# Why is the absolute zero a rational number in Celcius?

From the question "Why is the absolute zero -273.15ºC?" I understood that 1°C is the 100th part of the difference of melting and boiling temperature of water (this is my high school physics, maybe there is a more accurate definition). So we linked the value of 1°C to a physical quantity.

Then we measure another physical quantity (the absolute zero) and it turns out this is exactly 273.15°C less the melting temperature of water. So how come those two quantities have a rational ratio?

If we look at other constants (gravitational, Planck or whatever) they all are irrational, aren't they?

What did I miss? Is it because the melting point of water is not exactly 0°C (maybe something really close to 0 like 0.000565...)?

• Note that the speed of light is now and integer when expressed in SI units. Think about how that came to be. Metrology is an interesting business. – dmckee May 1 '14 at 17:14
• "how come those two quantities have a rational ratio?" It's by definition. See the Wikipedia page on Celsius. – DumpsterDoofus May 1 '14 at 17:19
• You need to realize humans defined these things arbitrarily. I could construct a system where I defined the melting point of water in some scale to be 15(an integer) and not 15.000001298219812981 or whatever. – user18764 May 1 '14 at 17:20
• I think I get it. We can fix by definition absolute zero at -273.15°C and melting temperature of water at 0°C. As a consequence when measuring boiling temperature of water this will not be exaclty 100°C, is that correct? – Paolo May 1 '14 at 17:28
• Note that whenever measurements are involved, we can't tell a rational from an irrational number. In any interval of non-zero size there will always be an infinite number of rational numbers, and every measurement comes with an interval of uncertainty. Only definitions are exact, if sometimes somewhat arbitrary. In this light, asking about the rationality of ratios between some measured constants doesn't make sense. Asking whether they form nice, easy, short, small rational numbers, on the other hand, makes sense, although I guess the answer to be “no” in most cases. – MvG May 2 '14 at 0:38

it turns out this is exactly 273.15°C less the melting temperature of water.

Actually, "Kelvin" and "degrees Celsius" are defined such that there are 273.16 degrees between absolute zero and the triple point temperature of water. Degrees Celsius are defined as $K - 273.15$.

The freezing point of water is a measured quantity and is not exactly 273.15K nor 0°C and isn't necessarily a rational number.

• I assume that there is a historical reason for defining the triple point to be at $0.01$ degrees Celsius rather than simply zero. Could you tell me more about this? – Danu May 1 '14 at 18:43
• The Kelvin was defined relative to the triple point in 1954. See: bipm.org/en/CGPM/db/10/3 I believe the old standard was 0 degrees C is the freezing point of water, with the triple point being measured (approximately 0.01 degrees C), but that it was realized the triple point is more precisely measurable than the freezing point of water at a standard pressure. You don't have to refer to a standard pressure by using the triple point. – DavePhD May 1 '14 at 18:55
• Nice, so really now it'd make more sense to define that as 0 in Celsius, given that it doesn't matter in everyday applications anyhow. – Danu May 1 '14 at 20:51
• there is a proposal to redefine Kelvin and Celsius relative to the Boltzmann constant (k) and Joule. See this reference especially page 1760 thermophysics.ru/pdf_doc/Boltzmann1.pdf – DavePhD May 1 '14 at 20:58

As a metrologist, I am glad of this interest in correct notation, often not enough pondered also among metrologists but essential for understanding with each others.

I would say first that any numerical value of an experimental result is always expressed as a rational, not irrational, number, because the number of digits is always bounded by the position of the measurement uncertainty level, e.g. 1.2345(x) or 1.23456(xx), according to guidelines for correct notation of uncertainty –where x indicates the uncertainty applied to the least significant digit(s). However, it is true that the numerical value of the measured quantity is in itself an irrational number, unless we have a specific reason to understand that is a rational one.

In this respect, the numerical value assigned to any ‘fundamental constant’ (e.g. the Planck constant), whose numerical value we can only obtain through measurement, is always a rational number with an associated uncertainty (though sometimes the latter indication is dropped, in which case it is correct to add the three dots to indicate an irrational number).

In other words, all experimental numbers are ‘uncertain numbers’. However, when used in a definition they must instead be written as exact numbers: for this purpose they are ‘stipulated’, indicating with that term the decision taken to agree on an exact value (it is a consensus value that does not modify at all the intrinsic uncertainty of our knowledge).

So it is 273.16 K exactly for the triple point of water. So is 299792458 m/s exactly for the speed of light in vacuum. So it would be for any other constant, should it be stipulated in future. Incidentally, for the speed of light the notation can be misleading: it is not an integer number, as it would become clear when writing it as 299792.458 km/s.

Also the value 273.15 K in the definition of the Celsius scale (t/°C = T/K – 273.15) is an exact one. This a tricky case. In fact, the definition implies that, when T = 273.16 K, thus t = 0.010 °C exactly. On the other hand, when one measure T = 273.1500(x) K, thus t = 0.0000(x) °C: however, in fact, this is not anymore necessarily the freezing point of water, exactly for the same reason why the normal boiling temperature of water is no more 100 °C but 99.974(x) °C.

Should in future temperature unit be defined using the Boltzmann constant, also the temperature of the triple point of water will not anymore be exact, not being anymore used in the definition of the kelvin. However, in order to avoid a discontinuity in the more precise temperature measurements due to the change between the old and the new unit, it should be assumed, at least initially, that the value will be 273.16000(15) K, where the uncertainty in parenthesis arises from the uncertainty of the Boltzmann constant determinations (1 ppm). Obviously, future determinations could depart from that value, since the value 273.16 K was stipulated before 1954, when the precision of the measurements was lower than 0.0001 K.

• Many good points here: if only all contributors on this site shared your appreciation of measurement error and the importance of reporting it... $*sigh*$ – Floris Sep 11 '14 at 0:05

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