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Imagine you can see the whole Number Line and every one of the infinite individual points it comprises. Imagine you want a quick and easy way to distinguish those points corresponding to rational numbers from the ones corresponding to irrationals. What you're going to do is ID the rational points by draping a bright-red hankie over each one; that way they'll stand out. Since geometric points are technically dimensionless, we don't know what they look like, but what we do know is that it's not going to take a very big red hankie to cover one. The red hankie here can in truth be arbitrarily small, like say .00000001 units, or half that size, or half that half,...,etc. Actually, even the smallest hankie is going to be unnecessarily large, but for our purposes we can say that the hankie is basically infinitesimally small-call such a size (infinitesimally small symbol). So a hankie of size (infinitesimally small symbol) covers the N.L.'s first rational point. Then, because of course the hankie can be as small as we want, let's say you use only a (Infinitesimally small symbol)/2-size hankie to drape over the next rational point. And say you go on like that, with the size of each red hankie used being exactly (1/2) that of the previous one, for all the rational numbers, until they're all draped and covered. Now, to figure out the total percentage of space all the rational points take up on the Number Line, all you have to do is add up the sizes of all the red hankies. Of course, there are infinitely many hankies, but size-wise they translate into the terms in an infinite series, specifically the Zeno-esque geometric series (1/2^0 +1/2^1 + 1/2^2 +1/2^3 +1/2^4 + ...; and, given the good old a/1-r formula for summing such a series, the sum-size of all the infinite hankies ends up being 2*(Infinitesimally small symbol). But (Infinitesimally small symbol) is infinitesimally small, with infinitesimals being (as we mentioned in Section 2b) so incredibly close to 0 that anything times an infinitesimal is also an infinitesimal, which means that 2*(Infinitesimally small symbol) is also infintesimally small, which means that all the infinite rational numbers combined take up only an infinitesimally small portion of the N.L.-which is to say basically none at all-which is in turn to say that the vast, vast bulk of the points on any kind of continuous line will correspond to irrational numbers, and thus that while the aforementioned Real Line really is a line, the all-rational Number Line, infinitely dense though it appears to be, is actually 99.999...% empty space, rather like DQ ice cream or the universe itself.
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