# Should the ° symbol move after the temperature unit to indicate addition/subtraction?

When I was in university, I had a physics lecturer who insisted that when adding or subtracting a certain change in temperature to a known temperature that the unit of the change in temperature be written as C° or F° instead of °C or °F.

For instance: 15°C + 4C° = 19°C

I've never seen anyone ever use this convention anywhere else, so was this something he made up himself or is there some historical foundation to it?

• Interesting. I've never seen this notation. – valerio Nov 15 '16 at 12:21
• I've never seen this either. – gj255 Nov 15 '16 at 12:34
• Do you have any references for the use of this notation? – Farcher Nov 15 '16 at 22:59
• No references. It was literally just one physics teacher in university 18 years ago who required this notation! – hurleytom Dec 6 '16 at 12:47

I don't use or teach this distinction, but let me be the devil's advocate and explain why it's not a stupid idea. Consider this scenario"

Your boss is a grumpy old man of a physic professor and he has asked you to help him rewrite an old set of lab instructions with all SI units instead of the English traditional unit it was originally prepared in. At one point you find in the instructions the sentence "We find that the new temperature is $73\,\mathrm{F} + 42\,\mathrm{F} = 115\,\mathrm{F}$."

How do you re-write the sentence?

Obviously you will convert the last figure with $T_C = \frac{5}{9} (T_F - 32)$ to get the Celsius reading that corresponds to a Fahrenheit reading of 115. And you'll do the same thing with one of the figures on the left, but the other one will need to be converted with $\Delta T_C = \frac{5}{9} \Delta T_F$ (without the zero offset, right?). Only which of the two on the left do you use the zero offset for?

The right answer is that you use it for the figure that represents the reading on the thermometer, and you don't use it for the one that represents a difference of readings, but as written (or with all the units written $^\circ\mathrm{F}$ there is no visual indication to indicate which figure represents which kind of quantity.

This convention suggests using $^\circ\mathrm{C}$ ("degreees Celsius") for readings taken from thermometers (temperatures) and $\mathrm{C}^\circ$ ("Celsius degrees") for temperature differences, so that you indicate which quantities are temperatures and which temperature shifts in the text.

My experience is sufficiently limit, however, that I have never seen the convention used in the wild (outside of a textbook).

This notation is not uncommon, but I've seen in only when expressing temperature differences.

Thus 90 ${}^\circ$C - 80 ${}^\circ$C = 10 C${}^\circ$ or 80${}^\circ$C + 10 C${}^\circ$ = 90 ${}^\circ$C.

Also, the unit for specific heat: cal/g/C${}^\circ$

I don't know if this is standardized by any organization, but I doubt it. I think authors use it for clarity. It does make some sense; whether or not it is a good practice is a matter of opinion.

• Do you have any sources that use it? – innisfree Nov 15 '16 at 13:50
• you say it makes some sense. What would that sense be? (He asked, with genuine curiosity). Why does it have to be $80^\circ C+10C^\circ$? Why not $80C^\circ$? If it makes sense to differentiate which temperature is being used to offset an existing one, what do you do in standard units (Kelvin), with no degree symbol? This, to me, seems too arbitrary and pointless to have a basis in science or necessity. I call shenanigans – Jim Nov 15 '16 at 13:57
• @innisfree I have (or had, I'll have to look) textbooks that do this. Checkout this Wikipedia entry – garyp Nov 15 '16 at 15:04
• @Jim Don't shoot the messenger! I'm just saying that I've seen it, and it makes some sense. I'm not saying that it makes a lot of sense. In practice, I find that it confuses beginning students (I've taught from a textbook that uses it). See also this Wikipedia entry – garyp Nov 15 '16 at 15:08
• Bah humbug. I'll stick to Kelvin and forget about all this – Jim Nov 15 '16 at 16:47

If you read the BIPM document about temperature you will find that the unit is degree Celsius with the symbol $^\circ\rm C$.

Also it would seem that your equation should be written as $15\; ^\circ \rm C + 4\; ^\circ \rm C = 19 \; ^\circ \rm C$.

• What's with the slashes? Why would you use inverse degrees C – innisfree Nov 15 '16 at 13:51
• @innisfree Quite right. I have edited my answer. – Farcher Nov 15 '16 at 14:01

I've never seen this notation used, and instinctively it seems too subtle to be very effective. It would be very easy to misplace the degrees symbol, which may be why your professor is very pedantic about the concept.

At the dimensional analysis level, the desire to distinguish relative temperatures and absolute temperatures is a real thing. Many automated units systems such as Boost.Unit distinguish between relative temperatures and absolute temperatures because they convert differently.

The equation to convert absolute temperatures from Celsius to Fahrenheit is $F=\frac{9}{5}C + 32$. However, if we are interested in relative temperatures, such as "5 degrees Celsius warmer", the equation is simply $\Delta F=\frac{9}{5}\Delta C$ (and note that, in this case, I elected to put a delta symbol in front of them... that's my chosen notation for this post, because it's even more noticeable than your teacher's notation).

Let's use your example: 15°C + 4C° = 19°C. As written, this is pretty benign, but what if the units did not all agree. Consider two similar expressions with mixed units: 15°F + 4C° and 15°C + 4F° If you have to convert these to have the same units, you need to understand whether you are converting an absolute temperature or a relative temperature. If you convert 15°C (an absolute temperature) to Fahrenheit, you use $F=\frac{9}{5}C + 32$. On the other hand, if you convert 4C° (a relative temperature) to Fahrenheit, you use $\Delta F=\frac{9}{5}\Delta C$. It is up to you to keep track of which equation to use.

It sounds like your professor has a system to notate whether you are talking about relative or absolute temperatures, and it sounds like he's pretty emphatic about it. Personally, I think the odds of a misplaced ° are much higher than the odds of mistaking a relative temperature for an absolute temperature, so I'm not a big fan. If I think I care, I'll use the $\Delta$ symbol to get my point across. However, he is your teacher, so choose wisely!