##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 theory## 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: $10\times824157\overline{824157}$? [1]: http://math.stackexchange.com/questions/1311/are-there-more-rational-numbers-than-integers