A definition of rational numbers
kunzmilan
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
elsiemelsi
tell us what number are
all their definitions are impredicative
just like yours
it is circular and self referential thus meaningless
all it says is a number is a number
quote
"Rational numbers have either a finite number of valid digits [ie
numbers]"
,all it says is a number is a number
thus meaningless babble
go read deans book
http://gamahucherpress.yellowgum.com/books/philosophy/Absurd_math_science4.pdf
The absurdities or meaninglessness of mathematics and science: paradoxes
and contradiction in mathematics and science which makes them meaningless,
mathematics and science are examples of mythical thought, case study of
the meaninglessness of all views
--
Message posted using http://www.talkaboutscience.com/group/sci.logic/
More information at http://www.talkaboutscience.com/faq.html
ju...@diegidio.name
> 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
elsiemelsi
An irrational number is a number with no period,
isay
again you cant tell us what a number is
all this is circular rubbish
untill you tell us what a number is we dont know what an irrational oR
rational number isSIMPLE
smn
> 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
ju...@diegidio.name
> you say
>
> An irrational number is a number with no period,
>
> isay
>
> again you cant tell us what a number is
> all this is circular rubbish
Can you count?
If you can count, then that's a natural number.
I guess even Colin Leslie would agree on this one.
-LV
translogi
>
> - 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.
elsiemelsi
Can you count?
If you can count, then that's a natural number.
I guess even Colin Leslie would agree on this one.
i say
that does not tell us what a number is - which colin leslie would say
until i know what a number is
i cant know what a natural
or
rational
or
irrational
number is
Frederick Williams
> quote
> "Rational numbers have either a finite number of valid digits [ie
> numbers]"
> ,all it says is a number is a number
> thus meaningless babble
"digits i.e. numbers" is silly because numbers and digits are quite
different things.
--
He is not here; but far away
The noise of life begins again
And ghastly thro' the drizzling rain
On the bald street breaks the blank day.
Frederick Williams
>
> you say
>
> Can you count?
Please reply to Julio's post not Kunzmilan's.
elsiemelsi
digits i.e. numbers" is silly because numbers and digits are quite
different things.
i say
you dont know the definition of digit
http://www.thefreedictionary.com/digit
a. One of the ten Arabic number symbols, 0 through 9.
b. Such a symbol used in a system of numeration.
clearly a digit is a number
A N Niel
<dd5cb308-ef26-4bdb...@c58g2000hsc.googlegroups.com>,
translogi <wile...@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
fishfry
<dd5cb308-ef26-4bdb...@c58g2000hsc.googlegroups.com>,
translogi <wile...@googlemail.com> wrote:
> On Aug 9, 9:59 am, smn <smnewber...@comcast.net> wrote:
> > On Aug 9, 1:33 am, kunzmilan <kunzmi...@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)
>
Still imprecise. Are 1/2 and 2/4 different rational numbers, or the same
rational number?
Dave
> Still imprecise. Are 1/2 and 2/4 different rational numbers, or the same
> rational number?
The definition doesn't speak to that. It just asserts that if 2*x = 1
and 4*y = 2, then x and y are rational numbers, which we can represent
as ordered pairs, (1, 2) and (2, 4), or more commonly, as fractions,
1/2 and 2/4. That 1/2 and 2/4 are members of an equivalence class
(that we can call the rational number 1/2) is a deduction that follows
from a definition of equality of fractions: two fractions a/b and c/d
are equal if and only if a*d = b*c.
Dave
porky_...@my-deja.com
so you define 1/2 = 0.5, without defining 0.5 or explaining why 1/2 =
0.5.
The standard definition of rational numbers as an ordered pair (a,b)
where a is either natural number or 0 and b is natural number, with
appropriate rules for addition and multiplication, works just fine for
everyone.
The issue of finiteness of decimal expansion of rational numbers comes
as a consequence of their definition, not as *their* definition.
Please before you start reinventing the wheel, at least check what
already has been done.
Virgil
<e379f14a-8947-4017...@m45g2000hsb.googlegroups.com>,
kunzmilan <kunz...@atlas.cz> wrote:
A real number r is rational if and only if there is some non-zero
integer s such that r times s is integral.
MoeBlee
> The absurdities or meaninglessness of mathematics and science: paradoxes
> and contradiction in mathematics and science which makes them meaningless,
> mathematics and science are examples of mythical thought, case study of
> the meaninglessness of all views
So the mathematics and science used to build the computer you're using
is meaningless?
MoeBlee
Frederick Williams
>
> you say
>
> digits i.e. numbers" is silly because numbers and digits are quite
> different things.
>
> i say
> you dont know the definition of digit
>
> http://www.thefreedictionary.com/digit
> a. One of the ten Arabic number symbols, 0 through 9.
>
> b. Such a symbol used in a system of numeration.
>
> clearly a digit is a number
You're not posting your replies in the right place.
Digits are symbols and thus concrete, numbers are abstract. Look:
'2' is a digit that denotes a number
'II' is a pair of digits that denote the same number
Just because 2 = II it does not follow that '2' = 'II'.
kunzmilan
Thanks for your interest and replies. I started badly. Thus, once
again:
We construct lists of rational numbers.
In the first list of rational numbers, we find 9 numbers 0.1 till 0.9.
In the second list of rational numbers, we find 99 rational numbers
0.01 till 0.99.
In the third list of rational numbers, we find 999 rational numbers
0.001 till 0.999.
In the n-th list of rational numbers, we find (10^n - 1) rational
numbers with n decimal places.
If we add a number to an existing rational number (by any technique)
from n-th list, we find it in the (n + 1)-th list.
Since some rational numbers are infinite sequences, there must exist
infinite many lists of rational numbers to include all these nonending
rational numbers.
Now, a question can be posed, if 0.5 and 0.50 should be considered as
identical or different rational numbers.
If yes, the last list counts all numbers lesser than 1.
If not, rational numbers are counted by the sum of infinite serie 9 +
99 + 999 + ...
Another question is, if all numbers in the last list are really
rational, since it will be difficult to find two finite natural
numbers corresponding to them.
kunzmilan
translogi
>
> - Show quoted text -
> Another question is, if all numbers in the last list are really
> rational, since it will be difficult to find two finite natural
> numbers corresponding to them.
there is even one finite natural number corresponding to them
if your number is a/b (and a is smaller than b)
(so in your case b= 10, 100, 1000 and so on )
N= b*b +a will uniquely identify one of them
(there are other formula's possible as well i know that Paut Smiths
book mentiones another one)
(Am i promoting here again?)
translogi
>
> - Show quoted text -
OOPS
made a mistake (and it was not my first)
for every number a/b with a and b positive there is a number N that
uniquely idenitifies if
N = (b+a) x (b+a) +a +1
i think you can change it for every integer by
M= 4*N + 2* sign(a) + sign(b)
the question is 1/2 and 2/4 the same number ?
I guess not.
but they do both refer to the same value.
Like 2 and the smallest prime also refer to the same value.
kunzmilan
If they are different, we can count rational numbers as the sum
9 + 99 + 999 + ...
Adding to each term 1, and still one 1, this sum can be approximated
as (10/9)*10^n. From it we subtract (n + 1) added ones.
kunzmilan