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I don't understand how an angle (radians) is considered a mere number, while degrees (for example) aren't.

I think that degrees are different in that they are defined arbitrarily, but I don't find the relation between being defined arbitrarily, and having a unit.

When do we consider quantities to be merely numbers? And when do we not?

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I don't think "mere number" is a technical term. Are you asking why radian measure is a natural way to quantify angles, or why angle measures in general are unitless, or what? – Henning Makholm Mar 3 '12 at 20:03
You have to put in mind radians are the length of the curve in the circumference with a specified radius equal to one then your unity is the radius of a an arbitrary circumference. – checkmath Mar 3 '12 at 20:20
@HenningMakholm I am asking why radians don't have units, while degrees have. Why do we put a little circle above degree angles, while we don't have to put anything with radian angles? – w4j3d Mar 3 '12 at 20:31
up vote 2 down vote accepted

Physical units like meters or kilograms are arbitrary and unknown, in the sense that we can't meaningfully answer a question like "how many is a kilogram?" (how many of what?).

On the other hand, with units like degrees or percents, we do know the numerical value of the unit: $1\% = \frac{1}{100}$, and $1^\circ = \frac{\pi}{180}$. The reason we use such units, even though we could just replace them with their numerical value, is simple: convenience (and habit).

In fact, some units that we might think of as unknown do in fact have numerical values. For example, chemists often measure amounts of substance in units of one mole, which is defined as the number of constituent objects (atoms, molecules, etc.) equal to the number of atoms in 12 grams of carbon-12. As it happens, we do know how many atoms there are in 12 grams of carbon-12, although not exactly: the number is known as Avogadro's number, and is approximately $6.02214129 \times 10^{23}$. So $1 \operatorname{mol} \approx 6.02214129 \times 10^{23}$.

In fact, there have been recent proposals to define an exact value for Avogadro's number (which would then make the kilogram a derived unit equal to the mass of $\frac{1000}{12}$ moles of carbon-12); one somewhat popular proposal is $602{,}214{,}141{,}070{,}409{,}084{,}099{,}072$ $=$ $84{,}446{,}888^3$, which is well within measurement error of the currently known value, and has the nice feature of being a perfect cube.

(Also, quite a few "fundamental" physical units, like the meter and kilogram mentioned above, can in fact be given numerical values by expressing them in Planck units, which are (believed to be) in some sense natural to the universe we live in. So, in that sense, the line between the two kinds of units does get somewhat blurred.)

Of course, there's a related question I haven't answered, which is "why are angles naturally measured in radians, then?" One answer is that the math just works out that way: the trigonometric functions have some very nice and natural mathematical properties when defined in terms of radians (like the fact that the derivative of $\sin x$ is $\cos x$), which they would not have if angles were measured in any other units.

However, despite their naturalness, radians are not always the simplest choice for measuring angles. For example, in many situations, measuring angles as fractions of a full circle can make calculations simpler by getting rid of annoying factors of $2\pi$. Then again, sometimes it goes the other way, too.

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Perfect :) Thank you. – w4j3d Mar 3 '12 at 20:47

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