infinity - All sets of rational numbers are bigger than the set containing infinite integers - or are they?

Intro

This started with me learning the different types of infinity. I like to call them types instead of sizes due to the fact, that infinite is defined by being endless or "not-finite" - meaning not a size. (I do know the right definition is different sizes).

My Hypothesis

I started trying to match integers to rational numbers one-to-one (or bijection). I have found, in the top answer in this question, that:

Two sets $A$ and $B$ are said to have the "same size" if there is a some function $f:A\to B$ which is a bijection. Note that we do NOT require that ALL functions be bijections, just that there is SOME bijection.

The way I found it possible was by limiting the set of rational numbers to $[0,1)$. Now by reversing the order of the decimals I could map all rational numbers excluding fractions resulting in an endless repeating sequence of decimals, thus matching:

• 0.034 to 430


• 0.2331 to 1332


• ...

As said, this maps any rational numbers excluding fractions resulting in an endless repeating sequence of decimals. Now the method to mapping those.

($\overline{\text{Overline}}$ means repeated endlessly)

The workaround with this is made in two steps:

• $\frac47 = 0.571428\overline{571428}$

If we accept this as a number I assume that this following is acceptable as well:

• $824175\overline{824175}$

Question

Does this mean, that there are exactly as many rational numbers in the set $[0,1)$ as there are integers in the set $[0,\infty]$?

And furthermore that in the rational set $[0,1]$ contains one more number that the set of integers $[0,\infty]$?

Bonus Quention

Is this allowed: $10\times824157\overline{824157}$?

Winther made it clear that $824157\overline{824157}$ not is a real number. Thanks for that. However ... (I don't give up that easy)

If every fraction NOT ending in an infinite repeating is saved as already stated except set as next free even integer (multiplying with 2) - Just as in Hilbert's paradox of the Grand Hotel - with infinite new guests. Thus making:

• 0.1 -> 1 -> 2


• 0.2 -> 2 -> 4


• ...


• 0.5 -> 5 -> 10


• ...


• 0.01 -> 10 -> 20

The fractions not already mentioned (the ones with infinite repeating) will then get the uneven integers. The way to list these will then be done with Cantor's Diagonal listing:

• 1/1 is not in [0;1[ so moving on


• 1/2 is already represented (0.5 -> 5 -> 10) see above


• 2/2 again not in [0;1[


• 1/3 is not represented yet. 1/3 -> 1


• 2/3 is not represented yet. 2/3 -> 3


• ...


• 1/6 is not represented yet. 1/6 -> 5


• ...



• 4/6 is not represented yet. 4/6 -> 7

New Question

Is this proof then?