20200804, 14:50  #12  
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
24411_{8} Posts 
Quote:
You have provided a proof of theorem that there is one more odd natural number than there are evens. 

20200804, 15:52  #13 
"Jeppe"
Jan 2016
Denmark
2^{2}·41 Posts 
You consider d(N) as the difference between the number of odd natural numbers under N and the number of even natural numbers under N. Then d(N) does not converge for N tending to infinity, in the usual sense. Therefore you consider the sequence of arithmetic means of all d(i) for i up to N, and now it converges. The limit, if everything goes the way I think, reveals what definition of "natural number" you have. /JeppeSN

20200804, 16:14  #14  
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
10505_{10} Posts 
Quote:
Toto, I've a feeling we're not in Kansas any more. 

20200804, 17:44  #15 
Random Account
Aug 2009
U.S.A.
2·3·13·23 Posts 

20200804, 17:59  #16  
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
10505_{10} Posts 
Quote:
"if the counting stops on an even": if it does not, the extra odd number is that one which does not have an even counterpart. 

20200806, 10:42  #17  
Dec 2012
The Netherlands
1,579 Posts 
Quote:
If we work with fractions (or, more generally, fractional ideals in any Dedekind domain) then this problem disappears. 

20200806, 10:57  #18  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
3^{2}·23·29 Posts 
Quote:


20200808, 16:23  #19  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
3·5·17·19 Posts 
(in response to:)
Originally Posted by kriesel Quote:
Last fiddled with by kriesel on 20200808 at 16:26 

20200808, 16:40  #20 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
3×5×17×19 Posts 
For any given binary integer word size, of a positive whole number of bits n>0, there are exactly as many evens as odds.
https://en.wikipedia.org/wiki/Two%27s_complement. The difference being zero for any n, the slope of the difference function is zero, and the difference in the limit at n=infinity is also a difference of zero. It does not change if considering unsigned integers, since 0 to 2^{n}1 is also comprised of exactly as many evens as odds. Last fiddled with by kriesel on 20200808 at 16:44 
20200808, 20:10  #21 
"Oliver"
Sep 2017
Porta Westfalica, DE
110010110_{2} Posts 

20200808, 23:11  #22  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
12ED_{16} Posts 
Quote:
Would that be natural numbers, all integers, all rationals? Some of the online definitions for odd number are imprecise. https://duckduckgo.com/?t=ffnt&q=odd...&ia=definition Is x= 1 / 2 odd or even? In what bases? At https://www.mathsisfun.com/numbers/evenodd.html, "Any integer that can be divided exactly by 2 is an even number." x=1/2 = exactly 0.5 (base 10) =0.3 (base 6) is even, but x=1/2 = 0.3.. (base 7) so one is both even and odd, by that definition, depending on the number base in which the calculation is attempted. (They do continue on to give examples of odd numbers that imply integer computation only, no reals.) https://www.rapidtables.com/calc/mat...alculator.html is a selectable base calculator. Last fiddled with by kriesel on 20200808 at 23:12 

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