Is the fact that there are more irrational numbers than rational numbers useful?

Question

Although it is known that the cardinality of the set of irrational numbers is greater than the cardinality of the set of rational numbers, is there any usefulness/applications of this fact outside of mathematics?

Answer 1

The uncountability of the reals can be used to prove the undecidability of the language $A_{TM}$ consisting of pairs $(M, \omega)$, where $M$ is a Turing machine that accepts input $\omega$. The undecidability of $A_{TM}$ is often used to prove rather applied problems are themselves undecidable. This is practical in the sense that it's quite useful to know whether a problem is undecidable before you begin a quest for an algorithmic solution. Here are some undecidable problems.

Answer 2

It has many applications in computer science for example. The fact that $\Bbb{R}$ is separable (meaning it has a countable dense subset: $\Bbb{Q}$), means that many algorithms which work on the real numbers are possible through approximation.

Answer 3

Everything in "real" world is finite as the universe has finitely many particles. Therefore, if you consider "mathematics" any abstraction out of "real" world then no, there are no applications outside mathematics.

Now, solvig for instance PDE's that deals with problems of your day-by-day life is based on models that uses mathematical theories with many infinite cardinals. Is that useful?

Again, the fact that $\mathbb R$ is connected with respect the topology of the order is a pure mathematical fact. However, bilions of proofs use this fact. Also proofs of results that applies in "real" life. Is that useful?

Answer 4

Outside of mathematics, the distinction between the irrational number $\pi$ and the rational number $10^{-10000}\lfloor 10^{10000}\pi\rfloor$ is irrelevant. In everyday life, even the distinction between $\pi$ and $\frac{22}7$ is of little importance.