If you look at all the constants that exist in nature, \(\pi, e, \phi, \hbar\), etc, they are all irrational. Makes me wonder: is Nature not perfect? Is irrationality built into the Universe?

Or is the flaw in our number system? Could an alien civilization have a number system in which all of the natural constants are nice rational numbers?

According to me, that cannot be the case. We can approximate \(\pi\) using a ratio. And a ratio will be the same in all conceivable number systems. So, maybe an irrational number will remain irrational. Am I wrong?

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## Comments

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TopNewestWhile you are correct that \(\pi, e,\) and \( \phi\) are irrational, any physical constant with units like \(\hbar\) can't truly be rational or irrational because their values depend on arbitrary, man-made definitions of units. This is why \(c=299792458 \ \mathrm{m/s}\) is in fact a rational number in SI units: because it is defined to be this exact number by the definition of the meter. An attempt was made to define units in a way that is not based on humanity in any way, and in this system the values of all fundamental physical constants \(G, \hbar, c, k_e,\) and \(k_B\) are precisely \(1\). You can read more about this at http://en.wikipedia.org/wiki/Planck_units.

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I agree with what you said about units, though I would add that there is another fundamental reason why one cannot ask whether physical constants are rational or irrational. If you choose your units in a way that does not tautologically fix your constant to be equal to some chosen number, such as \(1\), as in the second part of your answer, then typically you cannot measure the value of your constant

exactly. If an exact measurement is impossible, then it does not make sense to ask whether a given value is rational or irrational.Log in to reply

True. So even dimensionless physical constants like the fine structure constant which are the same in all systems of units cannot be called rational or irrational.

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Actually, I believe the metre is defined using the speed of light and the second (and the kilogram using \(G\) ), but that's somewhat arbitrary.

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Kilogram is actually still defined by a standard kept in a vault in Paris. Defining 1kg in any other way so that it doesn't change with time is very difficult (there is work going on measuring how many silicon atoms or gold atoms are in 1kg, but that is very cumbersome). With that said, it is believed that the current kg isn't stable as every time it is brought together with its replicas for control measurements their relative weights have changed.

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I had wondered the same thing. But the fact that \(e^{i \pi} + 1 = 0\) makes me believe that somehow we did choose it right after all.

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Those are mathematical constants, not physical ones, which means that unlike for instance Avogadro's number and Planck's and Boltzmann's constants they would've popped out one way or another regardless of science's collective choises along the way, more or less identical to their current value.

That being said, I am among those who believe that Euler's identity demonstrates that we in fact chose... poorly. If aliens ever came to Earth and saw this they would most likely be thinking something along the lines of "Oh, those poor creatures, being stuck with the wrong constant." Then they would learn that we think of electrons as negatively charged, and be equally sympathetic. This is the curse of a science culture that defined things before it really knew what it was dealing with.

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Like you said, those mathematical constants would have popped out at some point, so in what way are we stuck with the wrong constant, or anything similar? As those constants are mathematical and not physical, I would say that those aliens have discovered that identity as well.

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I can assure you, the fact that our circle constant represents half a revolution, instead of a whole, makes introductory trigonometry a lot harder for people who aren't too comfortable with mathematics. Quick, how many degrees is \(\frac{\pi}{4}\) radians? Any sensible person without knowledge of radians would say that it ought to be \(90^\circ\), since that's a quarter of a circle. But no, it's a quarter of half a circle. This is what I mean when I say we chose poorly.

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I don't find it surprising, since 100% of real numbers are irrational. So if you pick a real number at random, it will be irrational with probability 1...

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A couple of points:

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What would you consider to be a fundamental physical constant? Would you say it is a constant that is not based on (trivial) man-made units? Would you consider \(\pi\) and \(e\) to be physical constants as well as mathematical ones?

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What would you consider to be a fundamental physical constant?

Would you say it is a constant that is not based on (trivial) man-made units?

Would you consider π and e to be physical constants as well as mathematical ones?

solutionof whatever the fundamental theory is can be described by geometry at everyday length scales, in particular Riemannian geometry. Therefore, we have the notion of a locally flat neighborhood around a point, and for these neighborhoods we can use concepts from Euclidean geometry like \(\pi\). This is, however, a property of the solution, not the fundamental physical theory. \(\pi\) doesn't appear in the action.Log in to reply

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solvingour equations. It's therefore a property of the solutions of our physical theories, not of the theories themselves.Log in to reply

If a number is irrational, does not mean that nature is imperfect. There is no number with privilege.

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There is an unfortunate historical connection between the words "rational", which is from the Latin word

rataionlils"of or belonging to reason", and "ratio", one of the meanings which is "relationship between two numbers", which back in Pythagoras's day (early Ionic Greek period), meant between two whole numbers. Pythogoras was a mystic, and believed that everything in nature can be described by ratios of whole numbers. This belief stemmed from his studies of "pure musical notes" which seemed "harmonious" to him and his followers. All nature was music, according to him. It's an appealing idea, but today even musical scales are no longer rational. It hasn't been since the advent of equal tempered or "well tempered" notes which largely replaced the older musical scales back in the 18th century, as popularized by Johann Sebastian Bach. The only thing rational about modern musical notes is the octave interval, which has the ratio of 2. Why? Because then all musical scales will play alike, except for consistent changes in frequency. Pythagoras, as it turns out, discovered to his horror that the square root of 2 cannot be expressed as a rational fraction, and so his entire theory that all nature is rational turned out to be irrational after all.There's absolutely no connection between the mathematical definition of "irrational" and any mental disorder associated with "irrational".

The famous mathematical constant pi is irrational. But, so? That constant appears everywhere in physics, it's fundamental to many things that makes for reality that we experience today. Is that "imperfect"? No. Pi is simply the ratio of the circumference of a perfect circle to its diameter in Euclidean space, and that ratio is perfectly understood and consistent. There are physical consequences to the fact that this ratio is mathematically irrational, as there are many other physical consequences to the fact many other constants are irrational, and we live with those consequences. Indeed, it's a good bet that if, somehow, there could exist an "universe" where all constants are rational, then there wouldn't be anybody such as you and I having this dialogue. It could be a very boring universe bereft of interesting things in it that live and breathe and think.

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hi

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Can there be antimatter for constants???

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Nature builds rational matter from irrational matter!!!!! It is UNIVERSALLY CORRECT!! I think so!!!

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This discussion is about mathematical information resulting from physical phenomena. Often, but not always, when discussing antimatter, values either remain the same (for some ratios) or get negated. Now, regarding your thought of "rational from irrational",

why?Log in to reply