[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.