A silly pseudo-math question...but it's bugging me

[DarrenS]

I saw this question on one of those lists of musings that gets forwarded by e-mail (of the "park on driveways" variety) This one bugs me for some reason:

"if it was zero degrees yesterday, and it's twice as cold today, what's the temperature today?"

What am I missing?

[Helen's Eidolon]

The fact that anything times zero is still zero.

[DarrenS]

Exactly! So the temperature is still zero...yet it's twice as cold !? So what gives?

[CurtC]

What would your answer be if it was 2 degrees yesterday, and twice as cold today? If you say 1, I would like an explanation.

[KneadToKnow]

Originally posted by DarrenS
I saw this question on one of those lists of musings that gets forwarded by e-mail (of the "park on driveways" variety)
I'm sorry, Derwood, but it's just a joke. Don't get all worked up over it.

[Smeghead]

Well, it's a stupid question to begin with. You don't measure cold. Temperature is a measure of heat. There's a lower limit on how much heat you can have (absolute zero) but there's no upper limit (yeah, OK, I guess you could measure the total amount of energy in the universe, blah blah blah, but that doesn't count, because that just happens to be the number for this universe. It could easily be different.)

The way I see it, if you ask how cold it is, you're asking how far away the temperature is from the theoretical maximum, of which there is none. Therefore, there is no numberical measurement of cold, and you can't double something that can't be numberically measured.

If you want, I guess you could define "twice as cold" as "half the heat" and divide the temperature (in kelvins) by two, but that rubs me the wrong way for some reason.

More than anyone wanted to know, I'm sure.

[DarrenS]

Well put, Smeghead. (Feels funny to call someone that!) You're right that heat and cold are inherently "asymmetric" - temperature can go up forever, but only down to zero K. I guess in common usage, if it's 10 degrees tomorrow, and you ask someone what it'll be tomorrow if it's twice as cold, most people would say 5 degrees. Yet that doesn't seem to work for zero !?? Yet there's nothing special about zero celsius...maybe this only makes sense if you express the temperatures in Kelvin?!

[Smeghead]

Exactly, Darren.

Half of zero degrees C is 270 something kelvin, so half that would be 130 or 140 something. (No doubt hard numbers will be quickly forthcoming) Zero C has no special significance, other than that water freezes there. So I stand by my previous stance - it's a stupid question. The riddle, I mean, not your OP. :D

[Steve-o]

.15 = something, so 0 deg. C = 273.15 K

[Lumpy]

The question states "twice as cold", which makes the Kelvin scheme problematic because it implies you're starting out from some temperature which is considered "not cold", and then adding twice as much "coldness".

I believe chemists sometimes refer to a standard temperature for laboratory measurements which is considered the official "room temperature"; something like 72 degrees Fahrenheit(sp). So if one day it was 50F, or 22 degrees below standard, "twice as cold" would be 28F or 44 degrees below standard.

I know, the whole question is silly but that's what makes it challenging.

[Crafter_Man]

As others have already mentioned, if you want to do linear math with temperature you've got to use an absolute temperature scale (Kelvin and Rankin are common).

So let's see... assuming "zero degrees" means 0 °C,

0 °C = 273.15 K

twice as cold = half as hot = 0.5*273.15 = 136.575 K.

Converting this back to °C gives us -136.575 °C.

That's a bit nippy.

[Arjuna34]

It depends on the context. If someone used the phrase "twice as cold" where I work, it would go without saying that they meant in an absolute sense (i.e. 0 degrees = absolute zero). You can use Kelvin scale, or if you want to avoid metric, Rankine. It would probably be more likely for someone to say "half as hot" than "twice as cold".

If someone outside of work said it was going to be "twice as cold outside", though, I'd interpret it to mean it would somehow "feel" twice as cold, and not interpret it literally. If it were 50 degrees F yesterday, and it was supposed to be "twice as cold", I'd expect it to be maybe 35, just because it would somehow feel that much colder, not the -230 degrees F that using an absolute scale would imply!

Arjuna34

[Luckie]

KneadtoKnow is quite correct. It's a joke,.... get it? *wink, wink, nudge nudge*

of course, invariably people will try to put a reasonable answer to this particular joke, there was a protracted discussion on it at my office a year and a half ago. One character insisted that the correct answer was 127K (half the distance to zero kelvin).

At first I maintained that, in so far as there IS an answer to the joke, the only reasonable one would take the approach Lumpy mentioned. Cold is purely perceptual, and that is relative to a "comfortable" temp for a typical human, say 68F. Twice as cold would then be -68F, or twice the temp distance from normal.

When i thought about it a little longer, i decided that was bunk too. -68 is way more than twice as cold as zero! The best i could come up with, is that twice as cold is whatever temperature is half as likely for that particular location ( if you are talking about the temperature outside as the OP was).

of course, that is bunk too. but that is the point of the joke.

-Luckie
--------------------------------
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htpp://mp3.com/RobRyland

[Lance Turbo]

Originally posted by DarrenS
You're right that heat and cold are inherently "asymmetric" - temperature can go up forever, but only down to zero K.

Cecil disagrees in this column. (http://www.straightdope.com/classics/a3_347.html)

Cecil gives the highest possible temperature as 10^32 K. So if yesterday it was zero C outside. And today it is twice as cold that must mean todays temperature must be twice as far from the absolute maximum temperature. That will be about - (10^32) C. Which is friggin' cold.

By this method it is impossible for it have ever been twice as cold as yesterday in human history, and possible since moments after the big bang.

[RM Mentock]

You have to get up early to put one over on you guys. Tomorrow, I'm going to get up twice as early.

[Phobos]

Originally posted by Lance Turbo
Cecil disagrees in this column. (http://www.straightdope.com/classics/a3_347.html)
Cecil gives the highest possible temperature as 10^32 K.

Sheesh! That Planck guy claimed all the good theoretical limits! :)

[cmkeller]

Tell me this much: if it's 2 degrees outside yesterday, and today it's 4 degrees, is it now twice as cold? Obviously not. The number attached to the zero point on the thermometer is meaningless in a statement like "twice as cold."

Most likely, when someone says this, it means "cold that is twice as uncomfortable" as the other day. So figure out at what temperature the speaker begins to be genuinely uncomfortable from cold, and then double the distance from that temperature.

[muppetsoup]

I think you guys are going about it wrong. Of course, it depends what the temperature was the day before yesterday. It's all relative. if it was 10 degrees the day before yesterday, then yesterday it was 0. today it's twice as cold as yesterday.. so -10 degrees. You can see where I'm going...

[HeyHomie]

Cold, as it relates to the weather, is relative both to the observer vis a vis both season and the locale. For instance, if it's January in Miami Beach, the locals might say it's "cold" if the temperature doesn't crack 60. If it's January in, say, Illinois, we'd strip naked and dance around in the streets if the temperature hit 60.

So, in the situation mentioned in the OP, we'll have to assume that the observer in a climate where winter temperatures are known to hit 0. So, arbitrarily let's say that the locals deem that it's "cool" at around 40 and "cold" at around "30."

Therefore, if it's 0 degrees today, then it's thirty degrees below the threshold of "cold." So if it's going to be twice as cold tomorrow, it's going to be sixty degrees below the threshold of "cold." IOW, it's going to be -30.

[Chronos]

How about this: Temperature T2 is said to be "twice as cold" as temperature T1 if a standard person exposed to temperature T2 would develop frostbite and/or hypothermia twice as quickly as that person would at temperature T1.
Does that work, or do we need to define "standard person", as well?

[Smeghead]

Originally posted by Lance Turbo

Cecil disagrees in this column. (http://www.straightdope.com/classics/a3_347.html)


Grrr. Cecil was using the approach I pooh-poohed in my earler post - adding up all the energy in the universe, blah blah blah. Sure, it's a maximum, but just for this universe. Absolute zero is absolute zero in any universe - either atoms are moving or they're not - so I don't think this upper limit is of the same caliber.

And as far as I'm concerned, all the argument on this subject just proves my point - it's a stupid question. I don't like questions that devolve into arguments about how to define the question, because then there's no real answer. But that's just me. :D

[DarrenS]

I guess it *is* an "ambigously" worded question. It implies that "cold" is a quantifiable measure. Hmmm I wonder if that generalises to other things quantities. E.g. if I have a stick that's 1 meter long, and someone else has one that's twice as long, that'd be 2 meters. But if there's one that's "twice as short", would that be 1/2 a meter? Or how about an interval "twice as short" as 1 hour?

[The Ryan]

So if this is a stupid question, what would a question that's twice as stupid be like?

[DarrenS]

Hey Ryan - a self-referential post - neat! ;-)

Just kidding.

[Arjuna34]

Originally posted by DarrenS
I guess it *is* an "ambigously" worded question. It implies that "cold" is a quantifiable measure. Hmmm I wonder if that generalises to other things quantities. E.g. if I have a stick that's 1 meter long, and someone else has one that's twice as long, that'd be 2 meters. But if there's one that's "twice as short", would that be 1/2 a meter? Or how about an interval "twice as short" as 1 hour?

It's ambiguous not because "cold" isn't quantifiable, but because it's unclear which scale to use. To use linear math with temperature, you have to use an absolute scale such as Kelvin or Rankine, or it doesn't make sense. However, the question is worded such that a normal scale such as Celsius or Fahrenheiht is implied, where "twice as cold" doesn't make sense.

A stick "twice as short" as a meter is 1/2 meter- everything's fine because the smallest distance is zero meters- i.e. it's an absolute scale. If you defined 0 meters as the length of a standard meter stick, and -1 meters as zero length, then you'd have the same trouble with "twice as short" as "twice as cold".

Arjuna34

[RM Mentock]

Hmm, that didn't work, did it?

[lolagranola]

Crafter_Man gave the exact same answer my Chem teacher gave us in class.

Apparently they asked the same question on the radio one day, and about 10 science majors phoned in with the same answer. Yes, my Chem teacher was the one that got through on the radio, hence the somewhat unrelated story in my class (ok, we were discussing temperatures in kelvin).

[Yossarian]

...even though his post didn't adequately address the difference between "heat" and "temperature". Here's my argument which nonetheless supports his answer.

q = Cp(T),

Where q = Heat (thermal energy) at constant pressure (in Joules),
Cp = heat capacity at constant pressure (J/K), and
T = temperature (in Kelvins).

I couldn't readily find the heat capacity of air, but run-of-the-mill nitrogen gas is 29.1 J/K and oxygen is 29.3 J/K, so I just picked 29.1. At 0°C (assuming the jokester is talkin' metric), there are 7.95 kJ of heat energy in the air. Taking "twice as cold" to mean "half as hot", we just take half of 7.95 kJ (=3.97 kJ), plug it back into the equation for heat energy and solve for T. The answer, as sayeth C_M is 136.575 K, or -136.575°C.

So, yeah--I know it's a joke... but it does have an answer and that's it. (File this one under "more evidence for my wife's claim that I am gradually losing my sense of humor".)

[DarrenS]

Thanks for all your considered replies. I'm continually amazed by the number of intelligent people that post to this board...

Yes, of course it's just a joke. But we can also have a sense of curiosity and think...hey...that kinda works?!? It's not something I lose sleep over but it is interesting to ponder...

-Darren.