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### Twice as Cold as Zero

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

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

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/
```
Associated Topics:
Elementary Temperature
Middle School Temperature

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