# 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