Hilbert's Paradox of the Grand Hotel
A similar observation is that there&#x27;s exactly as many natural numbers [0, 1, 2, 3, ...) as integer numbers (the same, but with negatives: ..., -2, -1, 0, 1, 2, ...).<p>Normally we&#x27;d need to count things to say there&#x27;s &quot;as many&quot; X as Y. But with infinities counting is a bit tricky. So, is the infinite count of natural numbers the same kind of infinity as count for integer numbers?<p>To check this, we need to see if for every number in one set we can assign exactly one number in the other set (and the other way around). It&#x27;s actually pretty simple:<p><pre><code>    0 -&gt; 0
    1 -&gt; -1
    2 -&gt; 1
    3 -&gt; -2
    4 -&gt; 2
    5 -&gt; -3
    6 -&gt; 3
    ...
</code></pre>
Since this mapping (&quot;bijective function&quot;) exists, we know that every number in one set has exactly one representative in the other set - so the set counts are identical.<p>What&#x27;s interesting, if we look into real numbers (think: double in C, but without problems with approximation), there&#x27;s much, much more of them. The infinite count of real numbers is much larger than the infinite count of integers. But let&#x27;s keep it for another comment.