How do we deal with irrationals in Physics?

Question

This question is really very basic: how do we deal with irrationals in Physics? If for instance, in some meaningful calculation we get a length of $\pi$ meters, or a force of $\pi$ Newtons, how should we understand this? One could say: "well, we understand it as a rational approximation", but then there comes another question: which rational approximation? We can approximate $\pi$ by $3,14$ or $3,1415$ and so forth, increasing the precision, but yet never getting really to $\pi$.

Another problem is: well, the equations gave us $\pi$, and this is not a physically meaningful thing, so is there something wrong with the equation? Because, in nature I think there really isn't irrationals lurking arround, so that this becomes a little complicated.

So: how do we deal with irrationals in Physics?

Answer 1

You just do. If a calculation gives you $\pi$, then you just say it gave you $\pi$. It is a real number and it is as good as any rational in describing any numeric quantity: $$\text{all physical quantities are described by }\mathbb{REAL}\text{ numbers}.$$ If you must insist on what real numbers are and how we can even conceive of such a number as $\pi$, then you need to look carefully at your definition. As far as physics goes, the definition of the real numbers is

The real number field $\mathbb R$ is the unique ordered field that is Dedekind complete.

This is independent of the multiple possible realizations of this set of axioms from a "more basic" set, and of the multiple representations which you might think of using (such as decimal expansions, continued fractions, Dedekind cuts, equivalent classes of Cauchy sequences, or what-have-you). In describing numeric quantities, we need a structure that allows us to

• add, substract, multiply and divide as usual,
• compare one number to another and always conclude that one is greater than (or equal to) the other, and
• take limits, infima, suprema, derivatives, integrals, and so on.

The definition above embodies all these requirements. Since there is a unique such structure (up to a natural isomorphism), we don't really care what that structure is meant to be.

"But, but, but...", I hear you say, "what about rational approximations? We can never measure an irrational number!" And yes, that is correct. But we can't measure rational numbers, either. Our measurements come with a central value and a finite, nonzero precision: they describe an interval.* If our calculations come out to an irrational number, then all we can say is whether that number is consistent with experiment, and it's exactly the same with a rational number. Say that some calculation predicted a value of 4/3 for some quantity: then, in your own words,

we can approximate $1.33$, $1.3333$, and so forth, increasing the precision, but yet never getting really to $4/3$.

Such is life, and such is the mathematical structure of physics.

_{*or a probability distribution, if you insist. Same difference, though.}

Answer 2

the equations gave us $\pi$, and this is not a physically meaningful thing...

What is "a physically meaningful thing" then? I suppose anybody would safely say that natural numbers are physically meaningful, you can certainly count separate entities.

But the trouble starts much earlier than irrational: what's a negative number supposed to be? I don't think you can give a sound physical definition, you need to resort to maths. Similarly for $\mathbb{Q}$: you can define a positive rational number as a tuple of natural numbers and describe how to add/multiple/divide those, but ab initio there's not much physical about this.

Incidentally, the completion to $\mathbb{R}$ is where things become physical again: unlike the rationals, the reals are a continuum. The usual mathematical definition goes via Cauchy sequences, which is actually more physical than one might think: it's a lot like measuring some quantity with ever-increasing precision. You get value after value in some sequence, and each time there's some uncertainty. But if you can limit that uncertainty for each measurement, and are able to make it go to zero, then the sequence of measurements converges towards the "true physical value" of the measured quantity. (Of course, this is never really possible due to quantum limitations.)

It boils down to this: maths by itself is never physically meaningful, you can only give it a physical meaning by mapping mathematical objects to measurement outcomes in some way. That rational numbers are in any way special and "more pure" is an idea going back to the Pythagoreans. From a modern point of view there aren't really good arguments to back this.