I just read somewhere that there are as many even numbers as natural numbers! Is there an intuitive way of coming to terms with this fact? Because honestly, it's quite baffling. Also, does the concept of odd/even apply to negative integers as well? By definition, it should. But I have never seen a negative multiple of two being used as an example of an even number.
[Math] As many even numbers as natural numbers.
number theory
Related Solutions
Yes. The generalization is provided by modular arithmetic. The properties you are observing all come from the fact that taking the remainder modulo $n$ respects addition and multiplication, and this generalizes to any $n$. More generally in abstract algebra we study rings and their ideals for the same reasons.
The notion of evenness and oddness of functions is closely related, but it is somewhat hard to explain exactly why. The key point is that there is a certain group, the cyclic group $C_2$ of order $2$, which is behind both concepts. For now, note that the product of two even functions is even, the product of an even and odd function is odd, and the product of two odd functions is even, so even and odd functions under multiplication behave exactly the same way as even and odd numbers under addition.
There are also huge generalizations depending on exactly what you're looking at, so it's hard to give a complete list here. You mentioned chessboards; there is a more general construction here, but it is somewhat hard to explain and there are no good elementary references that I know of. Once you learn some modular arithmetic, here is the modular arithmetic explanation of the chessboard idea: you can assign integer coordinates $(x, y)$ to each square (for example the coordinate of the lower left corner), and then you partition them into black or white squares depending on whether $x + y$ is even or odd; that is, depending on the value of $x + y \bmod 2$. Then given two points $(a, b)$ and $(c, d)$ you can consider the difference $c + d - a - b \bmod 2$, and constraints on this difference translate to constraints on the movement of certain pieces. This idea can be used, for example, to prove that certain chessboards (with pieces cut out of them) cannot be tiled with $1 \times 2$ or $2 \times 1$ tiles because these tiles must cover both a white square and a black square. Of course there are generalizations with $2$ replaced by a larger modulus and larger tiles.
As for matrices and vectors, let's just say that there are a lot of things this could mean, and none of them are straightforward generalizations of the above concept.
$2\cdot 3\cdot 5\cdot 7\cdot 13\cdot 17\cdot 19 = 881790$ doesn't appear to qualify - there are no $6$th, $13$th or $20$th powers in the LCM.
Best Answer
Let’s take the questions in reverse order.
Yes, the concepts of odd and even apply to negative integers: any integer $n$ is even if and only if there is an integer $k$ such that $n=2k$. The integer $k$ can be positive, negative, or zero. Thus, $-6=2(-3)$ is even, as is $0=2\cdot0$. An integer is odd if and only if it is not even, so $-7$ is odd: there is no integer $k$ such that $-7=2k$.
The answer to the first question is also yes. First, it’s very easy to put the set of natural numbers, $\Bbb N=\{0,1,2,3,\dots\}$, into one-to-one correspondence with the set $E=\{0,2,4,6,\dots\}$ of even natural numbers: the map $\Bbb N\to E:n\mapsto 2n$ is clearly a bijection. If you want to fine a bijection between $\Bbb N$ and the set of all even integers, you have to work a little harder. Here’s a picture of part of one:
$$\begin{array}{r} 0&1&2&3&4&5&6&7&8&9&\dots\\ 0&2&-2&4&-4&6&-6&8&-8&10&\dots \end{array}$$
This can be expressed as
$$n\mapsto\begin{cases} -n,&\text{if }n\text{ is even}\\ n+1,&\text{if }n\text{ is odd}\;, \end{cases}$$
and you can check that this really is a bijection from $\Bbb N$ to the set of all even integers.