Is Planck's Constant Irrational?

[Unredeemed]

My maths teacher was talking about irrational numbers and I asked if Planck's constant was one, but he said no. However, I don't understand how this can be as it does not seem to terminate. Can anyone help?

Thanks,
Jamie

 

[g_edgar]

Unless you know the number exactly (as you can with a constant defined mathematically, but as you cannot with a constant defined using measurements), you cannot tell if it is rational or irrational.

What is a MATHEMATICAL definition of Planck's Constant? One that does not require measuring physical quantities?

 

[LeonhardEuler]

It definitely depends on what units you choose to express Plank's constant in. In atomic units, for example, \hbar = \frac{h}{2\pi} is defined to have a value of one. In this case, Plank's constant itself is clearly irrational since it is equal to 2\pi. It would be possible to define the units in another way so that Plank's constant itself is one, in which case it would be rational.

In SI units, Plank's constant has to be measured rather than defined. Since there is always (presumably) experimental uncertainty, this would mean that it is impossible to know whether it is rational or not in that system. If you pick a number, say 5.274, and then you pick an uncertainty, say 0.000001, there will always be both rational and irrational numbers in the interval [5.274 - 0.000001, 5.274 + 0.000001], and this will hold no matter what numbers you choose as long as the uncertainty is more than 0. So we will probably never know.

 

[ice109]

what? in atomic units hbar = 1 not h. no physical quantity can be irrational.

 

[HallsofIvy]

What authority do you have for the statement "no physical quantity can be irrational"?

(if \hbar is 1, then h= 2\pi which is irrational.)

 

[jhooper3581]

Of course, Planck's constant is described in this http://en.wikipedia.org/wiki/Planck_constant" .

 

[Unredeemed]

But the stand-alone Planck's constant of 6.67 the 10^-19 is not irrational?

 

[Tac-Tics]

What authority do you have for the statement "no physical quantity can be irrational"?

(if \hbar is 1, then h= 2\pi which is irrational.)

What proof is there that both \hbar and h are both physical quantities? Maybe one is physical and the other is simply derived.


The real answer to this question that has been hinted at is that the numbers used in physics are entirely approximate. We don't know their exact values. But that doesn't bother anyone because most phenomena are modeled by continuous functions (where small deviations are unimportant).

 

[Elucidus]

Since the exact value of Planck's constant is not known, the rationality or irrationality of the constant cannot be presently determined.

Only constants that are exactly known can be categorized.

--Elucidus

 

[Unredeemed]

okay, thanks all, I understand now.

 

[CRGreathouse]

Only constants that are exactly known can be categorized.

...which follows from the density of irrationals and rationals in the reals.

 

[Dragonfall]

no physical quantity can be irrational.

Let's not forget \sqrt{2} now.

 

[ice109]

What authority do you have for the statement "no physical quantity can be irrational"?

(if \hbar is 1, then h= 2\pi which is irrational.)

because no in their right mind believes platonic objects exist in the world.

Let's not forget \sqrt{2} now.

where do we find sqrt(2) in nature?

 

[Dragonfall]

where do we find sqrt(2) in nature?

The length of the long side of a right triangle whose short sides are 1.

 

[lurflurf]

The length of the long side of a right triangle whose short sides are 1.

Thare are no triangles in nature, and if there were still would not be right or isosceles triangles.

 

[Mentallic]

The length of the long side of a right triangle whose short sides are 1.

But is this not a mathematically derived quantity? If we instead knew nothing about this triangle, we would try to measure the ratio of the hypotenuse to a side length as closely as possible, but would always have uncertainties. Thus, rational or irrational?

Just look at the history of pi. Thousands of years ago they would give the constant pi an approximate rational value. Until it was mathematically derived, it was unknown if pi truly was irrational or not.

 

[Preno]

The question doesn't even make sense. You need to specify which units you are using (and you need to be able to specify those units with arbitrary precision). Anyway, in almost all units, it is irrational, because almost all reals are irrational.

 

[Dragonfall]

Thare are no triangles in nature, and if there were still would not be right or isosceles triangles.

In that sense there are not any mathematical objects in nature. No wave functions, no tensors, no curvature of space-time.

Being a Platonist, I say there things exist.

 

[f95toli]

Since the exact value of Planck's constant is not known, the rationality or irrationality of the constant cannot be presently determined.

Although it is possible that we will one day DEFINE Planck's constant to have a certain value (this might happen in a few years time), in the same way as we've e.g. defined \mu_0 to be 4\pi*1e-7 or the speed of light be be equal to 299,792,458 m/s.

Would defining it a constant to have a specific value automatically make it a rational number?

 

[CRGreathouse]

Would defining it a constant to have a specific value automatically make it a rational number?

Depends on the definition, of course.

 

[Bohrok]

Only constants that are exactly known can be categorized.

--Elucidus

I have to ask now: is Avogadro's number rational? I would think so, but we don't know it's exact value, so we can't categorize it either way?

 

[Red_CCF]

My Professor stated that all numbers are a figment of our imagination and that they do not actually exist in the real world. Numbers are just symbols we use to represent something that is real

 

[Elucidus]

I have to ask now: is Avogadro's number rational? I would think so, but we don't know it's exact value, so we can't categorize it either way?

Just a comment:

Avogadro's Number is an integer (and a very big one) and therefore rational.

--Elucidus

 

[Elucidus]

Although it is possible that we will one day DEFINE Planck's constant to have a certain value (this might happen in a few years time), in the same way as we've e.g. defined \mu_0 to be 4\pi*1e-7 or the speed of light be be equal to 299,792,458 m/s.

Would defining it a constant to have a specific value automatically make it a rational number?

Firstly, what Physicists define a particular constant to be is a working definition. The actual constant is what it is - and may not be fully understood by scientists. The current definitions may in fact be off. The speed of light for example might be 299,792,458.000145669 m/s for all we know. Just because we say something is such does not make it true.

Secondly, there are many constants which are defined that are rational (zero), irrational (pi), or currently undetermined (Euler's gamma constant). Any "constant" derived from science could be anything and we may never have the scientific exactitude to accurately measure it to find out.

So defining something does not make it necessarily rational. Although the number that is the working definition might be.

--Elucidus

P.S. One might argue that if there exists a minimum quantum distance, then all distances in the universe are commensurable and consequently there are no such things as right angles, isosceles triangles, squares, or true circles. Unfortunately we may never know.

 

[LeonhardEuler]

Just a comment:

Avogadro's Number is an integer (and a very big one) and therefore rational.

--Elucidus

I don't think this is necessarily true. Avogadro's Number is defined as the number of Carbon 12 atoms in 12 grams of Carbon 12. The gram is defined as 1/1000th the mass of the kilogram, which is a specific platinum-iridium artifact in France. Since it is not made of pure carbon 12, there is no reason to suppose that the ratio of the gram to the mass of a carbon 12 atom is rational, and it is therefore possible (probable, even, it seems given that the rationals are a set of measure zero) that it is irrational. In truth it changes over time as the kilogram loses mass slowly due to evaporation and unknown causes that have reduced its mass over the years.

 

[Elucidus]

I don't think this is necessarily true. Avogadro's Number is defined as the number of Carbon 12 atoms in 12 grams of Carbon 12. The gram is defined as 1/1000th the mass of the kilogram, which is a specific platinum-iridium artifact in France. Since it is not made of pure carbon 12, there is no reason to suppose that the ratio of the gram to the mass of a carbon 12 atom is rational, and it is therefore possible (probable, even, it seems given that the rationals are a set of measure zero) that it is irrational. In truth it changes over time as the kilogram loses mass slowly due to evaporation and unknown causes that have reduced its mass over the years.

As I understand it, and since I am not a Physicist I may be mistaken, Avogadro's number counts the number of molecules or atoms of something one needs until that substance weighs as much (in grams) equal to its atomic weight. Even if you changed the units of measure, the constant is still measuring the "number" of discrete objects. If you had 12 gm of C¹², since it consists solely of C¹² then there must be an integral number of atoms. Unless I am grossly mistaken, the constant has to be an integer (even if the reference kg changes mass).

--Elucidus

 

[Dragonfall]

Keep adding carbon 12 atoms until you exceed 12 grams. Subtract 1.

 

[LeonhardEuler]

As I understand it, and since I am not a Physicist I may be mistaken, Avogadro's number counts the number of molecules or atoms of something one needs until that substance weighs as much (in grams) equal to its atomic weight. Even if you changed the units of measure, the constant is still measuring the "number" of discrete objects. If you had 12 gm of C¹², since it consists solely of C¹² then there must be an integral number of atoms. Unless I am grossly mistaken, the constant has to be an integer (even if the reference kg changes mass).

--Elucidus

That works as a practical way of thinking about it, but in practice, it doesn't work that way for the reason I pointed out. There is no good reason to think that you can have exactly 12 grams of carbon 12 because of the way the gram is defined. If you add atoms one by one, you will probably come to a point where you have slightly less than 12g, and if you add one more atom you will have slightly more. Few people ever worry about this because the most precise measurements people can make are nowhere near precise enough for this to be an issue.

 

[LeonhardEuler]

Keep adding carbon 12 atoms until you exceed 12 grams. Subtract 1.

But that would not fit the definition of Avogadro's Number. That is the floor function of it. Why not take the ceiling function of it (i.e. the number you get just after you exeed 12 g)?

 

[f95toli]

The current definitions may in fact be off. The speed of light for example might be 299,792,458.000145669 m/s for all we know. Just because we say something is such does not make it true.

No, the speed of light is -by definition- 299,792,458 m/s. Imroved measurement modify the length of the meter not c , since the meter is defined using the speed of light.

The same would be true for Planck's constant if that is ever defined as a constant; it would be defined as a real number which would -again by definition- be exact. Any improvements in the accuracy of the measurement after that would just re-define units realized using Planck's constant , in this case it would be the kilogram.

Btw, Avogadros constant is one of the constants that will possibly be defined as a constant in the SI in a couple of year.

 

[Bohrok]

If you add atoms one by one, you will probably come to a point where you have slightly less than 12g, and if you add one more atom you will have slightly more.

That's exactly what I was thinking. Atoms' masses are so small so we have no practical way of measuring so precisely, but I don't see a reason to believe that a whole number of ¹²C happens to weigh exactly 12 g. It could be that the closest one can get is barely under or barely over exactly 12 grams by half the mass of a neutron or any other small, positive number. And since it's nigh impossible to get two 12-gram samples of ¹²C with exactly the same percentage of isotopes, Avogadro's number will vary from sample to sample, which I hadn't thought of before...

 

[Elucidus]

If one defines units of measure on phyisical observations (as the speed of light), then yes it is possible to exactly know what the constant equals (since one is using the constant as the new "yard stick")

And I can see Avogardo's number being defined some day to be so many atoms of standard C¹² atoms (or some other standardizable atom) - and define the kilogram to be so many of those. This completely changes the conversation though.

Of course, there is always the issue as to whether all neutrons weigh the same etc.

But we digress. Some constants are rational, some aren't.

--Elucidus