Dimensions and units are separate things, though. For example, 1 minute and 1 second are different units of the same time dimension. Similarly, 1 rad and 1 degree are both dimensionless, but they are both different units.
replyTheoretically, you can define systems of measurement where a lot of seemingly separate things fall on the same units and dimensions. There are "natural units" in physics where you take the fundamental nature of particles and relativity into account and make some convenient choices for some physical constants such as the speed of light c := 1. Then the speed becomes dimensionless, length and time have the same unit and dimension (a length of 1eV is a time period of 1eV), and almost all units are simply derived from a measurement of energy (electron volt, not as basic as people would like, but useful enough).
replyIt is only equal to 1 by convention. If we instead considered the ratio of the diameter to the arc-length then rad would be 1/2.
replyPerhaps in the physics sense, but in computer science we do have the notion of types which does allow us to model the difference between an angle and other numerics.
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