Re: ga4- Integers found that form the sqrt(2) or transcendental numbers.

From: Guillermo Arango (abril10_at_geo.net.co)
Date: 12/29/04 

Date: Wed, 29 Dec 2004 13:46:49 +0000 (UTC)

On 27 Dec 2004, wrote:
>GA2,
>
>There's no logic in your statements. The statement that "reals are the
>quotient of two integers" borders the opposite of the accepted
>definition of "reals", and "Therefore all reals appear as the ratio of
>a real and an integer" is a corollary of the group of real numbers, not
>a definition of "reals". In short, one does not define a word (IE
>"reals") by referencing that word.
>
>In your other post, you refer to expanding the decimal of the numbers
>(IE sqrt(2)) until there is no more decimal expansion and thus one
>arrives at a quotient of integers. Last time I checked, sqrt(2) was an
>irrational number, and thus has no finite decimal expansion. As such,
>the assumption that it has renders the argument mute.
>
>You seem to have a decent grasp on integers and rational numbers.
>However, I would suggest reading a bit more about Real Numbers.

Thank you for your comment. My response to you is as follows:

A specialist wrote to me once this:
"The problem is that there will never be a moment when there is no longer
a decimal fraction if the number r was not rational to start with ...
A slight modification of your argument can be used to show that any
real number is the limit of a sequence of rational numbers - but that
is a classical and well-known result."

and I answer this:
In your answer you require that in my sequence r must be rational to start
with in order to have a decimal expansion to end. There is a way to do that
using the same example I used to show there are integers that have the
quotient of sqrt(2).

p/q = r = sqrt(18)/sqrt(9) = sqrt(2) but also, p/r = q = sqrt(18)/sqrt(2) =
sqrt(9)
in decimal numbers we have:
4,242640687119285146405066172629.../ 1,4142135623730950488016887242097... =
3 and
4,242640687119285146405066172629.../ 1,4142135623730950488016887242097...
= 3.000000000000000000000000000000... rational number

using p/q = r,
p/q = 10p/10q = 100p/100q = 1000p/1000q = ... = np/nq = r

the sequence follows,
42,42640687119285146405066172629.../14,142135623730950488016887242097...
= 424,2640687119285146405066172629.../141,42135623730950488016887242097...
= 4242,640687119285146405066172629.../1414,2135623730950488016887242097...
= ... = 3.000000000000000000000000000000... rational number

In this case, r is a rational number as require in your argument. The case
here is that continuing my sequence then q is an integer, because at the end
of the sequence it has no decimal expansion. But q is now an integer
multiple of sqrt(2) and this means:
14142135623730950488016887242097... / 10000000000000000000000000000000...
= sqrt(2)
Are not these two infinite integers (with the same number of digits) having
the quotient sqrt(2)?

multiplying numerator and denominator by 3, then:
4242640687119285146405066172629... / 30000000000000000000000000000000...
= sqrt(2)
As you can see we obtain the same integer numbers I initially got when I
used the sequence the first time. The conclusion is that r does not need to
be a rational number after all.

e.g.: irrational quotient as a result.
p = sqrt(18)
q = sqrt(9)
r = sqrt(2)
p/q = r = sqrt(18)/sqrt(9) = sqrt(2)
in decimal numbers we have:
4,242640687119285146405066172629.../3 = sqrt(2)
the sequence follows,
   42,42640687119285146405066172629.../30
= 424,2640687119285146405066172629.../300
= 4242,640687119285146405066172629.../3000
= ... = sqrt(2)

the sequence continues until there is no decimal expansion and a fraction
with two integers np/nq construct the quotient r.

This is precisely what I am intending to prove, that irrational and transcendental numbers are indeed rational numbers.

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