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Just as circumference of circle will remain $\pi$ for unit diameter, no matter what standard unit we take, are the speeds of light and sound irrational or rational in nature ?

I'm talking about theoretical speeds and not empirical, which of course are rational numbers.

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  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/2010/2451 , physics.stackexchange.com/q/52273/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Jul 23, 2014 at 6:43
  • $\begingroup$ Regarding rationality of numbers in measurement, David Z's answer is spot on. If you are trying to grasp whether the universe prefers integers, then yes it does. Things like the resonance frequencies of strings are in strict integral relationships (f, 2f, 3f, 4f). Quantum physics is also based on integers; the idea of quanta itself is that nature is lumpy rather than continuous. $\endgroup$
    – Phil H
    Commented Jul 23, 2014 at 9:09
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    $\begingroup$ π as a number is irrational. But π radians = 180 degrees, where π is irrational and 180 is rational. When you append the notion of units, the notions of rational and irrational become inapplicable. $\endgroup$
    – Timothy Shields
    Commented Jul 23, 2014 at 20:26
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    $\begingroup$ The speed of light is one of the physical constants (along with such things as Planck's constant (h) and the gravitational constant (G)), and given the current level of understanding there is no better answer than "because that's what it is". Given time, thought, and research it's arguably possible that we may someday understand WHY these constants have the values they do, but for now what we know is that they are what they are. If you really want to know why I suggest that you get your Ph.D. in physics, do the research, and tell the rest of us. $\endgroup$
    – Bob Jarvis - Слава Україні
    Commented Jul 24, 2014 at 12:00
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    $\begingroup$ The accepted answer is correct regards the question exactly as posed. However, what about the fine structure constant en.wikipedia.org/wiki/Fine-structure_constant? It is dimensionless and physical. $\endgroup$
    – Keith
    Commented Jul 25, 2014 at 5:15

6 Answers 6

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Something I posted on reddit answers this question quite well, I think:

"Rational" and "irrational" are properties of numbers. Quantities with units aren't numbers, so they're neither rational nor irrational. A quantity with units is the product of a number and something else (the unit) that isn't a number.

By choosing the unit you use to express a quantity, you can arrange for the numeric part of the quantity to be pretty much any number you want (though switching units won't let you change its sign or direction). In particular, it can be rational or irrational. And choices of units are a human convention, so it wouldn't make any sense to extend the idea of rationality or irrationality to the quantity itself.

You can use a natural unit system, where certain physical quantities are represented by pure numbers. For example, if you use the same units to measure time and space, $c = 1$. In such a unit system, it does make sense to say the speed of light is rational, but that's kind of a special case. That reasoning doesn't really work with other physical quantities. And you really do have to be using natural units. (Technically, you could make a natural unit system where $c = \pi$, but it would have very complicated and perhaps even inconsistent behavior under Lorentz transforms, so nobody does that.)

By the way, empirical measurements always have some uncertainty associated with them, so they're not really numbers either and are also neither rational nor irrational. A measurement is probably better thought of as a range (or better yet, a probability distribution) which will necessarily include both rational and irrational numbers.

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    $\begingroup$ And of course, in the geometric unit system, c=1 $\endgroup$
    – hdhondt
    Commented Jul 23, 2014 at 5:29
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    $\begingroup$ In the metric system, the meter is defined by the distance light travels through a vacuum in one second, so the speed of light in m/s is definitionally rational. $\endgroup$
    – Aaron Dufour
    Commented Jul 23, 2014 at 17:44
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    $\begingroup$ @Aaron the numeric part of the speed of light in this particular unit system, $c/(\text{m/s})$, is rational. But the speed of light itself, $c$, is not just a number so I don't think it's accurate to call it rational. $\endgroup$
    – David Z
    Commented Jul 23, 2014 at 17:52
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    $\begingroup$ Perhaps a better question would be "is the ratio of the speed of light to the speed of sound a rational or an irrational number?" I say "better" because that question at least has a precise meaning, of course, despite the fact that sound, unlike light, doesn't really have an "absolute" universal theoretical speed (as far as I know). $\endgroup$
    – Kyle Strand
    Commented Jul 23, 2014 at 19:52
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    $\begingroup$ I'm not sure in what sense a system with $c=\pi$ would have "very complicated and perhaps even inconsistent behavior under Lorentz transformations". A Lorentz transformation is simply a linear map on $\mathbb R^4$ preserving the quadratic form $(ct)^2-x^2-y^2-z^2$. Even leaving $c$ as a formal parameter, the theory is exactly what we teach in introductory courses. One can just as easily rescale $t$ such that $c=1$ or $c=\pi$ or any other positive real number; the same theorems in dimensional analysis ensure these are all equiconsistent. $\endgroup$
    – Logan M
    Commented Jul 24, 2014 at 6:07
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It depends on the unit you want to express it.

If you choose c/100 as the speed unit, c will be expressed with a rational number. If you choose c/π, you'll have an irrational one.

That depends on measure, not on nature.

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  • $\begingroup$ I had a college professor in Physics that set not only c=1, but also pi = 1 and 2*pi = 1. $\endgroup$
    – Oldcat
    Commented Jul 23, 2014 at 21:35
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    $\begingroup$ @Oldcat Let me guess. He/she was one of those people who just can't draw a decent circle on the whiteboard? $\endgroup$
    – Dawood ibn Kareem
    Commented Jul 24, 2014 at 12:11
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Well it's a tricky question in some way. You can for example consider the second as a rational number because its definition (a number of times the time needed for some atom to change state) is rational in nature (you can see it like this at least): you're technically just counting a number of occurrences of an event.

Then if you consider the speed of light it is the distance travelled by the light in this one second, you can also see it as a rational number. The meter is defined relatively to the speed of light as well (with an exact rational number)

For the speed of sound I guess it's harder to see it as rational as there's hardly "the" speed of sound since it depends on environmental parameters (no speed of sound in the void of space as everyone knows) so it's harder to associate it with something like a rational number.

I do agree with the previous answer saying that physical quantities are not really rational or irrational. in any case, it all comes down to how you see things.

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In the underlying physics, c = 1 (Planck units). 1 is rational. But your unit system might not have a rational length.

Speed of sound is rational in nature if macroscopic quantum mechanics holds (this is still open to debate that I will not enter). We should be able prove given macroscopic quantum that speed of sound is an integer multiple of Planck length / Plank Time because of the way particle interactions drive the speed of sound.

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I am also puzzled by this question. I would like to pose it in an alternative fashion. The question posed by kaka is simple and clear but the answers are too complicated.

Rational numbers and irrational numbers are mutually exclusive sets of numbers. Speed of light in vacuum has a constant numerical value,say in units of m/s. The question is does the numerical value fall in the set of rational numbers or the set of irrational numbers?

We talk of speed of mass to be 2m/s, sqrt2m/s etc., when we ask students to solve simple problems. Thus we treat speeds to be both rational and irrational numbers. But the numerical value of the speed of light which is a universal constant must fall either in the category of rational numbers or irrational numbers. To which category does it belong is the question.

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The circumference of a circle will be $\pi$ for a unit circle geometrically, but not physically; essentially because infinite precision doesn’t obtain in physics.

The question whether the speed of light or sound is rational or irrational is similarly ill-posed physically.

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