# Constants in nature are irrational

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?

Note by Ananay Agarwal
5 years, 1 month ago

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While 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.

- 5 years, 1 month ago

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.

Staff - 5 years, 1 month ago

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.

- 5 years, 1 month ago

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.

- 5 years, 1 month ago

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.

- 5 years, 1 month ago

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.

- 5 years, 1 month ago

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.

- 5 years, 1 month ago

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.

- 5 years, 1 month ago

We're stuck with $$3.14$$, not $$6.28$$. I mean, wouldn't you say the radius is more fundamental to a circle than the diameter? Thus the ratio between a circle's circumference to its radius should be the "real" circle constant. Euler's identity, written as $$e^{i\pi} = -1$$ basically says "$$pi$$ radians is half a revolution".

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.

- 5 years, 1 month ago

Right, I wasn't getting into $$\pi \text{ versus } \tau$$, but I totally agree with you. $$6.28$$ makes much more sense than $$3.14$$. Apart from that, I think that the more intelligent aliens have the constants $$\tau$$ and $$e$$ as well.

- 5 years, 1 month ago

I feel there is a problem with the nature of discussing what a civilization would perceive as "perfect" or not. Because perfection is purely a human invention, (even if taken from evidence in nature, such as observed symmetries and such ), attempting to predict what constants an alien race would use is complete speculation, and speculation with no real evidence to base it upon. Perhaps to an alien race, irrational numbers have an unseen symmetry that makes them appear perfect in comparison to the imperfect naturals.

- 5 years, 1 month ago

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...

- 5 years, 1 month ago

A couple of points:

1. This question, as mentioned by Ricky below, is applicable only to dimensionless numbers.
2. One must be exceedingly (exceedingly!) careful to differentiate between constants that appear in solutions of the fundamental physics equations vs. constants that appear in the equations themselves, independent of solution. For example, is $$\pi$$ a constant that appears in solutions only or does it appear in fundamental physics equations?
3. The constants that get inserted into fundamental physics equations are measured quantities, and hence, as John S mentioned, have error bars.
4. This is a theory dependent question. Some physics theories have lots of constants while others have only two and derive the rest . Therefore you need to put this question in the context of the theory you are talking about.
5. Good physicists argue about even defining the right number of constants of nature, regardless of whether or not they are irrational or rational.
6. I personally doubt fundamental dimensionless constants are irrational, as that would imply that physics has certain continuum behaviors that I find problematic. But that's just speculation.

Staff - 5 years, 1 month ago

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?

- 5 years, 1 month ago

What would you consider to be a fundamental physical constant?

• The fine structure constant is a constant that has physical meaning and is dimensionless.

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

• Yes, as it's dimensionless.

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

• No. We have numbers like $$\pi$$ in physics because the solution of 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.

Staff - 5 years, 1 month ago

What about e? It comes up naturally while solving the differentials.

- 5 years, 1 month ago

Precisely - e arrives when solving our equations. It's therefore a property of the solutions of our physical theories, not of the theories themselves.

Staff - 5 years, 1 month ago

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.

- 4 years ago

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

- 5 years, 1 month ago

hi

- 5 years, 1 month ago

Can there be antimatter for constants???

- 5 years, 1 month ago

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

- 5 years, 1 month ago

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?

- 5 years, 1 month ago