > two finite natural numbers giving the changed sequence,
> > since normal infinite natural numbers can not have infinite number of
> > digits.- Hide quoted text -
>
> - Show quoted text -

A rational number is a number that can be described as the quocient of
two natural numbers (except the dividor must be unequal to o)

a is a rational number iff there is are two numbers x and y and ({x
unequal zero) and (a * x = y))

Ra <--> Ex Ey(~(x=0) & (a*x = y))

the natural numbers are zero (0) and all numbers that are one more
than another number.
zero is the amout on things unequal to themselves

one is the successor of zero.
From: A N Niel on 9 Aug 2008 12:24
In article
<dd5cb308-ef26-4bdb-804c-f4b1854a83e9(a)c58g2000hsc.googlegroups.com>,
translogi <wilemien(a)googlemail.com> wrote:

>
> A rational number is a number that can be described as the quocient of
> two natural numbers (except the dividor must be unequal to o)

integers ... you must allow negative integers to get negative rationals

>
> a is a rational number iff there is are two numbers

integers

> x and y and ({x unequal zero) and (a * x = y))
>
> Ra <--> Ex Ey(~(x=0) & (a*x = y))
>
> the natural numbers are zero (0) and all numbers that are one more
> than another number.
> zero is the amout on things unequal to themselves
>
> one is the successor of zero.
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