Adjusting Our Units: Can Fundamental Constants Have Integer Values?

[Mu naught]

Is it possible to adjust the length of the meter, the length of time of a second, the amount of mass in a gram and the other basic SI units in a way such that all the fundamental constants of nature have integer values?

 

[Mu naught]

bump - was my meaning in this question clear?

 

[D H]

Which of h versus \hbar is a fundamental constant of nature? Both of them?

 

[Mu naught]

Which of h versus \hbar is a fundamental constant of nature? Both of them?

I would think h is more fundamental. When I say fundamental I really mean constants that describe something about nature - like mu_0, /epsilon_0 - constants that are properties of the universe and determine things like the speed of light.

 

[Andy Resnick]

Sure- in 'natural' units, c = h = G = 1.

http://en.wikipedia.org/wiki/Natural_units

 

[D H]

Mu naught asked for all the fundamental constants of nature to have an integral value, not a select few.

Anyhow, the answer is no. The fine structure constant, for example, is unitless. It thus has the same value regardless of selected system of units, and this value is about 1/137.035999679.

 

[Mu naught]

Mu naught asked for all the fundamental constants of nature to have an integral value, not a select few.

Anyhow, the answer is no. The fine structure constant, for example, is unitless. It thus has the same value regardless of selected system of units, and this value is about 1/137.035999679.

That is an irrational number correct? So you're saying you could set all fundamental rational constants to integers?

Are there any examples of irrational constants which are not unitless?

I just find this interesting from a philosophical perspective.

 

[Redbelly98]

A related, and maybe more reasonable, question is: how many fundamental constants could we define to be an integer--or unity, for that matter? We're basically asking how many independent physical constants are there.

Certainly we could have 5 of them as integers:

c
h or h-bar (pick one)
G
e or α or ε_o or μ_o (pick one)
k
​

As a bonus we could have both ε_o and μ_o as integers, since c²=ε_oμ_o. That makes 6 possible integer constants, though only 5 are independent.

Did I miss anything?

EDIT:
Oops, α can't be an integer.

 

[Bob S]

Here is a list of most of the fundamental constants.

http://pdg.lbl.gov/2002/consonepagerpp.pdf

Can all of these be made integers simultaneously?

Bob S

 

[diazona]

That is an irrational number correct? So you're saying you could set all fundamental rational constants to integers?

No, the unitless fundamental constants have fixed values. If one were a rational non-integer number, it would be a rational non-integer no matter what system of units you chose.

Are there any examples of irrational constants which are not unitless?

Well now that depends entirely on which system of units you use But generally speaking, every constant has an irrational numeric value, unless you pick a unit system specifically designed to make its value rational (and as above, this is not possible for unitless constants).

A related, and maybe more reasonable, question is: how many fundamental constants could we define to be an integer--or unity, for that matter? We're basically asking how many independent physical constants are there.

Certainly we could have 5 of them as integers:

c
h or h-bar (pick one)
G
e or α or ε_o or μ_o (pick one)
k
​

As a bonus we could have both ε_o and μ_o as integers, since c²=ε_oμ_o. That makes 6 possible integer constants, though only 5 are independent.

Did I miss anything?

I don't think so, unless you had some sort of weird unit system in which your units of, say, force were defined independently. Then you'd have to have a "fundamental constant" to convert between (force unit) and (length unit)(mass unit)/(time unit)^2.

 

[granpa]

Any three of the four constants: c, ℏ, e, or 4πε0, can be normalized (leaving the remaining physical constant to take on a value that is a simple function of α, attesting to the fundamental nature of the fine-structure constant) but not all four.

4π is a natural part of certain equations in 3 dimensions and should not be normalized out of those equations (even though it would simplify the equations it would also make them less intuitive).

 

[Andy Resnick]

Is it possible to adjust the length of the meter, the length of time of a second, the amount of mass in a gram and the other basic SI units in a way such that all the fundamental constants of nature have integer values?

Mu naught asked for all the fundamental constants of nature to have an integral value, not a select few.

Ah- I mis-read the question.

http://physics.nist.gov/cuu/Constants/Table/allascii.txt

I'm also going with 'no' on this one.

 

[Redbelly98]

Shoot, I glossed over DH's answer earlier:

Anyhow, the answer is no. The fine structure constant, for example, is unitless. It thus has the same value regardless of selected system of units, and this value is about 1/137.035999679.

Here is a list of most of the fundamental constants.

http://pdg.lbl.gov/2002/consonepagerpp.pdf

Can all of these be made integers simultaneously?

Bob S

Besides the fact that the fine structure constant is the same in any system of units, consider the masses of the proton and electron which are both on that list. Their ratio would have to be a rational number, but experimental uncertainty will always prevent us from knowing if that is so.

 

[Pengwuino]

Any three of the four constants: c, ℏ, e, or 4πε0, can be normalized (leaving the remaining physical constant to take on a value that is a simple function of α, attesting to the fundamental nature of the fine-structure constant) but not all four.

And there is your problem, the constants of the universe have to play nice with each other.