Irrational Constants

The pure mathematical constants such as 0, 1/4, 1, sqrt 2, e, phi (golden section), pi, etc., can be proven to be rational or irrational as the case may be. This is a matter of definitions and logic.

The physical constants such as c, G, h, etc., are known only to a finite number of decimal places. If you stop there, they are all rational approximations of what could be either rational or irrational numbers. I don't know (yet, anyway) of any means to "prove" which they are. I suspect this will be forever impossible, but maybe some metaphysicians or theologians might weigh in on this. This group of constants (if indeed they are constants) is a matter of empirical observation, and is subject to margins of error, albeit very small.

"..maybe some metaphysicians or theologians.."

No, this is the sort of thing that is being done now by people who use logic and reason. Many of the arbitrary constants of the past are now understood as relationships between more fundamental constants.

If there's some connection, it'll be uncovered by hard work and the logical beauty of mathematics, with experimental physics used to select between the various consistent possibilities.

OOPS!, in my sudden burst of righteous indignation I didn't notice the smiley.

I think I'm with you on the logic/reason/math/physics/other science. Per the smiley, I suspect that the only metaphysics worthy of discussion is mathematics. Various philosophers might dissent from this view.

The original question perhaps leads into some deep and interesting waters. I surely don't have all the anwers--indeed, not very many--but some hints may well turn up in this thread. I hope to see a variety of views.

A geometric explanation of the forces associated with these constants might lead to an ability to prove logically that they are irrational. Given that gravity can be expressed with differential geometry, we might be able to prove that G is irrational.

It would be interesting to see if there was a proof proving that G was irrational. I think it may be possible since G basically says "how much space curves" given some amount of mass. Since the curvature of space due to mass is continuous, I actually think that means G has to be irrational.

Otherwise, without a geometric expression of a force, I don't see how we can prove the irrationality of the constant associated with that force. That means c and h are troublesome.

Excellent points! (GA) If two constants are related to each other by an irrational factor (as might be geometrically possible), then either or both are irrational (unless one is the reciprocal of the other or some such circumstance). I don't remember if convention calls for G to be uniform, with local curvature belonging to a second parameter, or if G is considered to vary with the curvature. If the latter, and continuous, then G would (almost) always be irrational. But yet further: if everything is quantized (digital physics again), then all would be rational! (Some of the numerators and denominators might be huge, though.) Ain't metaphysics fun?

My thoughts are .......... yes, both pi and e (amongst others) are irrational.

They're also transcendental numbers.

Exactly, and while we're at it √2 is irrational but not transcendental.

Cheers............Codey

Does it repeat?

No. If it repeated it would be rational.

Well, if it doesn't repeat and it doesn't terminate, then it's transcendental.

From wiki/Transcendental_number

a transcendental number is a number (possibly a complex number) that is not algebraic, that is, not a solution of a non-constant polynomial equation with rational coefficients.

In most cases, how would we ever know? We currently know G only to 4, 5, or 6 figures (depending on who you believe); how long before we figure out the 12th digit, for example?

Thinking a bit more and I would say that c, h, and G can't be irrational numbers because they're measured constants, and the uncertainty principle puts a limit on the accuracy, or number of digits we can know. With units of seconds, kg, and meters in a base ten number system, 10e-35 is the most accurate we could ever be knowing these constants. Yeah, c, and especially h and big G, being know to such a degree seems such a long way off!

As you say, if we take the physical constants c, G, h, etc., to be as measured or as written down, they will be rational numbers. But as to what the "real" values are...

From a probability standpoint, the rationals are countable; the irrationals are not. To the extent that one can say "pick a number, any number," the chance of hitting a rational is zero or arbitrarily close to it.

On the other hand, if Ed Fredkin's "digital physics" is an ultimate truth, then maybe all existent numbers are rational, and all the irrationals are in some kind of Platonic fantasy-land.

Then we run into the problem of "Light travels x meters per second... Uh, x.138... Uh, x.3184765... Uh, x.31847659312..." You get the idea.

GA. Measuring the value for something as fundamental to our understanding of the universe as the speed of light, in terms of arbitrary units (1 meter = 1/40,000 of our little planet's estimated circumference, 1 second = 1/86400 of our planet's estimated rotation period) might accidentally result in a rational number but the odds are strongly against it. Certainly the irrational numbers greatly outnumber the rational ones, since there are an infinite number of irrationals between any two consecutive rational numbers.

The number of irrational solutions to the Pythagorean theorem is much larger than the number of rational solutions. It has always boggled my mind that we were so fortunate that there was a small whole-number solution (3, 4, 5) to this theorem. If this had not been the case, then the odds of this lucky and fundamental discovery might never have occurred.

since there are an infinite number of irrationals between any two consecutive rational numbers.

That's a bit misleading: if you think about it no matter how you decide to "count" the rational numbers there is also an infinite number of rational numbers between any two consecutive rational numbers. (You can only really define consecutive in terms of the way you have decided to count them.)

The whole point is that the rational numbers are countably infinite (you can define a one to one mapping from the set of natural numbers to the set of rational numbers) whereas the irrational numbers are uncountably infinite.

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As an aside I've just discovered a new theorem:-

irrationals outnumber rationals by an infinite ratio,

therefore .........

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........ wait for it.......

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men don't exist.

Apologies to any ladies present.

GA. Exactly. But I would have said:-

the chance of hitting a rational is zero. (full stop)

However irrational (pun inadvertent) it seems: once you start to conceive how much bigger uncountably infinite is than countably infinite you realise just how ZERO that zero is.

I basically agree, and when I wrote that line at first I too stopped at the period. I added the codicil as sort of a nod to an argument like this: Pick any number.... Okay, I pick 2. Now I have accomplished a feat that has/had a probability of zero. Well, have I now mastered the impossible?

I even waffled on taking the codicil back out, but I thought that it might open up a sideline of discussion. Maybe this is just a matter of convention; anyway, I hope the whole structure of mathematics doesn't crash on my head!

First, a definition: a 'rational' fraction is one that can be calculated by dividing two whole numbers; it need NOT be finite.

For example, 2/3 = 0.3333333 ... The latter is rational since it's equal to 2 divided by three, but it's not finite.

As concerns whether some numbers such as pi perhaps being irrational: yes, there are such things. No matter which units are used to measure length, distance, area, etc., the ratio between a circle's area and its radius squared (assuming a flat plane) will ALWAYS be pi. And Pythagoras proved way back when that pi CANNOT be calculated by dividing two whole numbers. Pythagoras also used mathematics to calculate pi to a fair number of decimal places.

Some seemingly-irrational contants actually are rational: their values are just artifacts of the measurement units used. For example, if we hypothesize being able to measure precisely the distance that light travels in a precisely-measured second in precisely-flat space, then we might wind up with a figure with an infinite number of decimals in the 3xx.dd million metres in vacuum and flat-space. Or, we could define that length to be one light-second, in which case, the 'length' becomes one light-second, which is perfectly rational.

So, irrationals: they can't be measured precisely. Pseudo-irrationals: they can be measured, in which case one just has to change units (from metres to light-seconds, say).

Cheers!
DZ

For example, 2/3 = 0.3333333 ...

I trust you actually meant " 1/3 = 0.3333333 ... " or "2/3 = 0.6666666..."

Mathematics and measurements are abstract, (read as 'man-made'). Constants, such as pi, are measured observations of relationships (such as a circles circumference in 'relation' to its diameter). While we 'observe' gravity, we do not yet have a basis of comparison or relationship that defines what it is or where it comes from. Any and all 'theories' of gravity are tryimg to find these fundamentals. Equating a gravitational concept to another constant would yield the 'holy grail' in mans quest for describing the universe we live in. (Well, most of us anyway.)

In the middle of a Blackhole all are irrelevent.