# Half as cold

If it’s zero degrees outside and it becomes twice as cold it was before, what is the temperature?

Usually it’s presented as a joke. I’d like to consider some alternative interpretations.

1) It could literally mean the number denoting temperature divided by two. This has a number of unfortunate features. For one, it depends on the temperature scale. Half of zero is zero, but if you’re at zero Celsius you’re not at zero Fahrenheit. Worse, if the temperature is less than zero then half the temperature is going to be hotter.

2) It could mean the number for the temperature divided by two, but only in the Kelvin scale. Since it’s not possible to drop below absolute zero, you don’t have to worry about half the temperature being warmer. The scale is a little inconvenient though – half of room temperature in Kelvin is colder than dry ice.

3) It could mean the average speed of the gas molecules in air divided by two. Unfortunately this means the number for temperature actually drops by a factor of 4 on the Kelvin scale.

4) It could mean half of the thermal radiative power output via the Stefan-Boltzmann law. Twice as cold means a Kelvin temperature falling by a factor of roughly 0.84, which would mean a day at the freezing point of water falls to negative 46 degrees Fahrenheit / negative 43 Celsius.

I think the last one is my favorite, as it keeps the temperatures human-scale. It’s a little contrived though.

Comments? I’d love to hear if you have any suggestions for better interpretations.

1. #1 Dunc
February 19, 2009

“Twice as cold”? That could mean “double the difference between the current temperature and standard temperature.” (The wording of the question indicates that it’s already cold, so the current temperature must be below that considered comfortable by humans.) The question then becomes which standard temperature? Given that we appear to be talking about human perception of comfortable temperatures, I’d go for the NIST standard of 20°C. So if it’s 0°C, twice as cold would be -20°C.

Or, you could argue that “twice as cold” doesn’t refer to a measured temperature at all, but rather to the perception of cold. Then you’d need to devise some means of measuring the perception of cold… Time to the onset of hypothermia, perhaps?

(Sort-of-related anecdote: I used to work on a helpdesk dealing with office accommodation for civil servants. One day I got two calls from two adjacent rooms complaining about the temperature – one complaining that it was too hot, the other complaining that it was too cold. The temperature in both rooms was within 1°C of each other, and the person complaining it was too hot was in the cooler room. Man, I hated that job…)

February 19, 2009

You could go with a neurobiology interpretation and declare that it’s the temperature at which cold-sensing nerves are stimulated twice as much as before, or the part of the brain that reacts to cold temperatures becomes twice as active.

The catch is that it’s entirely subjective and difficult to study, probably resulting in the subjects rising up against the researchers and then going out for a cup of cocoa.

3. #3 dean
February 19, 2009

I think your cases speak to the matter that (from a statistical point of view) this is an ill-defined question.
Temperature (degrees F or degrees C) is not a ratio variable, since there is no fixed zero, so it isn’t reasonable to speak of “twice as hot” or “thrice as cold”. This addresses your case 1.

Since there wasn’t any indication given HOW the notion of temp is being measured, there isn’t any answer that can be made independent of adding assumptions.

4. #4 Peter
February 19, 2009

When in doubt, switch scales. If it’s 0 deg.C it is 32F , twice as cold would be 16F (yes, technically half, but this is a colloqialism we’re talking about here…) on the other hand, if it’s 0F, it is @-15C, so twice as cold would be… Very cold indeed.

5. #5 humorix
February 19, 2009

Bien sûr, si l’on considère que tout ce qui est au dessus de zéro est chaud et que le froid ne commence qu’à zéro degré (glace), la réponse est zéro (puiqu’on reste dans le ‘froid’/on ne mélange pas torchon & serviette).
+2°-> 3 x + froid = -4° (écart 6°)
+2°-> 2 x + froid = -2° (écart 4°)
+1°-> 2 x + froid = -1° (écart 2°)
+0,5° > 2 x + froid = 0° !
(et non -0,5° qui ferait -1°)

6. #6 gort
February 19, 2009

Cold is a relative term and depends on context. In the implied human context, I think that the cold scale begins at the point where a person’s body starts needing assistance to comfortably maintain the body temperature. Twice as cold could be measured either by the change in the amount of assistance the body needs (e.g., shirt sleeves v. sweater v. jacket v. winter parka) or by the change in the discomfort/damage to the body (frostbite in 1 hour v 30 minutes).

7. #7 Andrew
February 19, 2009

@4: I think it’s okay to assume ‘twice as cold’ and ‘half as hot’ to be synonymous, much like conductance=1/resistance.

Personally, I interpret ‘twice as cold’ to mean ‘colder, but I’m too ignorant to tell you how much colder’. The phrase is like the bogosort algorithm: anyone knows how to use it also knows why they shouldn’t.

8. #8 Joshua Zelinsky
February 19, 2009

Another possible definition: Ask people how cold it feels with 0 being not at all cold and 10 being as cold as they can feel. Doubling would then double under this scale.

Or another possibility: Twice as cold at any temperature means it is so cold that someone’s brain has been damaged to the point where they are trying to use something as hideously ill defined as twice as cold. (And yes I know this is normally meant as a joke, but I’ve heard at least one idiot use this sort of statement in a serious fashion).

9. #9 Jon Claerbout
February 19, 2009

Understanding absolute zero: In 1695, 150 years before Lord Kelvin’s absolute temperature scale, 120 years before Sadi Carnot’s PhD thesis, 40 years before Anders Celsius, and 20 years before Gabriel Farenheit, the French physicist Guillaume Amontons, deaf since birth, took a mercury manometer (pressure gauge) and sealed it inside a glass pipe (a constant volume of air). He heated it to the boiling point of water at 100 degrees C. As he lowered the temperature to freezing at 0 degrees C, he observed the pressure dropped by 25%. He could not drop the temperature any further but he supposed that if he could drop it further by a factor of three, the pressure would drop to zero (the lowest possible pressure) and the temperature would have been the lowest possible temperature. Had he lived after Anders Celsius he might have calculated this temperature to be -300 degrees Celsius.

10. #10 D. C. Sessions
February 19, 2009

“Cold” in human terms is the degree to which the environment sucks heat from the human body. With a nominal surface temperature of about 30C, 10C is “twice as cold” as 20C.

dcs, channeling my inner ski patroller

11. #11 dWj
February 19, 2009

This is close to what #1 and some others have mentioned, but if you’re going for human scales, I would go with “the human body loses heat twice as quickly”, which probably means “twice as far from body temperature”.

12. #12 humorix
February 19, 2009

While ignoring the present temperature, it is not stupid to say that today it is twice more cold than yesterday (!).

13. #13 rob
February 19, 2009

we should drop the temperature scales we have and use the proper units: electron volts. the temperature is really a measure of the average kinetic energy, so our units should reflect that.

imagine the weather person describing the day:

“it is a wonderful 1/40 eV today, partly cloudy with no chance of rain.”

oh, and we should also convert to the metric system.

(ha ha ha ha, like that will happen)

“I’ll convert to the metric system when you pry the English system from my cold, dead hands.”

14. #14 rob
February 19, 2009

oh, apparently there was a story about how the metric system is spreading across the US:

http://www.theonion.com/content/node/28768

15. #15 Eric Lund
February 19, 2009

One day I got two calls from two adjacent rooms complaining about the temperature – one complaining that it was too hot, the other complaining that it was too cold.

I presume you are aware that those are the two available settings on the climate control system of any institutional building ;-).

The question is definitely ill-posed. If you are in a lab setting, halving the absolute temperature (it need not be the Kelvin scale; there is also a Rankin scale which is to Fahrenheit as Kelvin is to Celsius) is the way to view it, and is quite relevant for people whose research involves liquid helium. But if you are talking about a weatherman, I would agree with the departure from reference temperature idea. The NIST standard is as good as anything; when you are talking about how something feels then individual tastes come into play (the 30C which DCS suggested above I find to be uncomfortably warm).

Somewhat OT but related: When I first moved to New Hampshire mumble years ago I discovered that there is a legal standard for heating systems in rental housing units. If the heating system cannot maintain an average indoor temperature of at least 65F (about 18C), or your landlord who controls the thermostat does not set it to at least that temperature, you have cause for action against your landlord.

16. #16 humorix
February 19, 2009

If yesterday the temperature was zero degree (0°), if today it is twice as cold as yesterday, the temperature is less two degree (-2°).
The temperature is twice as low in the thermometer.
Thermometer millimeter = less two millimeters.

17. #17 Great William
February 19, 2009

There is no absolute zero. There also is no 180* down, since it would take energy to draw that. “180* down” is merely ‘the most inefficient thing that there is. Life is, and that cannot be changed.

If you’re looking for a place with no strings in it, you are a string, and you looking is technically a string and response. That place of absolute zero is invisible.

The perfect circuit is also invisible, and technically the exterior of the perfect circuit is ‘cold’ from being not measured, while the inside is ‘the desired temperature’.

But since there is no absolute zero, the problem becomes a /0 error. That is merely a measure of ‘how little temperature is there’? Which is down to a 0.0bar1.

That can produce a relative temperature, but it is neither the perfect circuit nor is it absolute zero. The parameters are “8” infinity. A chunk of fat floating in the air.

This is the extent of the scientific method of relativity to being measured.

…For this reason there is no suicide, nor homosexuality, nor death, nor parasites that will ever prosper. Every wave leads to or is ruled blind by a particle, and there is one particle that creates all things.

18. #18 bbbeard
February 19, 2009

Well, a thermodynamicist might tell you that the temperature variable in statistical mechanics is beta=1/T. In this way of looking at things, absolute zero corresponds to beta going to infinity (which is why you can’t get there from here). Higher beta means a colder temperature. So “twice as cold” makes perfect sense (and corresponds to half the absolute temperature).

BBB

19. #19 Not Great William
February 19, 2009

Great William,

Thanks for that ‘helpful’ information. By the way, how did your run for the Mass. House of Representatives turn out?

20. #20 nanoAl
February 19, 2009

Being from canada, I know cold.(so clearly what i say goes on this matter :)) It doesn’t really start to feel cold until about -5C give or take (specific measurements could be made). When they say its 0C outside and ask you what twice as cold is, you can tell them that it is in fact comfortably warm and that they are a moron. however, twice as cold -6, is -7 and twice as cold as -10 is -15. It is also possible that cold isn’t linear, it could be quadratic. or more generally, a polynomial in (x+5).

21. #21 humorix
February 19, 2009

The zero centigrade is called the ice. The ice is a part of the cold. There is no warm ice. The cold begins in zero. The warmth begins in one.
Each has the temperature of cold and warm, but the thermometer in his.
It is him who says: Cold = ice = 0°

22. #22 rob
February 19, 2009

as it turns out, you *can* have a temperature lower than absolute zero. that is, you can have a system with a temperature that is negative kelvin. interestingly, this negative temperature is *hotter* than infinite temperature.

http://en.wikipedia.org/wiki/Negative_temperature

23. #23 Bing
February 20, 2009

Thanks so much. You can just rock me to sleep tonight.

HJ

24. #24 Åka
February 20, 2009

This reminds me of the girl in the sauna who said that it was more than twice as hot in there as outside. I asked her what she meant, and it was along the lines of your suggestion 1. She did not understand why I didn’t think this was a good definition of “twice as hot”…

I like 2 and 3, and my solution to the problem of the scale is just to say that “twice as cold” is not a very interesting temperature to talk about. Maybe that’s cheating? But the question is kind of strange anyway.

February 21, 2009

Strictly speaking, what dean said in #3 is probably the most accurate answer: C and F scales are interval level data, so the ratio of them doesn’t make sense. It does in the Kelvin scale, as temperature in Kelvin is a ratio level variable (halving the temperature corresponds to halving the thermal energy of the particles, which for gases is of coarse related to the root mean square velocity, to tie in with option 3). Of course, even in C or F the ratio of temperature differences in these scales should make sense, so one could always pick a reference temperature, say 37 deg C for body temperature, and so under that interpretation to say it’s twice as cold today compared to yesterday would mean that (Temp_today – T_ref)/(Temp_yest – T_ref) = 1/2.

26. #26 CCPhysicist
February 21, 2009

The point is that the question is ill-posed, so your idea that we should re-pose it is an excellent one.

I really like the suggestion at #1, because it works regardless of temperature scale used. If it is actually freezing, a person using the Fahrenheit scale would come up with -4 F, the same as -20 C. However, even then you have the problem that the question of what is “normal” is quite subjective. I think that 20 C (the temperature we select during the winter, 68 F) is “cold”! I’d use 25 C (77 F).

I disagree with #25’s reference temperature. Humans are mammals, producing 100 W of thermal energy at a typical basal metabolic rate. Most humans find a 37 C ambient air temperature uncomfortable because we don’t radiate heat very effectively. (Our “sigma” must be awful.) We do much better with convective heat transfer in air cooler than our body temperature.

27. #27 humorix
February 22, 2009

Thus of less 2 millimeters.
Whether it is in Celsius or Fahrenheit.

28. #28 jth
February 23, 2009

The answere would be 0 degrees, if one removes half their cloths.

29. #29 Andrew
February 23, 2009

I think the human body reports temperature as different intensities of ‘hot’ and ‘cold’ senses, so you could define the difference that way, as a sort of bistimulus thing.

I’m stumped as to which of the above comments are serious and which are attempts at humour. Measuring temperature in electron volts? I can see the reasoning, but it’s a bit like trying to measure height in kilograms.

30. #30 opony
February 23, 2009

I sleep outside tonight.I try to feel what temperature it’s at the moment

31. #31 Donalbain
February 23, 2009

Number of jumpers you need to wear.

32. #32 humorix
February 26, 2009

We wait for the conclusion of Matt Springer…

Unless he died (twice) from cold!!

33. #33 Andres Villarreal
July 12, 2009

It seems several people have forgotten that a measurement for cold has to take into account measurable phenomena:
– The temperature at which your body starts to burn additional fat to keep you warm is somewhere around 30 C, whether you are accustomed to it or not.
– The humidity of the air is critical. The efficiency at which the air takes heat from your skin changes dramatically with humidity.
– We must talk only about naked, dry adults of normal weight and size if we are going to expect any kind of working definition.