# Is Planck's Constant Irrational?

1. Sep 9, 2009

### 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

2. Sep 9, 2009

### 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?

3. Sep 9, 2009

### 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.

4. Sep 9, 2009

### ice109

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

Last edited: Sep 9, 2009
5. Sep 10, 2009

### 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.)

6. Sep 10, 2009

### jhooper3581

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

Last edited by a moderator: May 4, 2017
7. Sep 10, 2009

### Unredeemed

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

8. Sep 10, 2009

### Tac-Tics

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).

9. Sep 10, 2009

### 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

10. Sep 10, 2009

### Unredeemed

okay, thanks all, I understand now.

11. Sep 10, 2009

### CRGreathouse

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

12. Sep 10, 2009

### Dragonfall

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

13. Sep 12, 2009

### ice109

because no in their right mind believes platonic objects exist in the world.
where do we find sqrt(2) in nature?

14. Sep 12, 2009

### Dragonfall

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

15. Sep 12, 2009

### lurflurf

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

16. Sep 12, 2009

### Mentallic

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.

17. Sep 12, 2009

### 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.

18. Sep 12, 2009

### 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.

19. Sep 12, 2009

### 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 $\mu_0 to be 4\pi*1e-7$ 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?

20. Sep 12, 2009

### CRGreathouse

Depends on the definition, of course.

21. Sep 12, 2009

### 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?

22. Sep 12, 2009

### 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

23. Sep 12, 2009

### Elucidus

Just a comment:

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

--Elucidus

24. Sep 12, 2009

### 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.

25. Sep 12, 2009

### 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.