## 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][1], 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: $10\times824157\overline{824157}$? # EDIT # [Winther][2] 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][3]. 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][4]: - 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? [1]: https://math.stackexchange.com/questions/1311/are-there-more-rational-numbers-than-integers [2]: https://math.stackexchange.com/users/147873/winther [3]: http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel [4]: http://