Re: Repeating decimals are irrational

wij <wyniijj5@gmail.com> writes:

On Tue, 2024-03-26 at 15:01 -0700, Keith Thompson wrote:

wij <wyniijj5@gmail.com> writes:

On Tue, 2024-03-26 at 13:18 -0700, Keith Thompson wrote:

[SNIP]

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.
 

 
Then, I would ask you to provide proof, not just assertion that you are right.
My model is not perfect, but it does not matter, I am sure now you cannot really
distinguish good or bad.

 
Do you understand what a mathematical limit is?  Can you explain it?
 
If you understand limits, you should be able to understand that
0.333..., where the "..." denotes the limit as the number of 3s exceeds
without bound, is exactly equal to 1/3.
 
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