# Exact meaning of 'degree'

I wish to know if there is an exact meaning of degree in physics/math/chemistry.

It is used in many cases and it is not clear to me which requirements must have an unit of measurements for carrying this word in its name.

For example, in temperature it is used for Celsius but not for Kelvin. In angles is used for deg but not for radians. In the mentioned cases 'degree' was used when some arbitrary limits were imposed. But it is also used for terms like 'degrees of freedom' or in the quantification of alcohol in a drink.

• I've understood "degree" to qualify a unit that doesn't have the property that $0\,\mathrm{unit} = 0 ,$ or otherwise has some non-scalar qualities. Having trouble finding a reference for this, though. – Nat May 21 at 2:50
• Well, as you've pointed out in your question, there isn't one exact meaning because degree means different things in different contexts. – innisfree May 21 at 3:14
• See e.g., en.wikipedia.org/wiki/Degree for a list of meanings – innisfree May 21 at 3:20
• @user1420303 $pV=nRT ,$ then holding $n$ and $V$ constant,$$\frac{p_2}{p_1}=\frac{\frac{nRT_2}{V}}{\frac{nRT_1}{V}}=\frac{T_2}{T_1} \,,$$so if temperature doubles such that $\frac{T_2}{T_1}=2 ,$ then $\frac{p_2}{p_1}=2 ,$ meaning pressure doubles, right? But, does this hold when $T_1=10\sideset{^{\circ}}{}{\mathrm{F}}$ and $T_2=20\sideset{^{\circ}}{}{\mathrm{F}} ?$ This is, does $\frac{20 \sideset{^{\circ}}{}{\mathrm{F}}}{10 \sideset{^{\circ}}{}{\mathrm{F}}} = 2?$ I'd tend to see the degree-symbol as a warning like, "Hey! This isn't really a scalar unit!". – Nat May 21 at 3:50
• @user1420303 Rotation-units seem like an interesting case. I mean, you can say that they're scalar-ish in that, for example, $2 \times {200}^{\circ}={400}^{\circ} ,$ but then it's going too far to assume that they are scalar if we say that ${400}^{\circ} \neq {40}^{\circ},$ since they're all $\text{mod-}{360}^{\circ}\text{'d} .$ Since scalar-units are a subset of all-units, it's probably most accurate to say that all units are degree-units in the sense that it's consistent, but then that scalar-units are a particular subset where we can neglect the scales on which the degrees are. – Nat May 21 at 4:01

## 1 Answer

This is actually more of a linguistic question than a physics one. And, like most linguistic questions, there are no normative rules describing when the term is used. We try to capture the use of these terms after the fact, but there's no prescribed rule.

The best way to approach this is from the etymology of degree. We find it is used to describe steps of a process. So if a subject permits division into steps, "degree" is often a word that follows shortly behind.

One conjecture I have seen is that "degree" often appears when the most natural division is simply too big to be used. For example, the natural unit for angles is, well, full circles (what we now call 360 degrees). However, this is too large to be generally applicable, so we map a uniform scale to this to describe smaller divisions. In the case of angles, maps of the heavens (1 rotation per year) across the 365 days quickly turns to a much more convenient 360 degrees. In the case of Celsius, treating 0 as "coldest water" and 1 as "hottest water" gets divided up into 100 even divisions. One might extend this conjecture to suggest that Kelvin doesn't have "degree" used with it because the divisions were already tied to small things (degrees Centigrade), and did not need to undergo this uniform division process again. (though I will note that "degrees Kelvin" is not an uncommon phrasing)

This is not a solid rule, but this conjecture does suggest why there doesn't seem to be a solid rule. It suggests historical abilities to work with these quantities drives the linguistics.

• And the convenient unit of "degree" gets subdivided into minutes and seconds which are not minutes and seconds of time... – user207455 May 21 at 12:22