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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?

Luis Dias
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    Look up the completeness of $\mathbb{R}$ – mathworker21 Sep 07 '17 at 22:35
  • There are extensions of the real numbers to include infinitesimals (as well as infinitely large numbers). Look up nonstandard analysis. However your notion/intuition of the "numberline" (which is undefined here) may exclude some or all of those extensions. – hardmath Sep 07 '17 at 22:47
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    @mathworker21 What about completeness ? $\mathbb Z$ is complete too, yet has holes relatively to $\mathbb R$. This is a delicate argument. – zwim Sep 07 '17 at 22:51
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    This is not a stupid question. It is a very good question. How can we show that the reals "fill up the number line"? This question has motivated some fine work in mathematics. You'll need to make sure that you understand / clarify both your ideas of "the entire number line" and "the Real numbers". – Jim H Sep 07 '17 at 22:54
  • @hardmath I agree, note that surreal for instance form a class, while reals are a set. But the definition of the numberline stays fuzzy... – zwim Sep 07 '17 at 22:56

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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