While temperature conversion factors are rational numbers, are length and mass conversion factors rational too ?
I may be mistaken, but I believe the conversion factor for converting mass (kg) to weight (lbs) depends on the local acceleration due to gravity (which is empirically determined).
Yes. How could it be otherwise? Are you going to define a pound to 1/e times the ??? in kilos. The reason for the question mark is that pounds and kilos are different kinds of units. Pound is a unit of weight and kilo a unit of mass. The pound is defined, IIRC, as the weight of 1/2.206 kilos under some standard conditions (maybe where g = 980 cm/sec/sec). Your weight on the moon is only about 1/6 your weight on earth, but your mass doesn’t change.
The inch is defined to be precisely 2.54 cm. What would it mean to define it to be 2.540134967… and what would be the point? Not only are they rational, but they are given only to a finite number of decimal places.
I think that that is quite an interesting question. The meter for example is a millionth of the circumference of the earth. The yard was developed completely separatley, so the ratio between them could be either rational or not. A couple things would help us to understand which is more likely.
1, How many rational numbers are there compared to non rational numbers (is their a mathematician in the house ? If I remember correctly the ratio of non rational/rational numbers is infinity, which would make it a certainty that any accurate and independently developed systems would be linked via a non rational ratio.)
2, Was there any jiggery pokery to the lenght of a yard to make the ratio more simple (The French have got a platinum bar which is 1 meter long, I think that they measure it every so often, and if it’s not exactly a meter they change the tape measure so that it is)
For length, imperial units are defined in terms of metric units. So 1 in = (127/50)cm is an exact rational relation. As a result, 1 mile = (25146/15625) km is also exact.
According to Treasure Troves, there is a unit of mass called a pound (or more explicitly a pound-mass) equal to 1/32.1740 slug. A slug is defined as 1 lbf / (ft/s[sup]2[/sup]), where lbf is a pound of force, the more commonly used meaning of “pound”. A pound of force is itself 4.448 Newtons. I don’t know if all these numbers here are exact, but if they are, then yes, the conversion between lbf and kg is rational.
When I was growing up, I was taught that 2.54cm/in was an approximation, not a definition, and the real number for the conversion was not a rational number.
Was I misinformed? Or did it change?
One of the two; it depends on when you were in school.
No, that’s impossible. An irrational number cannot be expressed as a ratio; a conversion factor is manifestly the ratio of two exact numbers. I think you’re referring to a non-terminating decimal number, which is quite common, and not at all the same thing.
Still assuming you mean non-terminating decimal numbers instead of irrational numbers, I don’t think this is the case, although I’m sure there are more non-terminating decimal numbers than terminating decimal numbers in any given interval.
I think the idea was that the platinum-iridium bar DIDN’T change; there was no need to go changing tape measures and what-not. When measurement precision increased to the point where this was not true, the technology changed (defining length by a wavelength of light) and chamged again (defining length by the speed of light and and an interval on an atomic clock). And yes, they redefined the inch to be exactly 2.54 centimeters, which implicitly changed the yard.
If the relations in my last post are exact, then 1 lbm = (2780000/6129174) kg, which, according to my calculator, is only accurate to four decimals. However, the reciprocal of neither result is equal to 2.206. Closer to 2.2047. FWIW.
Hari Seldon, it would be possible for this to be an irrational number if the units were not defined in terms of each other, as the length units are.
Well a Irrational number can still be a ratio just like PI is the ratio of Circumference to Diameter of a Circle. Of course one (or both) of the numerator or denominator may not be rational.
Regarding, lb to kg conversion, I was referring to lbm (poundmass). If you like you can also consider lb to N conversion factor.
I disagree. Suppose, for example, I define a unit of mass called a ZUT that is equal to the mass of my coffee cup. To convert ZUTs to kilograms, I need to know either the mass of my coffee cup in kilograms, or the mass of the platinum kg standard in ZUTs. Either measurement will have some inherent error, so the conversion factor will necessarily be approximate, but it won’t be irrational.
Now, if I find that 1 ZUT = 0.173405988344676 kg, +/- 0.000000000000002 kg, I could use an irrational conversion factor of 0.17340598834467606006000600006000006…, but why not simply use 0.173405988344676? By definition, that’s the best approximation we’d have. On the other hand, I could go ahead and define 1 ZUT as 0.17340598834467606006000600006000006… kg, but why would I want to do that?
zut, I think the point is that base units are never defined to be +/- anything. They are exact. That’s why they are base units.
When was the change?
If you define one unit of measure in terms of another (1 inch = 2.54 cm, for example), then the relationship between the units is exact, by definition. If, on the other hand, the units of measure are defined seperately, then their relationship is not exact. In my silly example, the base unit kilogram is exactly defined by the mass of a particular platinum bar, and the base unit ZUT is exactly defined by the mass of my coffee cup. However, my coffee cup is NOT exactly defined in kilograms. Thus, converting from kilograms to ZUTs necessarily involves some error, because the only way to convert is to determine the mass of the ZUT standard in kg (or the kg standard in ZUT units), the accuracy of which is limited by measurement equipment.
Let’s try again. When units are defined in terms of one another you have an internally consistent set of standards.
What you are doing is converting across sets of standards that are not at all defined by one another. Apples to oranges.
I think you and Hari Seldon are talking about two different things.
EM: Well, I wasn’t responding to Hari Seldon, I was responding to Achernar, who said “it would be possible for this [conversion factor] to be an irrational number if the units were not defined in terms of each other.” Otherwise I agree with you.
I understand what you’re saying, but I disagree. The gram and the amu are both defined in terms of completely different things. However, I know of no reason to think that their ratio is not an exact number to an arbitrary number of decimals, even if we don’t have that ratio written down somewhere. And if it is an exact number, it would be mighty coincidental if that number were rational, I think.
OK, Achernar, you’re taking a slightly different tack than I was. I think I still disagree with you, but let me try to recap the different cases here:
I was interpreting a “conversion factor” (re the OP) as “the number one would use, which is as accurate as possible, when converting grams to amus” (or whatever). In the case of units defined in terms of different bases, this conversion factor must have some measurement uncertainty built into it. Thus this “conversion factor” might as well be a rational number. I think you agree with this?
You were interpreting a “conversion factor” as “the absolute, actual value of one base unit in terms of another.” How many amus there really are in the platinum kg standard, for example, measuring it “perfectly.”
Even in case 2, I disagree that there can be an exact conversion. My understanding is that the exact weight of a platinum bar, or weight of a carbon atom, or length of a certain wavelength of light, is subject to some fundamental quantum uncertainty, so that there is no precise weight or length which can be exactly expressed to an arbitrary number of digits. If an exact measurement doesn’t exist, then an exact conversion factor doesn’t exist, and again we might as well use a rational number.
I fully admit that the preceding argument is outside my area of expertise, and I will cheerfully admit that I’m incorrect if you or anyone else could clearly explain why this principle does not apply. Conversely, if I’m actually right, I’d appreciate it if someone with a bit more knowledge could attempt a clearer explanation.
In case 1, I agree that the conversion “might as well” be rational, but it could just as easily be irrational, from a physical standpoint. We could pick either one, and the only reason to pick a rational one is that terminating decimals are easier to write than non-terminating ones. However, I think it’s silly to say that the number is either rational or irrational, since it’s not defined accurately enough either way. Is Avogadro’s Number an integer? We’ll never know, but that doesn’t make the answer yes.
In case 2, I too had considered the quantum uncertainty, and I don’t know. A simple example without all this mucking about with units would be the fine structure constant. Is there some theoretical limit to how accurately we can measure it? I agree that it’s ridiculous to say that there’s an exact number which we can never measure; if we can’t measure it, it doesn’t exst. At the very least, there’s no problem classically.
>> The meter for example is a millionth of the circumference of the earth.
Not in my world. To begin with the circumference of the earth is about 40 million meters. Furthermore, the definition of the meter has nothing to do with the size or shape of the planet and a lot to do with certain electromagnetic waves.
>> The yard was developed completely separatley, so the ratio between them could be either rational or not
No. the yard today is defined as three feet or 36 inches. The inch is defined as 0.0254 meters. As a consequence the yard is defined as 91.44 cm exactly.