Is Planck's Constant Irrational?

Dragonfall

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

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 [latex]\mu_0 to be 4\pi*1e-7[/latex] 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

Depends on the definition, of course.

Bohrok

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

Just a comment:

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

--Elucidus

Elucidus

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

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

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

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

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

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

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