to continue the discussion, there are many kinds of infinities, and some are larger than others.
one of the smaller kinds of infinities is "countable". we say a collection of countably infinite things is countable if, intuitively, we can count them (as the parent says, put them in 1:1 correspondence with the natural numbers 1, 2, 3, ....).
there are only a countably infinite number of rational numbers, since each rational number has the form p / q, where p and q are (perhaps negative) integers. if we made a large 2d grid of all integer grid points, we could regard each rational number p/q as a grid point (p, q). Then we can count the grid points by starting at the origin of the grid (0, 0) * and spiraling outwards. This will eventually count every grid point, so this gives us a way to count all the rationals.
as the parent post says, by Cantor's diagonalisation argument we can demonstrate that we can't count the real numbers, so there are a lot more reals than rationals. it's a strictly bigger kind of infinity.
even if we start inventing new notation for particular reals we care about (e.g. pi, e, pi^e, door, super(door, |^|^bat)man, ) -- whatever you like provided it is well-defined, we can only name at most countably many reals, leaving a remainder of uncountably many un-named reals. we can single out any particular real that can be well-defined, and mint a new name for it, but we can only do this for at most countably many such reals, while the bulk of the reals escape naming.
* the origin (0, 0) corresponds to 0 / 0 which isn't a rational number, and some rational numbers such as 4 have multiple representations as coordinates. For example we could write 4 as 4 / 1 or 8 / 2 or 40 / 10 or -16 / -4 ... so strictly speaking by demonstrating we can count all of the 2d integer grid points shows that there are at most a countably infinite number of rationals. but since each natural number is a rational, and there are countably many rationals, we know there's at least a countably infinite number of rationals.