The value of a physical quantity is normally expressed as an implied product of a numeric value and a unit of measurement.
There are three categories to consider:
There is no explicit unit of measurement included. Examples of this would include index of refraction of a medium and the specific gravity of a substance (which is ratio of the density of the material divided by the density of some reference material, usually water at some specified temperature). In this category, there is an implied measurement unit of 1 . It is usually not written because 1 times any number is that same number, so it is pointless to write the “times 1”. The value of an index of refraction is simply a number, and that number is all you write for the quantity value. That number is the numerical value of the physical quantity. It is only slightly more complicated for specific gravity, because you are dividing one density by another, and both values should be expressed in the same units of measurement, and the division of one by the other cancels out those units, leaving you with 1 as the overall measurement unit.
For plane angles, there is a relationship between the length s of the arc of a circle, the radius r of that circle, and the angle a subtended by the arc at the circle center:
a = s/r
with the angle a being measured in the unit of radians. (To write the formula for some other angular unit requires incorporating a numeric factor, which is basically a conversion factor from radians to degrees,) Thus, if you have a circle of radius 3 m and an arc of 6 m on that circle, the the angle subtended or formed is:
(6 m)/(3 m) = 2, but we said this is the number of radians, so it is 2 rad.
Notice, we are dividing a length by a length (both the arc length and the radius being lengths), so if we use the same measurement unit for both lengths (regardless that unit being meters, feet, parsecs, or anything else), the two units cancel each other out upon division. This means that the unit we are calling radian is like with specific gravity in #1—it has the value 1. Indeed, we see the formula gives us 2 and we know that it is 2 rad, and the only way we can have them be the same, 2 rad = 2, is if the unit radian is actually just a funny, special name for the number 1. Why do we give the number 1 a special name here, unlike in category #1? That is because some inexperienced people find the concept of radian to be strange and inconvenient. They would rather use degrees, or arcminutes, or arcseconds, or semicircles, or some other such unit, and they all have different sizes. For example, a full circle is 2π rad and it is also 360°. Therefore, since both equal one circle of rotation, they must be equal to each other:
360° = 2π rad. Divide both sides by 360 to get:
1° = (2π/360) rad = (π/180) rad. Now, we saw above that rad = 1, so:
1° = (π/180) rad = (π/180) × 1 = π/180.
Thus, like the radian, the degree is also a number—not 1 though, but rather π/180, which cannot be “thrown away” because π/180 times a number does not yield back the original number.
Thus, 30° = 30 × π/180 = π/6 = π/6 × 1 = π/6 rad.
This is the explanation as to when we express an angle in degrees, we must write the ° symbol or spell out degrees, whereas when we express the angle in radians, we may either explicitly write rad or we may leave it off. Unfortunately secondary school geometry textbooks do not seem to understand this point and typically leave off the mandatory ° symbol. That usually gets straightened out when radians are presented—typically later in the second year of algebra or in trigonometry, but it becomes something necessary for students to unlearn the incorrect and learn the correct. Thus, if an angular unit is included, you can convert that angular unit into a real number and multiply by the numeric part of the physical quantity value to the the numeric value of the physical quantity. (And absence of angular unit implies radians, which have numeric value 1, so the numeric value of the quantity is just the numeric value that is present.
Solid angles work similarly, involving area divided by area. The steradian (sr) is the unit that has value 1.