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A few days ago, my math teacher (I hold him in high faith) said that $2\pi$ radians is not exactly $360^{\circ}$. His reasoning is the following.
His logic was the above, give or take any mathematical formalities I may have omitted due to my own lack of experience.
Is the above logic correct?
Update: This is a very late update, and I'm making it so I don't misrepresent the level of mathematics teaching in my education system. I talked with my teacher afterwards, and he was oversimplifying things so that people didn't just use $\pi=3.14$ in conversions between degrees and radians and actually went to radian mode on their calculator when applicable. In essence, he meant $2\times3.14 \ne 2\pi^R.$
