TITLE: How can we be certain that ℝ fills the entire number line?
QUESTION [3 upvotes]: It can be a stupid question, but how do we know that ℝ fills the numberline   
Initially we thought that fractions were enough, that is until we found the irrational numbers. Can't it happen again with a new type of number? Like infinitesimal numbers?

REPLY [2 votes]: It's true basically by definition!
The origins of the real number system are heavily rooted in Euclidean geometry, and the geometric methods of using line segments to mark distances, and using geometric constructions to do arithmetic with line segments.
In other words, once we developed the idea of putting an origin on a Euclidean line and that quantities could have signs, the notion of "real number" literally means "point on the number line".
Translated into modern terms, we always knew about the irrational numbers, we just hadn't realized they were irrational at first!
We have rigorous proof that decimal notation is capable of expressing every real number.

Incidentally, the numbers alone don't make up a line — you have to remember some geometric concepts (or their arithmetic analogs) to go with the numbers in order to retain its line-ness. For example, you might remember they were lying in a Euclidean plane. Or, you might remember the ordering relation and the subtraction operation.
So, in this sense the real numbers don't "fill" the number line, since knowing all of the points on the line aren't enough to know it.

Now, there is another dimension to your question you may not have realized; you ask whether the number line is filled, but one could ask which number line!
There are, in fact, other number lines one may be interested in considering. For example:

You could consider not the entire Euclidean plane, but just the points that can be constructed with straightedge and compass. In this restricted 'constructible' geometry, the number line consists only of the constructible numbers
You could consider a "nonstandard" model of Euclidean geometry; its number line will be some form of hyperreal numbers, which will have infinitesimals.
Basically, given any notion of number system you can interpret geometrically rather than algebraically to get its own related number line
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