Universal Constants

[RocketManJames]

Not probability, but I figure posting here will bring this question to the right audience.

Of all the unitless Universal Constants, are there any that are rational? If we exclude 0 (null) and 1 (identity) as possible candidates, are there any?

I specifically ask for unitless constants, because those constants that have units are based purely on the definition of the underlying units.

The unitless universal constants that come to mind are: pi, e, gamma, and the Golden Ratio. What other unitless universal constants are there, and are any of them rational?

-RMJ

[PairTheBoard]

How about the area of a unit sided equilateral triangle.

opps. guess that one doesn't work.

PairTheBoard

[Duke]

At first thought I'm thinking that we just call these "numbers," and only call it a constant with a name if it is indeed transcendental.

~D

[Siegmund]

[ QUOTE ]


Of all the unitless Universal Constants, are there any that are rational? If we exclude 0 (null) and 1 (identity) as possible candidates, are there any?


[/ QUOTE ]

Under most people's definitions, no.

Generally a requirement for being considered a fundamental constant is that said constant is not expressible in terms of simpler numbers.

Once you admit 1 to the list, all rational numbers are expressible in a finite number of basic operations (a bunch of pluses and one division) on 1s.

Similarly, your list of constants will probably include either e, or ln 2, or ln 10, or 1/e, but only one of them (and it won't include any special values of the trigonometric functions, since those are all expressed in terms of e and i.)

Even the Golden Ratio isn't on some people's lists because it's an algebraic number.

I imagine that i is probably on most people's lists, even some of the people who exclude the golden ratio.

Other leading candidates are key numbers from physics like the fine structure constant, Feigenbaum's constant, and Apery's constant = zeta(3). For more candidate's, see MathWorld's constants page (http://mathworld.wolfram.com/topics/Constants.html).

[bakamono90]

The only constant is change.