Proper way to write the units to indicate that they include an offset?

Question

Past a certain point of complexity, I get rather confused with physical units, so I am asking a physicist for help.

I have a code that represents temperature, with a resolution of [0.5 °C], whose value ranges from 0 to 255 as the temperature goes from -41 °C to the max.

The equation is:

Temperature code [units?] = 82 + 2 * Actual temperature [°C]

i.e.:

0 = -41.0 °C
1 - -40.5 °C
2 - -40.0 °C
...
81 = -0.5 °C
82 = 0 °C
83 - 0.5 °C
...

Are the units of the "Temperature Code" in the included graph one the following?

[0.5 - 82 °C]
[0.5 - 41 °C]
[-82 + 0.5 °C]
[-41 + 0.5 °C]
[0.5 °C - 82]
[0.5 °C - 41]
[0.5 + 82 °C]
[0.5 + 41 °C]
[82 + 0.5 °C]
[41 + 0.5 °C]
[0.5 °C + 82]
[0.5 °C + 41]

If not, what are the units?

EDIT: CLARIFICATION.

I know what the conversion is.

I know that the internal units of the code are "counts".

I know that, if there were no offset, the units of the code would be [0.5 °C].

What I don't know is how to include the effect of the offset in the units.

Why do I need to know?

• To document the code correctly
• For the sake of doing a strict units analysis in data conversions, to confirm that the code converts correctly; for that, each variable must be documented with the correct units
• To help others understand the code

Answer 1

255 values sounds like the value that can be contained in a single byte.

The person who created this "code" wanted to be able to represent "reasonable" temperatures with a single byte - they decided they wanted resolution better than 1°C, and they wanted to go down to "about as cold as you can get".

This means that the conversion is as follows:

From "C" to "code":

code = 2*(C+41)

from °C to code:

C = 0.5 * (code - 82)

This makes the maximum temperature that can be represented 86.5 °C - the value you get when code=255

Usually, instruments do not measure physical quantities directly in physical units: somewhere "below the covers" they measure something else - something you can translate to units. A spring balance measures displacement - you then convert displacement to Newtons.

In this case, you might have a thermistor with some circuitry and finally an ADC - measuring some voltage and expressing it as a single byte. You have a "calibration curve" which is the formula I gave above. If you actually measured the response of the device carefully you might find that the factors are slightly different.

In either case, the "units" are really contained in the calibration factors. So in my first equation, the number 2 has units "ADC/°C". And the unit on your vertical axis should just be "ADC units", "byte units", "device units", or whatever you feel comfortable calling it.

Answer 2

The units are probably in degrees Celsius.

Whenever you add physical quantities together, they must have the same units. You can't add meters to kilograms (although you can multiply them or divide them). The result of such a thing would be nonsensical. However, you can add meters to meters.

In your case, you are multiplying degrees by 2. If '2' is unitless, this is simple. The result is in degrees Celsius. You are then adding a number to this quantity, so this quantity must also be in degrees Celsius. The result is, therefore, in degrees Celsius.

However, if the '2' is not unitless, you have a problem. It may be a single unit (kilograms) or a combination of multiple units (kilograms times seconds divided by meters). In this case, the answer will be in this strange combination of units multiplied by degrees Celsius.

---

The question has been edited since I wrote up my answer quite a few hours ago. At this point, my answer addresses properly the original question (to the best of my knowledge), but fails to address the current question, because it's a bit unclear to me what exactly the OP is asking. As it stands, it appears to be more of a programming question than a physics question. I'm also not a physicist (yet!), but I hope that doesn't invalidate my contribution.

Answer 3

For the sake of doing a strict units analysis in data conversions, to confirm that the code converts correctly; for that, each variable must be documented with the correct units.

Then you're straight outa luck (or some cruder version of SOL). This unsigned eight bit integer contains a value that represents a temperature in a non-standard unit. The value might be

• A custom representation dreamed up a long time ago to represent temperature when every byte of storage on a computer was precious, or

• A custom representation dreamed up recently to represent temperature in a transmitted data stream where every single byte of transmitted data oftentimes still is precious, or

• The output of a digital temperature sensor that has been digitized using an eight bit analog to digital converter (ADC) with a digital step of 0.5°C and a zero value of -41°C.

Whatever the case may be, your unsigned eight bit temperature value is not in any standard temperature scale. This means you cannot meet your organization's artificial requirement to document each variable with the correct units. Some variables just are not represented in a standard SI or customary unit. This problem with custom representational units is a rather common occurrence when one works at the low level of processing data from a sensor, from an archaic archive, or from a transmission stream.

Just because you have to deal with that custom representation of temperature does not mean that you have to inflict that pain on everyone else. One thing you can do is to hide that custom representation to the users of your code. Data encapsulation is a 40+ year old concept. Use it! Hide that non-standard representation from the users of your code.

Answer 4

While it is not a general answer, there is an engineering standard that holds that thermometer readings in non-absolute scales and temperature differences in the same scales have both different written notation and different spoken reading.

The prescribed convention is:

• Thermometer readings get the degree symbol before the unit as in $$0 ^\circ\mathrm{C} = 32^\circ\mathrm{F}$$ which is read

  "zero degrees Celsius equal thrity-two degrees Fahrenheit".

• Temperature differences reverse the order of the degree marker and the unit, so that discussing the relative size of the scale increments one writes $$1 \,\mathrm{C}^\circ = 9/5 \,\mathrm{F}^\circ$$ and reads

  "one Celsius degree equals nine-fifths of a Fahrenheit degree".

I've only ever seen this in textbooks in the physics world, but I'm told there are engineering disciplines where it is expected.