There are more even numbers than odd numbers

[xilman]

Originally Posted by storm5510

All of this seems to make something very simple into something very complex. It is not. If you take them in pairs, (one of each type), the count will be the same for odds and evens, if the counting stops on an even.

(1,2)(3,4)(5,6)(7,8) and so on.

And if it does not?

You have provided a proof of theorem that there is one more odd natural number than there are evens.

[JeppeSN]

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

[xilman]

Originally Posted by JeppeSN

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

This thread is in "Miscellaneous Math" ...

Toto, I've a feeling we're not in Kansas any more.

[storm5510]

Originally Posted by xilman

And if it does not?

You have provided a proof of theorem that there is one more odd natural number than there are evens.

Alright, I will take your word for it, and not belabor the issue. One question though: Where does the extra odd number reside?

[xilman]

Originally Posted by storm5510

Alright, I will take your word for it, and not belabor the issue. One question though: Where does the extra odd number reside?

You have already told us.

Originally Posted by storm5510

All of this seems to make something very simple into something very complex. It is not. If you take them in pairs, (one of each type), the count will be the same for odds and evens, if the counting stops on an even.

(1,2)(3,4)(5,6)(7,8) and so on.

"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.

[Nick]

Originally Posted by retina

https://en.wikipedia.org/wiki/Fractional_ideal

I don't understand?

The asymmetry that lies behind your initial post stems from the fact that the exponents in ordinary prime factorization are restricted to non-negative integers.
If we work with fractions (or, more generally, fractional ideals in any Dedekind domain) then this problem disappears.

[retina]

Originally Posted by Nick

The asymmetry that lies behind your initial post stems from the fact that the exponents in ordinary prime factorization are restricted to non-negative integers.
If we work with fractions (or, more generally, fractional ideals in any Dedekind domain) then this problem disappears.

I'll have to think about that.

[kriesel]

Originally Posted by retina

4, 8, 16 ,32, 64

(in response to:)
Originally Posted by kriesel

Do tell. Give a list of five of them that are not immediately preceded by an odd number each.

4, 8, 16 ,32, 64 are each immediately preceded by 3, 7, 15, 31, 63, respectively; Mersenne numbers 2ⁿ-1 precede powers 2ⁿ.

[kriesel]

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 2ⁿ-1 is also comprised of exactly as many evens as odds.

[kruoli]

Originally Posted by kriesel

...preceded...

We are not talking about proceeding like 1, 2, 3. Someone gave a specific rule to follow, so this person set how to "proceed". In this case, all of retina's examples are perfectly valid.

[kriesel]

Originally Posted by retina

Every odd number is half of an even number.

Some even numbers are not double an odd number.

Therefore there are more even numbers than odd numbers.

The original post does not indicate a number system explicitly.
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/even-odd.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.