Arbitrary meter (metre)

[FrankM]

The methods used to determine the "size" of the original meter resulted in a length that can be considered somewhat arbitrary, as it wasn't exactly 1/10,000,000 the distance between the equation and the pole, this running on a line through Paris.

 

What impact would it have had on science if the meter had been slightly shorter, and its length had accidently resulted in matching a measured speed of light value of 314159265 meters/second?

 

How would scientists have reacted if they had to use a value of c=Pi(10^8) in equations dealing with physical law?

[Qfwfq]

No essential difference at all. Especially when natural units are used.

[FrankM]

Qfwfq - Not everyone thinks that way, which is why I presented this as a post subject.

 

I had a recent communications with a professor who has a very strong opinion to the contrary.

 

If Pi were artificially incorporated into definitions so the speed of light had Pi in it, it would be a physically meaningless appearance of pi, and when it canceled or combined with other quantities that had physically meaningful factors of pi, it would be a mess.

 

I had mentioned that Stoney units had been created in 1881 to eliminate the unwieldly numeric value of the speed of light (SOL) from equations. There are a number of other systems that are called "natural units" that provide a similar process. I guess equating the SOL "c=1" is not physically meaningless to the professor but having "c=pi(10^8)" is. The primary purpose of "natural units" is to equate the various unwieldy numeric values of the physical constants into "equation friendly" values.

 

One must realize that if the meter had been a shorter value, all of the derived units which are used to provide values for other measurements of "physical properties" would be related, thus if Pi cancelled it would be meaningful.

[Qfwfq]

Suppose you had an equation with a factor of [imath]\frac{c}{\pi}[/imath] in it. Suppose you write c as being [imath]c=\pi a[/imath] where a is the value you like, according to the units chosen for c. The above equation's factor could be recast as a, but why would this be an inevitable mess? It would be a matter of free choice and can be done for whatever units c is expressed in, even if a doesn't have a round value. If a factor such as the above [imath]\frac{c}{\pi}[/imath] were especially important, a symbol would likely have been chosen for the ratio, much like for Planck's constant: [imath]\hbar=\frac{h}{2\pi}[/imath]. If this isn't a mess, why would it be for c?

[Pyrotex]

The professor is right. Having c = pi * 10^8 m/s would be frought with peril.

 

First of all, because it would be impossible for c = pi * 10^8. Pi is a transcendent number... an infinite string of non-repeating digits. At best, an accident of history might have c with the first n digits matching pi, where n might be 6 or 8 or 10.

Assuming the first digit, 3, to be a given in mks units, the odds of the next 6 digits "accidently" being the same as pi, would be 1 chance in a million.

The odds of the next 9 digits being the same as pi, would be 1 chance in a billion.

The odds of ALL digits being the same as pi would be zero.

So, if you took the bait, and express c as pi * 10^8, you would be introducing error. You could NOT in principle cancel c with pi as discussed, because, REALLY, what you would have would be c = py * 10^8, where py is just an approximation of pi.

 

Second, it would introduce confusion. Pi exists in many formulas for physical or geometric reasons. Either you had to integrate over a circle or a sphere to get your final equation, or you were dealing with physical or mathematical entities known to have pi as a common factor or divisor. Pi in an equation means something.

 

If c= pi * 10^8 by accident of history, it would be meaningless. Canceling out your accidental pi (actually an approximation to pi as proven above) with a meaningful pi would lead to misunderstandings as to the origin of functions and equations.

[Qfwfq]

Pi is a transcendent number... an infinite string of non-repeating digits. At best, an accident of history might have c with the first n digits matching pi, where n might be 6 or 8 or 10.

How do you know the exact value of c in m/s ain't transcendental? :evil:

 

Either you had to integrate over a circle or a sphere to get your final equation, or you were dealing with physical or mathematical entities known to have pi as a common factor or divisor. Pi in an equation means something.

Just like in the case I compared it with, the [imath]2\pi[/imath] down there is the full circle in radians...

[Pyrotex]

How do you know the exact value of c in m/s ain't transcendental? :evil:

NOW who's being careless?? :hihi:

Measured quantities cannot, by definition, be referred to as transcendental. Transcendentals, by their nature, are calculatable to any number of digits, even millions of digits. They do not depend on crass physical measure, or engineering accuracy, but on pure mathematical principals.

Just like in the case I compared it with, the [imath]2pi[/imath] down there is the full circle in radians...

Nope. Apples and Oranges.

[FrankM]

The numeric value for the speed of light I had actually presented to the professor, using the shortened meter, was 314159264 m/s, which matched the precision of the current SOL using the SI meter. That number is not transcendental, but when it appears in an equation where it might be a dividend or divisor to a value of Pi, the difference would outside of the precision of the defined value for the SOL.

 

If the size of the length of the meter was one-half of that which produced the 314159264 m/s value, it would be 628318530 m/s, which would pose an equally difficult problem as that value would not be transcendental either, just close to a number that is transcendental. Every wavelength, regardless of size, can be expressed as having a 2Pi radian length, which is a transcendental value. That is not a measured value, it is a mathematical value.

 

Measured quantities cannot, by definition, be referred to as transcendental.

I am curious where you found that definition. I have been trying to identify if there are basic axioms of "Measurement Theory" which describe what can or cannot be a "unit of measure".

 

We do know that the current SI meter is a defined value not a measured value. Every time there was an advance in measurement technology the old platinum meter could be measured to a greater degree of precision, and this technically could go on ad infinitum, which would place the physical measurement in the position of being a transcendental number.

 

Are we better off having a numeric value for the speed of light that is "defined" to a limited precision or having it equated to a number with unlimited precision?

[Qfwfq]

Measured quantities cannot, by definition, be referred to as transcendental.

Not by definition, but the matter is more subtle. There is no way to rule out that two quantities, given a definition of each, could have that ratio. It is conceptually possible for any type of quantity as much as for lengths. The lack of perfect measurements prevents experimental verification of two quantities having that exact ratio but the ratio cannot be such a number. Unless the one quantity is defined in terms of the other, it makes less sense to believe the ratio will be rational than that it will be irrational or transcendental. It doesn't make much more sense the other way around, either. The crux is that by virtue of measurement alone, the quantity can't be determined as being rational or not, transcendental or not.

 

Any irrational number is an accumulation point for rational numbers. If all measurements of a given quantity q in given units u have each given a number and uncertainty compatible with [imath]\pi[/imath] (or a rational multiple of it), there are no logical grounds by which there must be a precision sufficient to make the distinction. You can't assert that it is, but you can't assert it isn't either and might never be able to.

 

Nope. Apples and Oranges.

The [imath]2\pi[/imath] sitting underneath Planck's arse is exactly the angle in radians of a full cycle, it has just as much meaning as in your point.

 

I am curious where you found that definition.

He did not really give a definition.

 

I have been trying to identify if there are basic axioms of "Measurement Theory" which describe what can or cannot be a "unit of measure".

Any repeatable quantity.

 

We do know that the current SI meter is a defined value not a measured value.

I disagree with the distinction. The old bar at Sèvres gave a repeatable length, within a precision reasonable for the time. The krypton spectral emission line gives a more precisely repeatable length. Each of these is a definition of a length, which is what a unit is. Any such definition is useless without being compared to other quantities of the same kind, which means a measurement. Metrology is a chain, from the original, or primary, standard (definition) down to the piece of cloth being sold; in a sense, you don't "measure" the primary standard at all.

 

Are we better off having a numeric value for the speed of light that is "defined" to a limited precision or having it equated to a number with unlimited precision?

The best thing is to choose units so that c = 1.

[FrankM]

The best thing is to choose units so that c = 1.

 

You cannot choose units (plural) that will result in c=1. If you chose a unit that made c=1, it would be quite large, and it that unit was called a meter we would have c = 1 meter/sec.

 

I think the whole concept of "natural units" has been distorted by those that claim "1" is the only natural number. Is a natural number that is scaled up or down by factors of 10 still a natural number? Is pi still a natural number when scaled as pi(10^8)?

 

If scaling is okay, then c = 100,000,000 units/sec would be a nice round number, which would make the actual unit "size" just shorter than an archaic unit of measure, the foot. There is an intelligent way to chose the "size" of a basic unit of measure, but don't expect to find the intelligence to do so in the committees that created and maintain SI units.

 

I have a couple of questions into a professor who teaches "The Theory of Measurement". This area of study is also referred to as "Measurement Theory" (MT). This persons doctoral thesis was about measurement theory and he has written a number of articles on this subject. Apparently Helmholtz started this area of study, so it hasn't been around that long.

 

One of my questions into the professor is whether MT has a different approach to determining what is suitable for a unit if it is for commercial purposes as opposed to scientific inquiry.

 

Pyrotex called it a definition. "Measured quantities cannot, by definition, be referred to as transcendental."

[Qfwfq]

You cannot choose units (plural) that will result in c=1. If you chose a unit that made c=1, it would be quite large, and it that unit was called a meter we would have c = 1 meter/sec.

This does not mean that you can't. In fact, the case of c = 1 is very simply the choice of using the same units for timelike lengths as for spacelike ones. The fact that we perceive a second as being rather brief and a kilometre as rather long is our own shortcoming, inevitably due to a physiological cause; our nerve signals propagate very slowly and the times for things to get through our brains is long.

 

I think the whole concept of "natural units" has been distorted by those that claim "1" is the only natural number.

They don't claim this. :winter_brr:

 

Pyrotex called it a definition. "Measured quantities cannot, by definition, be referred to as transcendental."

That's exactly what I had quoted, I just disagreed with it. A measurement cannot distinguish between the quantity being a rational or irrational multiple of the chosen unit.

[Pyrotex]

...I am curious where you found that definition. I have been trying to identify if there are basic axioms of "Measurement Theory" which describe what can or cannot be a "unit of measure"....

It came from a similar conversation in a graduate physics class at Mississippi State University in about 1971.

 

The mere fact that a measurement may, indeed, have an arbitrarily long string of non-patterned digits (ALSONPD), has NOTHING whatsoever to do with the value being "transcendental" or not. A measurement cannot, in principle, be PROVEN to have ALSONPD. A true Transendental number CAN. Transcendentals are derived purely from pure mathematics, like pi, and e. It can be PROVEN that no matter how many digits you calculate, the number is not, will not be and can not be the solution of an algebraic equation.

 

A measured number derives its existence from a physical act that is inherently limited by an unresolvable residue of error. Transcendentals are not. This means that if you measure something to be 3.141596 meters long, and that is as accurate as you can, then there ARE NO DIGITS OF VALUE AFTER THAT. The number ends there, period. Arguments to the contrary are meaningless.

 

Of course, you could make a better measuring device and add a couple of digits of accuracy, but you're still stuck with the number coming to an end. Any digits you speculate after that are figments of your imagination.

 

Sorry guys, but you can't win this one. From Wikipedia:

In mathematics, a transcendental number is a real or complex number which is not algebraic, that is, not a solution of a non-zero polynomial equation, with rational coefficients.

 

The most prominent examples of transcendental numbers are pi and e. Only a few classes of transcendental numbers are known, and proving that a given number is transcendental can be extremely difficult.

The hilite is mine.

[Qfwfq]

This is tru, but my point was that you can't conclude that the measure of a quantity in a chosen unit CAN'T be [imath]\pi[/imath]. It makes no more sense to say it CAN'T than to say that it IS by virtue of a measurement.

 

It is however possible to make it so by definition of the unit. Just as c can be made the unit of velocities by definition, so could you choose [imath]\frac{c}{\pi}[/imath] or [imath]\frac{c}{10^8\pi}[/imath] which would give the scenario Frank was suggesting. Any choice such as the above ones, along with a choice of unit length, consequently define a unit of time.

[FrankM]

When the meter was first established you have to consider what the French knew and did not know. They did not know that the "size" of the meter would be used to establish a numeric value for the velocity of electromagnetic waves. They did not know that numeric value, which we symbolize with c, would be used in almost all equations dealing with physical law.

 

Also, everyone thinks of c in terms of velocity, meters/sec, but the "size" of the meter is that wavelength that occurs c times per second, or c = 299792458 Hz. The French knew about wavelengths and that wavelengths had associated frequencies, but they didn't know "light" had associated wavelengths and frequencies.

 

We now know that frequency is associated with an energy level. If the French had known what we know now, they could have established a "size" for the meter that would have an "equation friendly numeric" and also be associated with a specific energy level. SI already defines the SI second in terms of an energy level (a frequency), but nobody uses the numeric value for that energy value in equations, or even considers equating it to a "natural unit" with a value of 1.

 

Had the French been knowledgeable, they would have chosen a "size" for the meter that produced an equation friendly numeric value that could be used for the velocity of electromagnetic waves and its equivalent frequency would be meaningful as the basis for a "unit of energy". That same equation friendly numeric value (frequency) would be used as the number of counts that define the duration of the "unit of time". Everything would be tied together with one meaningful equation friendly numeric value.

[Qfwfq]

Obviously the Sèvres crowd did not know SR or hep and maybe it's a relief they didn't. If they did, housewives would be asking shops for hundreds of Avogadro's numbers of gigaelectronvolts of tomatoes! :clock:

[FrankM]

The Sèvres crowd was trying to be practical and I doubt they would have chosen a miniscule quantity as a primary unit of measure. They were trying to improve on the units of measure that were being used in trade. They haven't been completely successful even in those countries that have adopted metric, as certain merchandise is still bought and sold in archaic units of measure.

 

Not all scientists agreed that the meter was suitable for a scientific unit of measure. During the debates on whether metric units should be adopted, James Clerk Maxwell stated (1873), "The most universal standard of length which we could assume would be the wavelength of a particular kind of light... Such a standard would be independent of any changes in the dimensions of the earth, and should be adopted by those who expect their writings to be more permanent than that body.”

 

Maxwell wanted a unit length that was relevant for the purposes of scientific inquiry, whereas the metric crew wanted one unit of length to serve both commercial and scientific use. Since the meter is essentially arbitrary, it ill serves the scientific community.

[Qfwfq]

The definition of the metre actually was changed to being a large multiple of the wavelength of a specified krypton atom transition. It's current definition is given in terms of c and of the second, which n turn is defined by a large multiple of the period of a specified caesium transition. This causes c, in m/s, to be exactly 299,792,458 by definition.

 

Pity the metre and the second had both already been defined too precisely, or they could have tweaked the new definitions to make c exactly 300,000,000 in m/s. At that point, c = 1 would be in nice, round multiples of the metre and submultiples of the second. Still, astronomers have it easy by measuring distance in light-years and in light-seconds (which of course, strictly, just means measuring distance in years and in seconds).