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This is probably a silly question, but, for integers...

$even$ x $even$ = $even$
$even$ x $odd$ = $even$
$odd$ x $even$ = $even$
$odd$ x $odd$ = $odd$

There are 3 times as many combinations that form even numbers than odd numbers, so why aren't there more odd numbers than even ones?

I know the truth can be seen by examining an integer number line, but I want to know why the above argument fails.

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    $\begingroup$ $0$ times anything is $0$; why aren’t there more zeroes than other numbers ? $\endgroup$ Commented Jun 17, 2020 at 14:20
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    $\begingroup$ Perhaps you should consider addition? $\endgroup$ Commented Jun 17, 2020 at 14:37
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    $\begingroup$ Now... it seems to me that your argument is that "Look here, if I consider $\Bbb N\times \Bbb N$ and I look at a finite square in it of the form $\{0,1,2,\dots,n\}\times \{0,1,2,\dots,n\}$ and I associate with each point a number which is the product of their coordinates, the limit of the ratio of these who result in an odd number is $\frac{1}{4}$" but this does two things wrong. First, the interpretation of the grid points as the product of their coordinates is not a bijection with the natural numbers, it is not injective. You have $36 = 36\times 1 = 18\times 2 = 9\times 4 = \dots$ $\endgroup$
    – JMoravitz
    Commented Jun 17, 2020 at 14:54
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    $\begingroup$ Next, even if it were in bijection, it is a rearrangement of sorts and rearrangements of things like this do not necessarily preserve relative density. I could have rearranged the numbers on the number line: $1,5,2,10,3,15,4,20,6,25,7,30,8,35,9,40,11,45,\dots$ where the even positions are occupied by increasing multiples of five and odd positions are occupied by increasing non-multiples of five. The relative density of multiples of five in the above rearrangement is $\frac{1}{2}$ despite the relative density normally being $\frac{1}{5}$ $\endgroup$
    – JMoravitz
    Commented Jun 17, 2020 at 14:57
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    $\begingroup$ Now... the answer of $\frac{1}{4}$ as the limit of the relative density of points in the square grid whose product of coordinates is odd is a fine answer for that question, but do not confuse that question with the question of the relative density of odd numbers in the natural numbers as commonly portrayed in their usual order. Those are both different questions. $\endgroup$
    – JMoravitz
    Commented Jun 17, 2020 at 15:02

3 Answers 3

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We can make this even more "surprising"

Every positive integer $N$ can be uniquely written as $$N=2^n\cdot m$$ with non-negative integer $n$ and odd $m$. Only if $n=0$, $N$ is odd, so "almost all" numbers should be even.

What is the catch ?

If we get to a fixed limit (instead of considering all positive integers), we will se that the upper bound for $m$ gets smaller with increasng $n$, so the number of even numbers is not larger (at least if the limit is even, otherwise we have one more odd number).

For infinite sets this leads to counterintuitive facts as "there are as many posite integers than numbers of the form $2^n$" with positive integer $n$" although very few positive integers are a power of $2$.

The even numbers are a proper subset of the positive integers, so in some sense there are "as many positive integers as even numbers". We only get to the correct fraction $\frac{1}{2}$, if we consider finite cases and increase the limit.

"Hilbert's hotel" might be a good introduction to get a feeling for this stuff.

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What you have observed is correct, but it is not about the Natural numbers, it is about product of natural numbers with themselves.

multiplying ${1,2,3,4,5,6,7,\cdots}$ with itself we get the following :

$$\begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & \cdots\\ 2 & 4 & 6 & 8 & 10 & 12 & 14 & \cdots\\ 3 & 6 & 9 & 12 & 15 & 18 & 21 & \cdots\\ 4 & 8 & 12 & 16 & 20 & 24 & 26 & \cdots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \\ \end{bmatrix} $$

The ratio of odd to even for above should be 1 to 3 and that is based on your observation of result of multiplication of evens and odds.

This seems to be the intuitive way of interpreting your observation.

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It's a property of multiplication. Addition does not have this special ordering. To get deeper into what happens we should look at a very simple example with squares.

Here we have two numbers; one odd number = $3$ and one even number = $2$. It does not really matter what numbers we choose, as long as its one even and one odd number. Setup some squares to see whats going on:

enter image description here

Here we've just added the same odd number to itself many times, and do the same thing to the even number. Take a look at what happens. The even number never changes to even. But the odd number changes each time. To conclude so far; for the odd case it looks like a discrete switch that turns odd to even and even to odd. For the even case, this switch is never touched. This was the most discrete example I could think of.

Im sure one can analyse this and find out deeper things.

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