Number ray: Representation of the set of all ►real numbers by means of a line.

The line consists of infinitely many points, each of which corresponds to a natural number. Of course, this at once raises the question of how a number of points, with no extension, randomly placed next to one another could form a continuous, extended line. This question is not at all trivial. The answer is that it depends on the number of points. Infinitely many points do not suffice. Their degree of infinity must be higher than א0 (see ►Cardinals) in order for a continuous line to be created rather than just an infinitely dotted line.

Gaps in the Number Ray

In the above section we can see where the integers -1, 0, 1, 2, 3 are located. There are infinitely many fractions, that is, rational numbers in between the integers. The rational numbers are infinitely close to one another — which means that the distance between any two of them can be of any degree of smallness. Nonetheless, there is still a huge gap between each two rational numbers no matter how close they are to each other. This gap is filled with the help of the irrational numbers. Only the latter are able to turn the number ray into a genuine continuum, that is, into one continuous line.

The pattern of rational and irrational numbers is independent of the scale; it is repeated in each magnification that we may use to access a section of the ray. This property is called a ►fractal. As we can see here, the seemingly simple number line really has a highly complex structure. We now know that the irrational numbers are not ►countable and thus are one level (or even several levels, see continuum hypothesis) "more infinite" than the integers and rational numbers with their smallest possible degree of infinity א0. This is why you will always hit an irrational number when you blindly touch any point on the number ray. Given the sheer quantity of irrational numbers, the probability that you may hit a rational number or even an integer is negligible.*

* The probability that any given number is irrational is 99.999...%. But this is identical to 100%; for the proof see ►Cantor Dust.

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