Are physical constants irrational?

[Magic Pancake]

Well, as the title asks, are the physical constants of the universe (such as plank's constant or the gravitational constant) rational or irrational numbers? I sure can't see a ready fraction in any of them, but then again these numbers are determined experimentally, so how can you tell for sure.

How about the universe itself? Is there a countable number of states for it?As a layman I understand that units of dimensions in the universe are quantified in various Plank units, so that would seem to say that these must be countable, but is this the case?

(Sorry if I mutilated some concepts.)

[Dr. Trintignant]

Eddington thought the fine structure constant was 1/136. Later, he changed his mind to 1/137. Both are wrong.

The short answer is that we don't know, and probably never will. Any number with a non-zero error bound can be approximated with a rational number, and most likely we'll never figure out what the physical constants are from first principles--only by imprecise measurements. They certainly "look" irrational, but that's infinitely far away from knowing that they're irrational.

[WhatRoughBeast]

Originally Posted by Magic Pancake

Well, as the title asks, are the physical constants of the universe (such as plank's constant or the gravitational constant) rational or irrational numbers? I sure can't see a ready fraction in any of them, but then again these numbers are determined experimentally, so how can you tell for sure.

Almost certainly. The thing is, their specific values are expressed in terms of units which are historical accidents, and (at least in terms of details) essentially random. Consider Planck's constant. Here is a quantity which, however fundamental, is expressed in units of J-s, or alternatively, kg-m^2/sec. Look up the historical foundations of each of the 3 units. Meter: based on the then-current understanding of the size of the earth. Kilogram: originally based on the meter (1 gram = 1 cc of water at triple point). Second: 1 sidereal day divided by (24 x 3600).

With units this arbitrary (from a universal perspective), why would you expect the numeric values of any fundamental constant to be other than essentially arbitrary? And since there are infinitely more irrational numbers than there are rational ones, well, you do the math.

The fine constant, of course, is the unitless joker, and it's anybody's guess how that one will turn out.

[Magic Pancake]

Originally Posted by WhatRoughBeast

Almost certainly. The thing is, their specific values are expressed in terms of units which are historical accidents, and (at least in terms of details) essentially random. Consider Planck's constant. Here is a quantity which, however fundamental, is expressed in units of J-s, or alternatively, kg-m^2/sec. Look up the historical foundations of each of the 3 units. Meter: based on the then-current understanding of the size of the earth. Kilogram: originally based on the meter (1 gram = 1 cc of water at triple point). Second: 1 sidereal day divided by (24 x 3600).

With units this arbitrary (from a universal perspective), why would you expect the numeric values of any fundamental constant to be other than essentially arbitrary? And since there are infinitely more irrational numbers than there are rational ones, well, you do the math.

The fine constant, of course, is the unitless joker, and it's anybody's guess how that one will turn out.

If the physical constants are rational, wouldn't it imply that the SI units, no matter how arbitrarily chosen, are multiples of them?

[Perpetual Student]

Why would one expect the fundamental constants to be rational any more than one would expect the ratio of the circumference to the diameter of a circle be rational?

[Modified]

Plank units turn a bunch of physical constants into rational numbers (1s).

[shadron]

Originally Posted by Perpetual Student

Why would one expect the fundamental constants to be rational any more than one would expect the ratio of the circumference to the diameter of a circle be rational?

On the other hand, why not? Is there some imbalance in the number of rational vs irrational numbers in the universe? What difference does it make to, say, the Plank constant that we happen to know pi is irrational?

[supaluminal]

Originally Posted by shadron

On the other hand, why not? Is there some imbalance in the number of rational vs irrational numbers in the universe? What difference does it make to, say, the Plank constant that we happen to know pi is irrational?

well, rational numebrs are countably infinite, irrational numbers are uncountably infinite, so from that alone, we shouldn't expect that there should be a bias towards rational numbers.

[xtifr]

Originally Posted by supaluminal

well, rational numebrs are countably infinite, irrational numbers are uncountably infinite, so from that alone, we shouldn't expect that there should be a bias towards rational numbers.

If you're going to test that theory, don't forget to account for the aleph-null hypothesis!

[Beerina]

Even the irrationals can be separated into those which have a formula and those which don't.

All the integers and rational numbers, and all the irrationals with a formula (which are enumerable) are an infinitely tiny set compared to all the irrationals without any possible formula.

[bjornart]

No, they're not. The speed of light, for instance, is one.

[Farsight]

Originally Posted by Magic Pancake

Well, as the title asks, are the physical constants of the universe (such as plank's constant or the gravitational constant) rational or irrational numbers?

Neither. See for example the NIST page on the fine structure constant. See the bit that says this:

Thus α depends upon the energy at which it is measured, increasing with increasing energy, and is considered an effective or running coupling constant. Indeed, due to e+e- and other vacuum polarization processes, at an energy corresponding to the mass of the W boson (approximately 81 GeV, equivalent to a distance of approximately 2 x 10^-18 m), α(mW) is approximately 1/128 compared with its zero-energy value of approximately 1/137. Thus the famous number 1/137 is not unique or especially fundamental.

It's a "running" constant, which means it varies. So it's neither rational nor irrational. It's often given as α = e²/2ε0hc, where e is the charge of the electron, ε0 is the permittivity of space, h is Planck's constant, and c is the speed of light. The e is described on the NIST website and elsewhere as "effective charge", but IMHO one should bear in mind conservation of charge. The effect of the electron's unit charge e might vary, but charge is conserved. So if α varies and e doesn't, that means ε0 and/or h and/or c vary too. You tend not to hear much about this sort of thing, but see this IOP physicsworld article Can GPS find variations in Planck's constant? Sadly what you do get to hear about is the "fine tuned" constants and the Goldilocks anthropic multiverse, which is woo.

[Tubbythin]

Originally Posted by Farsight

Neither. See for example the NIST page on the fine structure constant. See the bit that says this:

Thus α depends upon the energy at which it is measured, increasing with increasing energy, and is considered an effective or running coupling constant. Indeed, due to e+e- and other vacuum polarization processes, at an energy corresponding to the mass of the W boson (approximately 81 GeV, equivalent to a distance of approximately 2 x 10^-18 m), α(mW) is approximately 1/128 compared with its zero-energy value of approximately 1/137. Thus the famous number 1/137 is not unique or especially fundamental.

It's a "running" constant, which means it varies. So it's neither rational nor irrational. It's often given as α = e²/2ε0hc, where e is the charge of the electron, ε0 is the permittivity of space, h is Planck's constant, and c is the speed of light. The e is described on the NIST website and elsewhere as "effective charge", but IMHO one should bear in mind conservation of charge. The effect of the electron's unit charge e might vary, but charge is conserved. So if α varies and e doesn't, that means ε0 and/or h and/or c vary too. You tend not to hear much about this sort of thing, but see this IOP physicsworld article Can GPS find variations in Planck's constant?

Doesn't mean it can't be rational or irrational, just because it isn't actually a constant.

Quote:

Sadly what you do get to hear about is the "fine tuned" constants and the Goldilocks anthropic multiverse, which is woo.

No it isn't. Its speculative.

[edd]

Originally Posted by Tubbythin

Doesn't mean it can't be rational or irrational, just because it isn't actually a constant.

Not sure I follow you there Tubbythin, but it certainly doesn't change that the zero energy value is well defined as a single number and it seems to me that it's a valid question to ask about that number.

[Tubbythin]

Originally Posted by edd

Not sure I follow you there Tubbythin, but it certainly doesn't change that the zero energy value is well defined as a single number and it seems to me that it's a valid question to ask about that number.

I'm not sure I follow you, in turn.
I was saying that just because the fine structure constant may not be a constant, doesn't meant that we it is necessarily impossible to say whether or not it is (ir)rational (in principle).

[TubbaBlubba]

All fundamental physical constants are equal to one.

[phunk]

Originally Posted by bjornart

No, they're not. The speed of light, for instance, is one.

I think it only really matters for unitless constants. Anything with a unit can be defined as 1.

[TubbaBlubba]

Originally Posted by phunk

I think it only really matters for unitless constants. Anything with a unit can be defined as 1.

Yeah, there are those unitless buggers.

[ehcks]

Does it really matter if they're rational numbers in base 10 SI units?

[TubbaBlubba]

Originally Posted by ehcks

Does it really matter if they're rational numbers in base 10 SI units?

The unitless bastards are the same in any system of units, and irrational numbers are irrational in all bases, AFAIK.

[davefoc]

Originally Posted by WhatRoughBeast

Almost certainly. The thing is, their specific values are expressed in terms of units which are historical accidents, and (at least in terms of details) essentially random. Consider Planck's constant. Here is a quantity which, however fundamental, is expressed in units of J-s, or alternatively, kg-m^2/sec. Look up the historical foundations of each of the 3 units. Meter: based on the then-current understanding of the size of the earth. Kilogram: originally based on the meter (1 gram = 1 cc of water at triple point). Second: 1 sidereal day divided by (24 x 3600).

With units this arbitrary (from a universal perspective), why would you expect the numeric values of any fundamental constant to be other than essentially arbitrary? And since there are infinitely more irrational numbers than there are rational ones, well, you do the math.

The fine constant, of course, is the unitless joker, and it's anybody's guess how that one will turn out.

I think there is a fundamental logic error here. If a physical constant is rational it will be rational regardless of what units it is expressed in as long as the units are rational.

If the mass of an electron is a rational number if is measured in grams it will be a rational number if it is measured in troy ounces.

Are at least some ratio type constants guaranteed to be a rational number, for instance the ratio between the mass of an electron and the mass of a proton?

[ben m]

I don't think it's reasonable to label a physical constant rational or irrational.

The rational/irrational distinction is, in some sense, a statement about the infinite decimal expansion---if it eventually repeats forever, it's a rational number. If it doesn't, it's irrational. 1 + pi*10^(-10^(10^(10^1000000000000000))) is irrational, even though the non-repeating-zeroes behavior only shows up in the gazillionth digit.

What's the difference between a world in which "e" is really precisely 1.6*10^-19 (rational), and a world in which it's 1.6*10^-19*(1 + pi*10^(-10^(10^(10^1000000000000000))) ) (irrational)? Postulate some counting experiment. In the former case, this experiment expects N counts, and in the latter case it expects N+1 counts. You can only distinguish these values of "e" if the rational/irrational distinction begins being visible at the log10(N)-th digit.

But there's an upper bound on N---the visible Universe has a finite Bekenstein-Hawking entropy. You can't even imagine an experiment with a one count sensitivity to the "infinite" decimal expansion, because it would take all of the energy in the Universe to care about the Nth digit (for some large but finite N).

So, no, I don't think there is any physical meaning to the 10^100th (much less the infinite-th) digit of e, or h-bar, or alpha, so I don't think there is any sense in labeling these numbers as rational or irrational.

[Ryan O'Dine]

Originally Posted by Magic Pancake

...snip...
How about the universe itself? Is there a countable number of states for it?As a layman I understand that units of dimensions in the universe are quantified in various Plank units, so that would seem to say that these must be countable, but is this the case?

I'm in the middle of Brian Greene's latest The Hidden Reality, which addresses your second paragraph. Here's a quote:

Originally Posted by Brian Greene

...the number of distinct possible particle configurations within a cosmic horizon is about 10¹⁰¹²² (a one followed by 10¹²² zeros). This is a huge but decidedly finite number.
-- p.33 --

So within the region of the universe that could possibly affect you, there are indeed a finite number of possible configurations, or "a countable number of states," in your phrasing.

[Tubbythin]

Originally Posted by TubbaBlubba

The unitless bastards are the same in any system of units, and irrational numbers are irrational in all bases, AFAIK.

I suppose you could use base pi, then pi would be rational in that base. So I'm guessing irrational numbers are irrational in all rational bases only. But I am guessing.

[TubbaBlubba]

Originally Posted by Tubbythin

I suppose you could use base pi

I'm not sure what base pi even means. Do you count 0, 1, 2, .1415... or what? Is "base" even defined for irrational numbers?

[Perpetual Student]

Some of this stuff is just plain silly. The concept of base is defined only for integers. If a number is irrational it is irrational; base has no bearing of the meaning of irrational or rational regarding numbers. But many of the comments here are certainly irrational.

[Ziggurat]

Originally Posted by Magic Pancake

Well, as the title asks, are the physical constants of the universe (such as plank's constant or the gravitational constant) rational or irrational numbers?

One can pick units for which some constants are rational numbers. But even if one were to do this, some of them still must be irrational. For example, with the right set of units one could make hbar a rational number. But then h would necessarily be irrational. Conversely, if one makes h rational, then hbar must be irrational. Same thing for k and epsilon (two different forms of the constant for Coulomb's law): they're related by irrational numbers, so they cannot both be rational (but they can both be irrational).

Quote:

I sure can't see a ready fraction in any of them, but then again these numbers are determined experimentally, so how can you tell for sure.

You can't. But as mentioned above, since units are arbitrary and generally we don't pick them based upon fundamental constants, the probability is rather overwhelmingly in favor of any fundamental constant with units being irrational.

Quote:

How about the universe itself? Is there a countable number of states for it?As a layman I understand that units of dimensions in the universe are quantified in various Plank units, so that would seem to say that these must be countable, but is this the case?

No, that is not the case.

It's a common misconception that the Planck scales represent some sort of fundamental quantization of the universe. That is not the case. Or to be more precise, we have no evidence that this is the case. The significance of these scales is something rather different, namely the intersection of quantum mechanics and general relativity.

Let's think about length and mass/energy. In general relativity, a given mass will have an associated length scale, the Schwarzchild radius, which describes how small you need to get that mass in order to turn it into a black hole. If you're much further away from that mass than this length scale, then even if it IS a black hole, the gravitational field at that distance will still behave like Newtonian gravity. Which means you can essentially ignore general relativity. Conversely, if you're at distances from your mass of similar magnitude to this length scale, then even if it hasn't actually collapsed into a black hole, general relativistic effects will still be strong. So for a given mass, this distance characterizes the crossover from where you can ignore GR to where you must include GR. Now, the larger you make your mass, the longer this length scale will be.

We can do something very similar with quantum mechanics. Pick an energy, and you can associate a length scale, for example, the wavelength of a photon with that energy or the de Broglie wavelength of a particle. If we're at distances much larger than this wavelength, we can often ignore quantum effects, but below this length we definitely need to consider quantum mechanics. So that idea of a crossover is the same as with GR, but unlike GR, this length gets smaller as we increase out mass.

Now, let's start out by considering a very small mass/energy, much smaller than an electron. The quantum length scale is large, and the GR length scale is tiny. Now we start increasing the mass we're considering. The quantum length scale decreases, and the GR length scale increases. At some point, if we increase our mass enough, the two length scales will cross. This happens at the Planck mass, and the length is the Planck length. That's really the only thing either of those things definitely indicate.

But that's not quite the end of the story. If it were, they might be nothing more than a curiosity. The question is, what happens at these scales? And the answer is that we just don't know. The problem is that quantum mechanics and general relativity don't agree with each other. That's usually not a problem: almost everything in the universe interacts under conditions where we're in the quantum OR general relativity regime (or neither). So we can almost always ignore either quantum mechanics or general relativity, if not both. That means that we can get away without resolving that conflict. But if we got down to the Planck scales (which require fantastically large energy densities), then we wouldn't be able to ignore either GR or quantum mechanics, and we can't make predictions without knowing how to resolve that conflict. Now, some people speculate that some additional quantization of space itself happens at these scales, and that this additional quantization may resolve the conflict, but that's just speculation. We really don't know. And since we can't get anywhere near that density for practical reasons, we can't do direct experimentation to study such conditions either. But as it stands now, the Planck scales should be thought of as a limit to our understanding, not as a limit to space itself.

[Ziggurat]

Originally Posted by Tubbythin

I suppose you could use base pi, then pi would be rational in that base.

The "base" of a number system is a method of representing numbers with symbols. But it only applies to a certain category of number representation (Roman numerals don't have a "base"), and it is currently by definition integer only. If you want to introduce the concept of non-integer base, you need to define what you even mean by that, and I know of nobody who has done so in any sort of coherent form.

But even if one did, that would not make pi rational. Rationality is not determined by the compactness of a representation. One can easily form representations in which some irrational numbers have compact representations and integers have infinitely long representations (for example, just rescale: use a standard integer "base" but have the symbols represent multiples of an irrational number rather than integers), but doing so does nothing to change the actual nature of the numbers themselves. We chose our number system because it has a simple representation of integers, they aren't integers because they have a simple representation in our number system.

[Tubbythin]

Originally Posted by Ziggurat

The "base" of a number system is a method of representing numbers with symbols. But it only applies to a certain category of number representation (Roman numerals don't have a "base"), and it is currently by definition integer only. If you want to introduce the concept of non-integer base, you need to define what you even mean by that, and I know of nobody who has done so in any sort of coherent form.

Wikipedia has some stuff on the idea. Whether it has a consistent form or not is another matter.

Quote:

But even if one did, that would not make pi rational. Rationality is not determined by the compactness of a representation. One can easily form representations in which some irrational numbers have compact representations and integers have infinitely long representations (for example, just rescale: use a standard integer "base" but have the symbols represent multiples of an irrational number rather than integers), but doing so does nothing to change the actual nature of the numbers themselves. We chose our number system because it has a simple representation of integers, they aren't integers because they have a simple representation in our number system.

Point taken.

[Modified]

Originally Posted by Ziggurat

One can pick units for which some constants are rational numbers. But even if one were to do this, some of them still must be irrational. For example, with the right set of units one could make hbar a rational number. But then h would necessarily be irrational. Conversely, if one makes h rational, then hbar must be irrational. Same thing for k and epsilon (two different forms of the constant for Coulomb's law): they're related by irrational numbers, so they cannot both be rational (but they can both be irrational).

What if we allow irrational terms that can be determined to any desired precision mathematically (pi, etc.)? In other words, using the "best" base units possible (plank units?), how many physical constants can not be represented as equations involving only rational numbers and mathematical constants?

[Ziggurat]

Originally Posted by Modified

What if we allow irrational terms that can be determined to any desired precision mathematically (pi, etc.)? In other words, using the "best" base units possible (plank units?), how many physical constants can not be represented as equations involving only rational numbers and mathematical constants?

Given that the Planck scales are basically just ratios of other constants, if we used them as our units, that would guarantee that those specific constants would end up rational. But we have no reason to think that other constants (for example, the electron mass) would turn out to be rational in such units. And given the infinity of rational numbers, it's impossible to prove that one cannot do so. But we do know that you can't express all constants as an integer multiple of such scales. The Planck mass, for example, is far larger than any particle mass, so it would have to be a fraction of the Planck mass. We don't have infinite precision on the mass of an electron, so one could always find a rational fraction (actually, an infinitude of rational fractions) within our uncertainty, but that doesn't really demonstrate anything.

[Beerina]

Originally Posted by davefoc

I think there is a fundamental logic error here. If a physical constant is rational it will be rational regardless of what units it is expressed in as long as the units are rational.

If the mass of an electron is a rational number if is measured in grams it will be a rational number if it is measured in troy ounces.

Hmmmm. Are ounces and grams relatively rational? I suppose if they're both ultimately derived from a rational multiple of the same base unit, like mass equivalent of energy transitions of some electron, then obviously.

[Dr. Trintignant]

Originally Posted by ben m

So, no, I don't think there is any physical meaning to the 10^100th (much less the infinite-th) digit of e, or h-bar, or alpha, so I don't think there is any sense in labeling these numbers as rational or irrational.

It's at least conceivable, if not very likely, that some of the unitless constants will turn out have a computable value based on theoretical calculations, and from that we can decide if the number is rational or not. The googolth digit of pi is also totally irrelevant to our universe, but nevertheless we know it's transcendental.

[Dr. Trintignant]

Originally Posted by Beerina

Hmmmm. Are ounces and grams relatively rational? I suppose if they're both ultimately derived from a rational multiple of the same base unit, like mass equivalent of energy transitions of some electron, then obviously.

Most (all?) of the common customary units are currently defined as a rational multiple of metric units. The avoirdupois ounce is exactly 28.349523125 g.

[mijopaalmc]

If physical constants are derived by their relation to physical object and spacetime is quantized, physical constants are rational. Their mathematical abstractions, however, aren't.

[WhatRoughBeast]

The question is unanswerable, and the reason can be summed up in two words: experimental error.

We only know the value of fundamental constants, regardless of numeric base, through experiment, and all experimental results have some range of uncertainty, Therefor, any ratio of two fundamental constants will also have a range of uncertainty. Within this range there are an infinite number of possible rational values, and an infinitely larger number of irrational possibilities.

[nathan]

Originally Posted by Dr. Trintignant

Most (all?) of the common customary units are currently defined as a rational multiple of metric units. The avoirdupois ounce is exactly 28.349523125 g.

Right, but that's been achieved by (a) measurement and (b) changing the definition of the non-metric unit. There's plenty of opportunity for an irrational number to hide in the (impossible to measure exactly in new units) original definition.

For instance, an inch used to have some funky definition that was very nearly but not quite 25.4mm, but then its definition was changed to be exactly 25.4mm.

Even the metric definitions are mutable. It used to be that the metre was so many wavelengths of a certain kind of light, (after it stopped being the length of a specific lump of metal) now its a specific fraction of how far light goes in 1 second. Those aren't exactly the same. The speed of light had an error range when measured using the wavelength definition. The new definition simply decreed that error range to be zero.

[WhatRoughBeast]

Originally Posted by nathan

The speed of light had an error range when measured using the wavelength definition. The new definition simply decreed that error range to be zero.

Really? And how, exactly, does one measure a time duration of one second with zero uncertainty? I don't mean "to very high precision", I mean zero uncertainty.

[TubbaBlubba]

Originally Posted by WhatRoughBeast

Really? And how, exactly, does one measure a time duration of one second with zero uncertainty? I don't mean "to very high precision", I mean zero uncertainty.

It's not measured, it's defined. If a second is measured to be longer, the meter is longer.

[nathan]

Originally Posted by WhatRoughBeast

Really? And how, exactly, does one measure a time duration of one second with zero uncertainty? I don't mean "to very high precision", I mean zero uncertainty.

I think you're confusing the definition of a unit with producing a physical example of that unit. The definition of a metre really is what I said it is -- an exact fraction of the speed of light. Now to go and produce an example metre, you'd need to measure the speed of light, and that means accurate timing. That timing will have an error range associated with it, so your example metre will have an error range.

My Science Data Book (first published 1971) gives c as 2.997924590(8) 10⁸ m s^-1. That '(8)' is the +/- uncertainty in the final digit. The metre definition is given as 1650763.73 wavelengths in vacuo of the orange light emitted by 86Kr in the transition from 2p₁₀ to 5d₅.

You'll see that wavelength count is given to 1 part in 10⁸ and c has an error of about 1 in 10⁸. So at that point, they were kind-of equally accurate. The definition was changed because it's easier to measure the speed of light with current technology to equal or greater accuracy than it is to count the wavelengths needed by the earlier definition (it happens to be much simpler to state too). The current value for c is defined to be 2.99792458 10⁸ m s^-1 *exactly*.

Does that help?

NB If you go and look at the definitions for the SI fundamental units, you'll see all but one are in terms of measurements you can make some specified physical system. The Kg is the only one that's still defined in terms of a unique physical object. There are efforts to redefine the Kg in terms of things like 'number of atoms of Si', but that involves determining how to count 10^23 atoms. I'm not sure how successful those attempts currently are.