How rational infinity and integer infinity are of the same size?

Question

I was reading the book "One Two Three...Infinity" by George Gamow, too old book, yes I know but its not my fault if I am born late, nevertheless, he was talking about comparing infinities with nice examples and he successfully convinced me that even infinity is of the same size as of the integer infinity, he explained that by forming pairs like this, []

Then he started proving how fractional/rational infinity are also the same size of integer infinity, I will quote from his book, []

Now this means that the number 2 is associated with 1 fraction and the number 3 is associated with 2 fraction and the number 4 is associated with 3 fractions(so on and so forth), unlike infinities of even and integer here one element of one infinity is associated with one or more than one element of the second infinity(fractional),so that clearly makes rational infinity bigger in size, so if someone can correct the miscommunication between me and Mr.Galow, Thank you.

Answer 1

What this means is that there is a bijection between the integers and the rationals (written in lowest terms), by ordering the rationals by (a) the sum of their numerator and denominator and (b) by their denominator.

Here is the beginning of the positive part of this:

$$\begin{array} \, & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & \text{etc.} \\ \, &\updownarrow & \updownarrow & \updownarrow & \updownarrow & \updownarrow & \updownarrow & \updownarrow & \updownarrow & \updownarrow & \updownarrow & \updownarrow & \updownarrow & \updownarrow & \updownarrow & \cdots \\ \, & \frac11 & \frac21 & \frac12 & \frac31 & \frac13 & \frac41 & \frac32 & \frac23 & \frac14 & \frac51 & \frac15 & \frac61 & \frac52 & \frac43 & \text{etc.} \end{array}$$

and the non-positive part could then be $$\begin{array} \, & \text{etc.} & -3 & -2 & -1 & 0 \\ \, & \cdots & \updownarrow & \updownarrow & \updownarrow & \updownarrow \\ \, & \text{etc.} & -\frac12 & -\frac21 & -\frac11 & \frac01 \end{array}$$

Answer 2

Notice that even if associate a set of rational numbers to an integer, this set is finite, so we can redefine the correspondence to make it a 1-1 correspondence $\mbox{natural numbers}\leftrightarrow\mbox{rational numbers}$: Suppose that to $j\in\mathbb N$ we associate the set $$\{q^j_1,\cdots,q^j_{n_j}\}$$of rational numbers, then you can assign to the first $n_j$ natural numbers the rational numbers $q^1_k$ as $$k\mapsto q^1_k$$ then you assign to $n_1+1,\cdots,n_1+n_2$ the rational numbers $q^2_j$ as $$n_1+k\mapsto q^2_k$$ you assign to $n_1+n_2+1,\cdots,n_1+n_2+n_3$ the rationals $q^3_j$ as $$n_1+n_2+k\mapsto q^3_j$$ and so on. At the stage $p$, you assign to the natural numbers $n_1+\cdots+n_{p-1}+1,\cdots, n_1+\cdots+n_{p-1}+n_p$ the rational numbers $q^p_k$ as $$n_1+\cdots+n_{p-1}+k\mapsto q^p_k$$ this way you have your 1-1 correspondence