Re: ga10- Integers found that form the sqrt(2) or transcendental numbers.
From: Guillermo Arango (abril10_at_geo.net.co)
Date: 01/02/05
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Date: Sun, 2 Jan 2005 16:37:21 +0000 (UTC)
On 01 Jan 2005, Timothy Little wrote:
>Guillermo Arango wrote:
> On 31 Dec 2004, Timothy Little wrote:
>>I do have one question: what is your representation for 10 sqrt(2),
>>and how does it differ from that for sqrt(2)?
>
> I am not quite sure about your question, but in any case.
>
> 10 sqrt(2)= 14,142135623730950488016887242097...
> sqrt(2)= 1,4142135623730950488016887242097...
Show me the fraction of integers that gives sqrt(2). Then show me the
fraction of integers that gives 10 sqrt(2). Then tell me how they
differ.
ANSWER:
using the following formula:
pm/qm = r
where,
p = all the digits of r with no decimal comma.
q = 10^n-1
r = any real number.
n = the number of all the digits r has.
m = any integer different from 0
e.g.
p = 14142135623730950488016887242097...
q = 10^n-1
r = sqrt(2)
n = the number of all the digits r has.
m = 1
(14142135623730950488016887242097...)(1) / (1000000000000000000000000000000...)(1)
= 1,4142135623730950488016887242097... sqrt(2).
the numerator and the denominator have the same number of decimal places.
(14142135623730950488016887242097...)(10) / (1000000000000000000000000000000...)(1)
= 14,142135623730950488016887242097... 10 sqrt(2).
the numerator has 1 more decimal place than the denominator.
note:
when you are using the sequence I introduced earlier, you know that the denominator has the
same number of decimal places after the comma than the numerator because you start at the
decimal comma to construct the integers until you have no comma (the limit of the sequence).
> the sequence follows,
> 42,42640687119285146405066172629.../14,142135623730950488016887242097...
>= 424,2640687119285146405066172629.../141,42135623730950488016887242097...
>= 4242,640687119285146405066172629.../1414,2135623730950488016887242097...
>= ... = 3.000000000000000000000000000000... rational number
> we find that the limit of the numerator and denominator must be
> whole numbers, hence integers, by definition because the quotient is a rational number.
The definition says that if you divide an integer by a nonzero
integer, you get a rational. It does not say that if you divide x by
y and get a rational then x and y must be integers. Using a
definition the wrong way around can lead to incorrect conclusions.
ANSWER:
if you agree that in the example x = 4242640687119285146405066172629... and
y = 14142135623730950488016887242097... are whole numbers, and that
3.000000000000000000000000000000... is a rational number then,
you are dividing an integer by a nonzero integer, and getting a rational. (look at the end)
Also, we have two sequences (of numerator and denominator). Not every
sequence has a limit. In fact, it is fairly easily shown from the
definition of limit that these sequences do not have limits. Hence
any conclusions about the properties of their limits are vacuous.
- Tim
ANSWER:
because in the sequence I start with irrational numbers, it is fairly easy to imagine that
the sequence has no end. but let us work with the above example:
x/y = r
x = 4242640687119285146405066172629... {a multiple of sqrt(18)}
y = 14142135623730950488016887242097... {a multiple of sqrt(2)}
r = 3.000000000000000000000000000000... (also integer 3)
but also x/r =y
4242640687119285146405066172629... / 3.000000000000000000000000000000...
= 14142135623730950488016887242097...
three whole numbers, even 3.0000... (try all the properties of integers on the numerator, denominator
and the quotient and see how they all apply)
take y = 14142135623730950488016887242097... and divided by the integer
10000000000000000000000000000000... and you get sqrt(2). the integer in the denominator
has exactly the same number of decimal places than the numerator.
furthermore:
rational numbers are the quotient of ONLY a ratio of integers different from zero in the denominator. Please
look at this part, it is important.
In mathematics, a rational number (or informally fraction) IS A RATIO OF TWO INTEGERS, usually written as the vulgar fraction a/b, where b is not zero. The set of all rational numbers is denoted by Q, or in blackboard bold . Using the set-builder notation is defined as such:
Q = {m/n : m belongs to Z, n belongs to Z, n different from zero}
Each rational number can be written in infinitely many forms, for example 3 / 6 = 2 / 4 = 1 / 2. The simplest form is when a and b have no common divisors, and every rational number has a simplest form of this type.
The decimal expansion of a rational number is eventually periodic (in the case of a finite expansion the zeroes which implicitly follow it form the periodic part). The same is true for any other integral base above 1. Conversely, if the expansion of a number for one base is periodic, it is periodic for all bases and the number is rational.
source: wikipedia
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