TITLE: My teacher said that $2\pi$ radians is not exactly $360^{\circ}$?
QUESTION [17 upvotes]: 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.

$\pi$ is irrational (and transcendental).
$360$ is a whole number.
Since no multiple of $\pi$ can equal a whole number, $2\pi$ radians is not exactly $360^{\circ}$.

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 $2\times3.14 \ne 2\pi^R.$

REPLY [3 votes]: Your teacher reasoned incorrectly.
If his/her argument was right, you could similarly argue as follows:

$\pi$ is irrational. 
$2$ halves of the circle makes a circle. 
Since no multiple of $\pi$ can be a whole number, $2\pi$ radians does not equal $2$ halves of the circle.  

This is clearly wrong because two halves make the whole circle. 
We define, though somewhat arbitrarily, that $1$ degree is just one 360-th of the total angle of the circle, i.e. $1^\circ =\frac{\text{total angle of circle}}{360}$. Now since "the total angle of a circle" in radians is $2\pi$ we get that $1^\circ = \frac{2\pi}{360}$ and thus multiplying both sides by $360$ we get that $360^\circ = 2\pi$.