[heisenzombie]

Right, you can explain it but that doesn’t make it elegant. This is the equivalent of a function with a comment in it saying “Here be dragons!”.

One very desirable property of a units system is that incompatible quantities have incompatible units. So if I try to add 1 Joule (== 1 kg⋅m^2⋅s^−2) to 1 meter, I’m obviously doing something wrong and I just can’t continue.

But in SI I can take 1 Hertz (==1 s^-1) and add it to 1 radian/second (== 1 s^-1) and SI says sure, that’s 2 s^-1. But it’s not, that’s a totally unphysical thing to do and the answer is nonsense.

[fsh]

There is absolutely no guarantee that the sum of two quantities with compatible units makes any physical sense.

The issue here is that angles are inherently unitless geometrical quantities. "Radian" is merely a reminder that a number represents an angle. There is no way to make it a consistent physical unit.

[throw_a_grenade]

This is misunderstanding. There are many situations in SI where two units boil down to the same expression in terms on fundamental units, yet addition does not make physical sense.

More clear example: both Gy and Sv are J/kg (m^2 s^-2), but to add two values, one in each of those units, you need to multiply one of them by a coefficient (that coefficient depends on the kind of radiation and specific organ that absorbed this radiation). So somewhere in the math world lives a constant that is unitless 1, but x Sv/Gy, where x != 1 (again that x depends on circumstances).

Those situations just happen. It's how physics work.

[jonwiseman]

That sum to me is both physical and not nonsense, and I think your interpretation is wrong. 1 radian/second != 1 s^-1, but instead ≈ 0.159155 Hz. This is from interpreting a radian as a ratio, in which "doing" one radian in one second is only going over 1/2pi of a whole in a second. So 1 Hz + 1 radian/second, physically and sensibly to me, means ≈ 1.159155 Hz.

[whatshisface]

I am afraid you may have misunderstood them. It's not that you can't add a centimeter to an inch, it's that you have to convert units first. The unit symbols are intended to help indicate when a unit conversion would be necessary. What many find confusing is the fact that Hz and rad/s are both (in terms of basic physical units) advertised as s^-1.

To resolve the issue, you could define 1Hz as 2pi/s on the grounds that the Taylor series for sin(t) is in "units" of rad/s. You could also define one rad/s as 1/(2pi s), arguing that since the becquerel is 1/s and means "the number of times something happens per second," it would be inconsistent for frequencies to be denominated other ways, and that Hz is applicable to functions other than sin(t), including square and triangle waves, which have no relationship to angles.

[jonwiseman]

I might have misunderstood them, but I tried to argue that I was indeed converting units. Namely, 1 inch + 1 cm = 1.254 in (cm->in implicitly converted).

Also, defining a Hz as 2pi/s seems to be mixing units and quantities, like a Hz is 2 [pi/s] (where [] stores units), where pi is a unit equal to 6.28 [unitless]. Intead, defining a Hz as 2pi/s is to me changing the definition of Hz, unless it's like 2pi / 2pi [1 / s]. >You could also define one rad/s as 1/(2pi s) I was, specifically 1/2pi [1/s].

But as to your point that things like Hz are semi-meaningless unless attached to physical concepts, I'll agree with that (using a potentially different definition of semi-meaninglessness.)

My original statement that 1 Hz + 1 radian/second (i.e. 1 [1/s] + 1/2pi [1/s]) is both physical (semi-meaningless, meaning somewhat meaningful and fully meaningful when we flexibly attach it to a large range of physical concepts) and non nonsense (again, semi-meaningless until we attach to a physical concept) I think still hold.

The key to me is that we can attach physical concepts to units and measures flexibly (but not arbitarily). Hz is indeed applicable to non-sin functions, like square waves, but then we're all of the sudden talking about "the Hz of the nth harmonic of the square wave" (which has loads of relationship to angles).

[chrismorgan]

> 1 inch + 1 cm = 1.254 in

That should be 3.54cm.

[delta_p_delta_x]

> incompatible quantities have incompatible units

Okay; how do you add the unit of energy, one joule, to the unit of torque, one newton-metre?

Both have the same SI units: kilogram square metre per second per second. But one is a scalar quantity, whereas the other is a vector quantity.