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wij <wyniijj5@gmail.com> writes:On Tue, 2024-03-26 at 13:18 -0700, Keith Thompson wrote:wij <wyniijj5@gmail.com> writes:Correct, because that would mean no q can exhaust the conversion in long division, or subtraction.On Tue, 2024-03-26 at 09:45 -0700, Keith Thompson wrote:wij <wyniijj5@gmail.com> writes:On Tue, 2024-03-26 at 08:39 -0700, Keith Thompson wrote:wij <wyniijj5@gmail.com> writes:First of all, it is not really my definition (strict meaning of the wordSnipet from
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-en.txt/download
...
Real Nunmber(ℝ)::= {x| x is represented by n-ary <fixed_point_number>, the
digits may be infinitely long }
Note: This definition implies that repeating decimals are irrational number.
How does a definition that doesn't mention rational numbers imply
anything about which numbers are rational?
definition). What I showed is a reasonable proof of what the real number really
'practically' used world-wide (not the ones in academic theory)..
<fixed_point_number> is just a representation of real number specified for
convenience for math. proofs and discussion of numbers.
Your 'rational number' might mean a sub-class defined latter.
I don't see an answer to my question.
Let's list a common magic proof in the way as a brief explanation:
(1) x= 0.999...
(2) 10x= 9+x // 10x= 9.999...
(3) 9x=9
(4) x=1
Ans: There is no axiom or theorem to prove (1) => (2).
Note: If the steps of converting a number x to <fixed_point_number> is not
finite, x is not a ratio of two integers, because the following
statement is always true: ∀x,a∈ℚ, x-a∈ℚ
---End of quote
Is 1/3 a rational number?
Yes, by definition?
Is 1/3 a real number?
If 1/3 is a real number, what is its representation according to your definition?
Yes, 1/3 is a real number, it's n-nary <fixed_point_number> representaion is infinitely
long.
Infinitely long number is harder to explain by now. I think this part can be
skip for the moment (no present theory can make this very clear and satisfactory).
You didn't actually say what its representation is. Is it "0." followed
by an infinite sequence of "3"s?
Isn't the representation of 1/3 a "repeating decimal"? You stated
above that repeating decimals are irrational numbers. How do you
reconcile that with your (correct) statement that 1/3 is rational?
Ah, I see. I should make my statement more clear:
1/3 is representable in 3-nary <fixed_point_number> (e.g. 0.1)
"Infinite long" (for irrational) refers to numbers that is not finitely representable by any
n-ary <fixed_point_number>.
Are you now saying (for the first time, as far as I can tell) that a
number is rational if and only if it has a finite representation in
*any* integer base? For example, 1/3 has a finite representation in
base 3, and 1/7 has a finite representation in base 7 (both have finite
representations in base 21). So an infinite decimal representation
doesn't make a number irrational as long as it has a finite
representation in *some* integer base.
That's probably a workable definition. (I won't get into whether it's a
*useful* definition.)
Good luck if you try to conquer irrational
Still, rational numbers can be represented in decimal, and you already
acknowledged that "the digits may be infinitely long" for real numbers.
So 0.333..., where the sequence of 3s is unending, is a valid
representation of the rational number 1/3, yes? It's the number itself
that's rational, regardless of which of several valid representations
you choose to express it.
Do you agree so far?
I just understand the front part of your question.
No, 1/3 and 0.333(3) are not exactly equal. See the note "∀x,a∈ℚ, x-a∈ℚ"
The semantic of "repeating" already says the number 1/3 cannot be completely
exhausted by subtracting 3/10^n. The proposition "repeating decimal is rational"
is simple false by semantics.
And that's where you're quite simply wrong.
1/3 and 0.333(3) are exactly equal, and rational. "∀x,a∈ℚ, x-a∈ℚ" is
not violated; the difference is 0, which is also rational. You do not
have an internally consistent model of real or rational numbers.
You admit that you don't understand all of my question. I suggest that,
until you've learned enough to understand it, you're not going to be
successful in whatever you're trying to do.
I don't think you understand what "0.333(3)" *means*. Sure, you
understand that it's an unending series of 3s, but to understand its
value, you have to understand *limits*. I don't believe that you do..
You don't know how to represent 1/3 exactly in decimal notation. I do.
If so, consider the rational number that is the result of dividing 1 byIf your question were about 0.333...*3, simple answer is 0.999... (by infinite series)
3, represented in decimal as 0.333..., where the "..." denotes an
unending sequence of 3s. What is the decimal representation of the
number that is the result of multiplying that number by 3?
But in general, "..." is a place to play magic. Don't use it for controversial proofs
There is no magic. People who understand limits, which can be defined
rigorously, are able to understand how all this stuff works.
There are valid extensions to the real numbers (surreals, hyperreals,
etc.) -- but they were invented by people who understand the reals.
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