Twice as Cold as Zero

Date: 04/28/99 at 15:03:30
From: Daniel Carlson
Subject: 0 degrees x2

I've asked and wondered about this a lot but have found no good answer 
so far. 

What is the temperature if it's twice as cold [or warm] as zero 
degrees?

Date: 04/29/99 at 12:35:35
From: Doctor Nbrooke
Subject: Re: 0 degrees x2

Hello, and thanks for writing to Dr. Math.

You pose a very interesting question. What is the temperature if it's 
twice as cold as zero degrees? The Fahrenheit and Celsius temperature 
scales are "relative" temperature scales, in that they are both 
relative to a couple of specific temperatures, namely the freezing and 
boiling points of water.  

Let's make your question a little more specific: "What is the 
temperature if it's twice as warm as 0 degrees Celsius (the freezing 
point of water)?" To answer this question we need to convert this 
Celsius temperature to an "absolute" temperature scale, a scale with 
no negative temperatures. We'll use the Kelvin scale.  

0 degrees Kelvin is the point at which all motion in matter stops; it 
is known as "absolute zero." No temperature can exist that is lower 
than 0 degrees Kelvin. The formula to convert from Celsius to Kelvin 
temperature is K = C + 273.15, where K is the Kelvin temperature and C 
is the Celsius temperature. So 0 degrees Celsius is equal to 273.15 
degrees Kelvin. We can double this number to get 546.30, the 
temperature that is twice as warm as 0 degrees Celsius. We can then
subtract 273.15 from our answer to convert it back to the Celsius 
scale.

The temperature is 273.15 degrees Celsius when it is twice as warm as 
0 degrees Celsius.

Makes perfect sense, doesn't it? Thanks for writing, and write back 
any time.

- Doctor Nbrooke, The Math Forum
  http://mathforum.org/dr.math/   

Date: 11/09/1999 at 19:26:41
From: Lynn X.
Subject: Double of zero degrees?

I received an email with an interesting question and I would like 
to know the answer.

If someone tells you that it is zero degrees outside, and tomorrow it 
will be twice as cold, what will the temperature be tomorrow?

Date: 11/10/1999 at 11:37:55
From: Doctor Rick
Subject: Re: Double of zero degrees?

Hi, Lynn.

There is no really good answer to this question; the term "twice as cold" 
is not well defined. A physicist might want to define it one way, but 
someone deciding what to wear would have good reason for defining it 
in a very different way. To the best of my knowledge, no one has actually 
defined it quantitatively, in either of these ways or any other way. 
Still, I will tell you my ideas on two ways "twice as cold" *could* be 
defined if you insisted on doing so.

"Cold" is confusing in the same way that "short" or "small" is 
confusing. If I say that I am twice as short as you, what I probably 
mean is that I am half as tall as you. It isn't really a good way to 
talk at all; it's best to say "half as tall" because it's less 
confusing.

"Cold" is even more confusing than "short" because temperatures can be 
negative, but height can never be negative (not the height of a 
person, anyway). If "twice as cold" means "half the temperature", then 
if the temperature is negative, "twice as cold" is *warmer*! For 
instance, half of -10 degrees is -5 degrees, which is warmer than -10 
degrees. If "twice as cold" meant "twice the temperature", it would go 
in the right direction for negative temperatures, but not for positive 
temperatures.

Then, too, there's the problem that "twice as cold" will mean 
different things depending on whether you are using the Celsius or 
Fahrenheit scale. "Twice as cold" as 0 C is 0 C, but 0 C is 32 F, and 
half of this is 16 F, which is -8.9 C. 

One solution is to choose a meaningful zero point for temperature. 
Physicists determined that temperature has to do with how much energy 
there is in the air (or whatever you're taking the temperature of.) 
This energy is all gone when you get down to a temperature of -273.15 
degrees Celsius, or about -460 degrees Fahrenheit. This temperature is 
called "absolute zero." The scientists then invented two new 
temperature scales: Kelvin, which is the degrees C plus 273.15, and 
Rankine, which is the degrees F plus 460. That means that 0 degrees on 
either scale is absolute zero.

Therefore I propose that if you insist on saying "twice as cold," we 
should mean this by it: "half the temperature on the Kelvin (or 
Rankine) scale." By this definition, "twice as cold as 0 C" is half of 
273.15 K, which is 136.57 K. Converting this back to Celsius, we get 
-136.58 C.

That's very cold. I don't think it will ever be "twice as cold as 0 
degrees" in either Fahrenheit or Celsius. This definition of "twice as 
cold" might satisfy a low-temperature physicist, who is concerned 
about the amount of energy in a substance, but it does not agree with 
our everyday idea of cold.

Our everyday use of "twice as cold" might be something like, "It's so 
cold that I need to wear twice as much clothing." If I wanted to 
invent a mathematical meaning for "twice as cold" that would fit this 
idea, I would pick a temperature that is neither warm nor cold, 
perhaps 65 F. Then "how cold it is" would mean "how many degrees below 
65 F," and "how hot it is" would mean "how many degrees above 65." 
Then 45 F would be "twice as cold" as 55 F, and -65 F would be twice 
as cold as 0 F. (That's still very cold!)

You can see that this is a very different definition from the first 
one. It makes more sense according to our everyday idea of cold, but 
it has one problem: some international agency has to set that "neither 
cold nor warm" temperature. Judging from the battles some families 
have over thermostat settings, I don't think this will be easy!

So I say again, I would rather avoid the phrase "twice as cold" 
altogether.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/