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15 Feb 12
You have a temperature circuit controlled by two sliders. The temperature the circuit is set to is a/b + K where a and b are the slider settings.
Bend one slider into a circle and set the other slider to be a diameter of the circle.
Now slide the circular slider to the end, and the diameter slider so it touches the circumference of the circle.
The temperature the circuit is now set to achieve is Pi + K
15 Feb 12
Originally posted by iamatigerExactly? Or just near enough?
You have a temperature circuit controlled by two sliders. The temperature the circuit is set to is a/b + K where a and b are the slider settings.
Bend one slider into a circle and set the other slider to be a diameter of the circle.
Now slide the circular slider to the end, and the diameter slider so it touches the circumference of the circle.
The temperature the circuit is now set to achieve is Pi + K
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15 Feb 12
Originally posted by sonhouseIt's meaningless to even talk about something having a temperature of exactly pi degrees, for long enough for it to be measured at all. Sure, if you go from 3 to 4 you must, for some indivisible moment, have passed through pi; but in the time it takes to measure that temperature, you will also have passed through 22/7, 3.1415, 3.1416, 355/113, and Indiana.
Can you imagine the circuitry needed to make a temperature exactly PI C?
There simply is no such thing as a stable temperature. Temperatures may be stable enough for almost all purposes, but if you want to determine that it is exactly, not more or less, but exactly, pi, you need it to be stable enough to distinguish it from its rational neighbours, not in the third, not in the hundredth, not in the googolplexth, but in every single one of its infinite number of decimals. Brownian motion alone puts paid to that idea.
Richard
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15 Feb 12
1 edit
If you can't get exactly pi degrees, then you can't get exactly 1 degree either, or any other temperature you choose. However since temperature is a measure of the kinetic energy of the particles in the system I would have thought you could get things pretty exact with a small number of particles and/or a well insulated experiment.
By the way, internal Brownian motion does not change the temperature of a system.
15 Feb 12
Originally posted by iamatigerThat's right. I agree. You can never measure a temperature to its last decimal. You can only get a 'good enough' value of a temperature at hand.
If you can't get exactly pi degrees, then you can't get exactly 1 degree either, or any other temperature you choose.
Same things with lengths. How tall are you? To the last decimal? You don't know? Samo samo.
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15 Feb 12
Originally posted by FabianFnasWhich is why I say in the physical situation the ratio of C/F can never be exactly Sqrt(3)...Because to measure a true irrational would have to have infinite precision. Right or Wrong?
That's right. I agree. You can never measure a temperature to its last decimal. You can only get a 'good enough' value of a temperature at hand.
Same things with lengths. How tall are you? To the last decimal? You don't know? Samo samo.
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15 Feb 12
Originally posted by joe shmoI think it is wrong. If temperature varies continuously it must have had that that value at some point if is observed to be one side of it and then the other. Temperature cannot be precisely measured, but nevertheless at any instant in time the system is at some precise temperature.
Which is why I say in the physical situation the ratio of C/F can never be exactly Sqrt(3)...Because to measure a true irrational would have to have infinite precision. Right or Wrong?
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16 Feb 12
1 edit
Originally posted by iamatigerbut we measure temperatures essentially using integers, and no irrational can be expressed as a ratio of two integers, a and b, by definition...so how can the ratio of C/F = sqrt(3) ???
I think it is wrong. If temperature varies continuously it must have had that that value at some point if is observed to be one side of it and then the other. Temperature cannot be precisely measured, but nevertheless at any instant in time the system is at some precise temperature.
16 Feb 12
Originally posted by joe shmoWe measure distance in integers also, but you can have an irrational distance.
but we measure temperatures essentially using integers, and no irrational can be expressed as a ratio of two integers, a and b, by definition...so how can the ratio of C/F = sqrt(3) ???
How is temperature different?
Look at continuous probability scales.
For any continuously variable probability function, the probability that a value is exactly t is precisely 0, and yet, it must have some value. Is this a paradox? You might says yes, I say no.
16 Feb 12
Originally posted by joe shmoWrong, I would say.
Which is why I say in the physical situation the ratio of C/F can never be exactly Sqrt(3)...Because to measure a true irrational would have to have infinite precision. Right or Wrong?
You cannot measure an irrational temperature, but temperature are irrational by its very nature, even if they can have any temperature on the real line.
A temperature can have any value, even sqr(3), pi, or 1 or whatever. But we cannot measure a temperature with infinitly many decimals. We cannot know for sure that a temp is pi degress, only when it is near enough.
If you heat a specimen from 3 to 4 degrees, it must be pi degrees at some specific time, but we cannot know when.
We can define the temp scale from e = the freezing point of water, to pi = the boiling point of water, that's easy. The less easy is that the temperature when water boils and freezes is dependant of preasure, and only a well defined preasure gives the exact result. This exact preasure has the same difficulty as the temp itself - you can't measure it to its exact value, to the last decimal, only to a value quite ner, but not near enough. So this doesn't help you.
So I would say that your statement is wrong. It's impossible to measure the exact value of sqr(3) degrees centigrade.
16 Feb 12
Originally posted by forkedknightIt isn't.
We measure distance in integers also, but you can have an irrational distance.
How is temperature different?
Temperatures, lengths, times, masses, to take some examples, cannot be measured with irrational values. But they usually have irrational values. But the values are not exactly measurable.
16 Feb 12
Originally posted by forkedknightA good example of why something with a probability of 0 is not necessarily impossible!
We measure distance in integers also, but you can have an irrational distance.
How is temperature different?
Look at continuous probability scales.
For any continuously variable probability function, the probability that a value is exactly t is precisely 0, and yet, it must have some value. Is this a paradox? You might says yes, I say no.