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

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?

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What do you mean by "greater"? –  Saaqib Mahmuud Jul 7 '14 at 19:26
@Saaqib, edited. –  picakhu Jul 7 '14 at 19:27
Is the fact that $2^{10000000000000000000000000000000}+1$ is not the largest natural number useful outside of mathematics? –  Asaf Karagila Jul 7 '14 at 19:40
@BalarkaSen, could you please explain why it is "stupid"? –  picakhu Jul 7 '14 at 19:53
@picakhu People are voting to close this as "not about mathematics". I suggest you edit it to say something more like, "how is this fact used in applied mathematics"? –  NotNotLogical Jul 8 '14 at 0:23

4 Answers

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.

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For context, the study of Turing machines is pivotal as the theoretical underpinning of modern computer science. –  Andrew Coonce Jul 7 '14 at 23:40

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.

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I'm not sure if it's "of little importance", unless by everyday life you mean "going to the supermarket" or "binge watching Breaking Bad". In general if you have any interest in anything which includes distances which extend the solar system, you probably care a bit more about how precise your $\pi$ approximation is (probably not more than ten digits, but still. –  Asaf Karagila Jul 7 '14 at 20:32
@Surb: sure (though I don't think that's the best example), but this isn't really the point. Nothing in the physical world can be measured with less relative uncertainty than some $10^{-20}$, and even if it was $10^{-100}$ it wouldn't change the fundamental issue: the rationals are dense, so there are "sufficiently many" no matter how precise you measure. –  leftaroundabout Jul 7 '14 at 20:55

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.

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The fact that $\mathbb R$ is separable does not use that $\mathbb R$ is not countable. –  Andres Caicedo Jul 7 '14 at 19:42
@AndresCaicedo: ups, yes of course you are right –  sanjab Jul 7 '14 at 19:42
@AndresCaicedo But the fact that |ℝ| > |ℚ| is the reason that this is a useful optimization, isn't it? –  Aaron Dufour Jul 7 '14 at 21:10
@AaronDufour Not quite. Rather, it is the reason why such an approach is needed in the first place. –  Andres Caicedo Jul 7 '14 at 21:16

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?

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Do you think time is real? Do you think there are finitely many moments? –  MPW Jul 7 '14 at 21:10
I don't know if time "is" real, but I know that it is discretely measured by atomic clocks... –  user126154 Jul 7 '14 at 21:19
It's also measured by analog clocks with continuously sweeping set of hands, or by sundials with a similar continuous sweep. –  MPW Jul 7 '14 at 21:29
Our best known models of reality (the standard model and general relativity) both involve continuous space-time. The quantum amplitude is also a complex number rather than an integer. There's speculation that there may be an underlying discrete model, but it really is just speculation. –  Harry Johnston Jul 7 '14 at 21:31