Can Angles Be Assigned Dimension in Scientific Calculations?

[HappyFeynman]

Homework Statement

In class, we had determined that radians are a form of a count of distance in terms of radius units. We stated this is different from other units such as m, s, etc. I am having an issue in determining whether two values can be added. Lets say we have the equation (6.7 rad^2 * m) + (3 m) = z. Since we defined radians as a count of distance, we can put it in terms such as 1 rad^2 = m^2 / m^2 = 1, so it is dimensionless. Because of this, can I add the two values to simply state that z= 9.7 m?

Relevant Equations

1 rad^2 = m^2 / m^2 =1

(6.7 rad^2 * m) + (3 m) = 9.7 m

 

[BvU]

Radians are m/m, so in principle you are right.

 

[haruspex]

You might find this interesting: https://www.physicsforums.com/insights/can-angles-assigned-dimension/.
The bottom line is that rad² is certainly dimensionless, but if you find you are adding terms that have a mix of radians to an even power and radians to an odd power then you have very likely gone wrong. E.g. energy, $ML^2T^{-2}$, cannot equal angular momentum, $ML^2T^{-2}\Omega$, where $\Omega$ represents the 'angular' dimension.