[thedoc (OP)]

Pulkit Gambhir  asked the Naked Scientists:
   Hey guys

I had a question I was wondering if you could help me solve.

Often times reading about physics, I come across this concept of dimensionless physical constants. I am very curious to know how are the values of these constants determined. For instance as a high school student I never quite understood who & how managed to determine the value of Avogadro's constant.

Are all these constants only known from empirical expiremental calculations or do any of them have a closed form mathematical formula (presumable made from other mathematical constants).

And finally are all these physical constants known to be irrational numbers? If they are, then is there any theory as to why?

Thanks
Pulkit


 
What do you think?

[alancalverd]

Avogadro's constant can't be irrational - it's defined as the number of units in a mole, 6.022140857 × 10^23, i.e. an integer. You could in principle count the number of carbon atoms in 12 grams of carbon, or the number of hydrogen atoms in 1 gram of hydrogen, but the practical experimental determination is a bit more roundabout. Avogadro's hypothesis, as supported by Faraday's experiments, is that the numbers will be equal. If you are counting things, you will always end up with an integer, and all integers are rational.   

On the other hand the fine structure constant was for many years believed to be a rational number, 1/137, though nobody could work out why it should be. It now turns out that it isn't quite 1/137, so we have stopped looking for a reason why it should be - it just is what it is, from experimental measurement! 

Yet again, both π and e are necessarily irrational, can be proved so from their mathematical definitions, and can be calculated to a far greater degree of accuracy than they can be measured.

[evan_au]

Wikipedia helpfully has a list of dimensionless physical quantities. Many of these are dimensionless variables, comparing ratios of quantities which do have a dimension.

For example, the refractive index is the ratio of the speed of light in some material (in m/s) vs the speed of light in a vacuum. But there are many potential materials, which could be measured at many potential frequencies. You could measure the velocity in different units (eg miles/hour), but the ratio is dimensionless, and independent of the units used.

When you take a ratio, the dimensions cancel out, and you are left with a dimensionless number. See: https://en.wikipedia.org/wiki/List_of_dimensionless_quantities

dimensionless physical constants.

These are values which are considered basic to measurement.

Some of them are exact quantities because they are defined that way by the units we use - for example Avogadro's number of Carbon-12 atoms weighs exactly 12 grams, by definition. If Chemists were measuring in ounces instead of grams, Avogadro's number would be different.

But we only know Avogadro's number to about 8 decimal places, while it actually has 23 digits. So it doesn't have to be a whole number, but we may as well treat it as if it is a whole number.

Another one is defined to be irrational: the vacuum permeability constant µ₀, whose numerical value is 4π×10⁻⁷.

On the other hand, once we knew the fine structure constant to 5 decimal places, it became clear that it wasn't a whole number.

There are some constants which are believed to be so fundamental that they are independent of the units we use. For a list, see: https://en.wikipedia.org/wiki/Dimensionless_physical_constant