Home  18.013A  Chapter 1  Section 1.2 


Is $\mathbb{Z}$ countable?
Solution:
Though there are "twice as many" positive and negative integers as there are only positive ones, we can make a onetoone correspondence between $\mathbb{Z}$ and $\mathbb{N}$ . We can, in other words, assign a unique positive integer to each positive and negative integer.
How? Assign the positive integer $2n+1$ to the positive integer $n$ , and the integer $2n$ to the negative integer $n$ . The correspondence looks like this:
$\mathbb{N}$ : 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
... 
$\mathbb{Z}$ : 
0 
1 
1 
2 
2 
3 
3 
4 
4 
5 
5 
6 
6 
7 
7 
8 
... 
Sooner or later you get to every element of $\mathbb{Z}$ this way, though the elements of $\mathbb{N}$ grow faster than those of $\mathbb{Z}$ . The peculiar fact, but fact nevertheless is that it doesn't matter at all that the elements of $\mathbb{N}$ grow faster here.
