20201217, 05:20  #67 
Jan 2010
2×43 Posts 
made some renew for guide to add the more detailed remark about disadvantages.

20201217, 21:36  #68  
Jan 2010
2×43 Posts 
Quote:
Quote:


20210103, 08:12  #69 
Jan 2010
56_{16} Posts 
Side effect of research == is the rational number https://alg0z.blogspot.com/2021/01/squarerootof2isrationalnumber.html?m=0

20210103, 08:16  #70  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
6083_{10} Posts 
Quote:
Would you like to summarise what it says? 

20210103, 10:04  #71 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}·2,333 Posts 
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". 
20210103, 23:16  #72  
Jan 2010
2·43 Posts 
Quote:
at 1st glance, looks strange, but... according to the very principle of limits, approximation of continuous function cannot reach its final point. Here we could recall Achilles and the Tortoise 

20210103, 23:37  #73 
"Curtis"
Feb 2005
Riverside, CA
1245_{16} Posts 
No, 1/9 times 9 is 1, even in decimals.
0.9999repeating 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? 
20210103, 23:59  #74  
Jan 2010
2·43 Posts 
Quote:


20210104, 00:10  #75  
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}×2,333 Posts 
Quote:
Observe: a p/q is a rational number, where p is an integer and q is an integer.. 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? Hint: they are! 

20210104, 00:38  #76  
"Curtis"
Feb 2005
Riverside, CA
1245_{16} Posts 
Quote:
There is no sequence involved in the single number 0.9repeating, either. I didn't ask about 0.9, nor 0.99. 0.9repeating is neither of those numbers. Every member of your sequence is strictly less than 0.9repeating, anyway. You might figure out the flaws in your reasoning if you used words properly how do you define "continuous sequence"? 

20210104, 01:15  #77 
Feb 2017
Nowhere
10F5_{16} Posts 
(1) OP seems to be confounding "fractions" (rational numbers) and "decimal fractions," i.e. fractions that can be expressed with a poweroften denominator. Not all rational numbers are decimal fractions.
(2) OP also seems to think that invalidating a proof of A automatically proves ~A (notA). 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^{2} = 2*q^{2} 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. 
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