I was looking at a post on MathOverflow about "What is your favorite 'strange' function?" One of the answers mentioned Thomae's function claiming that the function was "continuous at all irrationals and discontinuous at all rationals."
This got me to thinking, how are irrational and rational numbers related on the real number line? Is every irrational number "surrounded" by two rational number "neighbors?" For instance, can we think of the non-negative real number line as:
Zero (rational), an irrational number infinitesimally close to zero, a rational number infinitesimally larger than the previous irrational number, an irrational number infinitesimally larger than the previous rational number, etc. ?
The kicker is that, among real numbers, there is no such thing as two numbers that are "infinitesimally close" to each other.
For example, consider the set $S$ of numbers greater than $0.$ Take any $s\in S.$ Now, note that $\frac12s\in S,$ and is strictly smaller than $s$. Likewise, $\frac14s$ is an even smaller element of $S$, and so on. In fact, given any fixed $s_0\in S,$ there are uncountably-many $s\in S$ that are strictly less than $s_0$! This may seem rather astonishing, but is a fairly natural property of uncountable Archimedean ordered fields like the real numbers.
As for how the rationals and irrationals are arranged, the answer is: "densely." In particular, given any real $a,b$ with $a<b,$ there are countably-infinitely-many rational numbers in $(a,b)$ and uncountably-many irrationals in $(a,b).$