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 SarK0Y 2021-01-03 08:12

Side effect of research == [TEX]\sqrt{2}[/TEX] is the rational number [URL="https://alg0z.blogspot.com/2021/01/square-root-of-2-is-rational-number.html?m=0"]https://alg0z.blogspot.com/2021/01/square-root-of-2-is-rational-number.html?m=0 :smile:
[/URL]

 retina 2021-01-03 08:16

[QUOTE=SarK0Y;568160]Side effect of research == [TEX]\sqrt{2}[/TEX] is the rational number [URL="https://alg0z.blogspot.com/2021/01/square-root-of-2-is-rational-number.html?m=0"]https://alg0z.blogspot.com/2021/01/square-root-of-2-is-rational-number.html?m=0 :smile:
[/URL][/QUOTE]For once I am pleased that the linked target displays nothing without JS, because just the title makes it look like a crank worthy post.

Would you like to summarise what it says?

 Batalov 2021-01-03 10:04

1 Attachment(s)
In a nutshell, he says:
Let's take a rational number p/q = 2/3.
"but definition of the odd/even has absolutely no sense for rational numbers," (direct quote)
so we cannot say that integer p=2 is an even number. it's neither even nor odd. It is 1.99999999999... End of proof.

Is that right, Evgeniy? 2 is not an even number? Would it make you feel better, if p/q = 1414/1000, "we cannot prove that integer 1414 is an even number"?

I attached his "proof".

 SarK0Y 2021-01-03 23:16

[QUOTE=Batalov;568165]In a nutshell, he says:
Let's take a rational number p/q = 2/3.
"but definition of the odd/even has absolutely no sense for rational numbers," (direct quote)
so we cannot say that integer p=2 is an even number. it's neither even nor odd. It is 1.99999999999... End of proof.

Is that right, Evgeniy? 2 is not an even number? Would it make you feel better, if p/q = 1414/1000, "we cannot prove that integer 1414 is an even number"?

I attached his "proof".[/QUOTE]
Åctually, no :) 2 is 2, but when you deal with rationals you cannot treat them like natural numbers. for example..
[TEX]\frac{1}{9}\cdot9 \neq\frac{9}{9}\cdot1[/TEX]
at 1st glance, looks strange, but...
[TEX]\frac{1}{9}\cdot9 \eq0.11111111111111..11\cdot9[/TEX]

according to the very principle of limits, approximation of continuous function cannot reach its final point. Here we could recall
[B][URL="https://www.cut-the-knot.org/WhatIs/Infinity/AchillesAndTortoise.shtml"]Achilles and the Tortoise[/URL][/B]

 VBCurtis 2021-01-03 23:37

No, 1/9 times 9 is 1, even in decimals.
0.9999-repeating is equal to 1- yet you claim it is not, and make fuzzy reference to a limit. If it's not equal to 1, how far away from 1 is it? Or, what number can you fit between it and 1?

 SarK0Y 2021-01-03 23:59

[QUOTE=VBCurtis;568233]No, 1/9 times 9 is 1, even in decimals.
0.9999-repeating is equal to 1- yet you claim it is not, and make fuzzy reference to a limit. If it's not equal [B]to 1, how far away from 1 is it? Or, what number can you fit between it and 1?[/B][/QUOTE]
0.9 < 0.99 < 0.999< .. < 0.999..99 it's continuous sequence/function. if you assume it can reach its final point 1 then you fall into bad situation like..
[CENTER][TEX]\lim_{x \to 1}\frac{1}{1-x}\eq \frac{1}{0}[/TEX][/CENTER]
:smile:

 Batalov 2021-01-04 00:10

[QUOTE=SarK0Y;568231]Åctually, no :) 2 is 2, but when you deal with rationals you cannot treat them like natural numbers. for example..
<<some red herring>>[/QUOTE]
Don't get distracted.

Observe: a p/q is a rational number, where p [B]is an integer[/B] and q [B]is an integer.[/B].
Done with rationals. Now all you have are integer numbers until the end of Euclid's proof.

Now, are you saying that it's not true that all integer numbers are either even or odd? [SPOILER]Hint: they are![/SPOILER]

 VBCurtis 2021-01-04 00:38

[QUOTE=SarK0Y;568237]0.9 < 0.99 < 0.999< .. < 0.999..99 it's continuous sequence/function. if you assume it can reach its final point 1 then you fall into bad situation like..
[CENTER][TEX]\lim_{x \to 1}\frac{1}{1-x}\eq \frac{1}{0}[/TEX][/CENTER]
:smile:[/QUOTE]

Neither side of your limit example exists, so your congruence is nonsensical- and irrelevant to whether 0.9-repeating is equal to 1.

There is no sequence involved in the single number 0.9-repeating, either. I didn't ask about 0.9, nor 0.99. 0.9-repeating is neither of those numbers. Every member of your sequence is strictly less than 0.9-repeating, anyway.

You might figure out the flaws in your reasoning if you used words properly- how do you define "continuous sequence"?

 Dr Sardonicus 2021-01-04 01:15

(1) OP seems to be confounding "fractions" (rational numbers) and "decimal fractions," i.e. fractions that can be expressed with a power-of-ten denominator. Not all rational numbers are decimal fractions.

(2) OP also seems to think that invalidating a proof of A automatically proves ~A (not-A). It doesn't. (Here, A is "The square root of 2 is irrational.")

OP, of course, did not invalidate the proof. What he actually did was (1).

Expressing the statement that the (positive) square root of 2 is rational as an equation in positive integers p and q,

(*) p[sup]2[/sup] = 2*q[sup]2[/sup]

invites a Euclidean proof that the square root of 2 is not rational, because the equation is impossible.

Euclid also proved a result now known as the Fundamental Theorem of Arithmetic, AKA unique factorization.

The equation (*) violates the Fundamental Theorem, because the left side is divisible by 2 evenly many times, while the right side is divisible by 2 oddly many times.

 SarK0Y 2021-01-04 03:59

[QUOTE=Dr Sardonicus;568248](1) OP seems to be confounding "fractions" (rational numbers) and "decimal fractions," i.e. fractions that can be expressed with a power-of-ten denominator. Not all rational numbers are decimal fractions.
[/QUOTE]

don't confuse numerical system & numbers itself == any proper numerical system can express any number. Another moment is how efficient that expression could be.
[QUOTE=Dr Sardonicus;568248]
(2) OP also seems to think that invalidating a proof of A automatically proves ~A (not-A). It doesn't. (Here, A is "The square root of 2 is irrational.")
[/QUOTE]
no, the very point is, you must uniform objects before you do operations on them. that proof messes w/ two different types of objects. for instance, 1/3 is odd or even? or let [TEX]p,q \in \mathbb{N}\; \frac{p}{q}=\epsilon,\; p=\epsilon\cdot q[/TEX] everything looks fine out there, right?:grin: But we have the damn grave problem, even two ones...
[TEX] p=\epsilon\cdot q[/TEX] exists everywhere for any p & q. second form has two troubling cases == 1st one w/ q = 0 and 2nd one is that..
[CENTER] [TEX]\displaystyle \lim_{q \to \infty}\lim_{p \to \infty}\frac{p}{q}\eq????[/TEX][/CENTER]

 SarK0Y 2021-01-04 04:00

[QUOTE=VBCurtis;568242]Neither side of your limit example exists, so your congruence is nonsensical- and irrelevant to whether 0.9-repeating is equal to 1.

There is no sequence involved in the single number 0.9-repeating, either. I didn't ask about 0.9, nor 0.99. 0.9-repeating is neither of those numbers. Every member of your sequence is strictly less than 0.9-repeating, anyway.

You might figure out the flaws in your reasoning if you used words properly- how do you define "continuous sequence"?[/QUOTE]
so [TEX]\ln(x)[/TEX] and [TEX] \frac{d\ln(x)}{x}[/TEX] do not exist, right?:smile:
[QUOTE=VBCurtis;568242]
There is no sequence involved in the single number 0.9-repeating, either. I didn't ask about 0.9, nor 0.99. 0.9-repeating is neither of those numbers. Every member of your sequence is strictly less than 0.9-repeating, anyway.

You might figure out the flaws in your reasoning if you used words properly- how do you define "continuous sequence"?[/QUOTE]
Oh, boy, really?:cmd::ermm:
[CENTER] [TEX]\lim_{n \to \infty}\left(1-\frac{1}{10^{n}\right)\eq0.9999..99[/TEX]

[/CENTER]

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