My teacher said that $2\pi$ radians is not exactly $360^{\circ}$?
Your teacher is wrong! The key point is that $360^\circ$ is not merely a whole number, but a whole number together with a unit, namely "degrees". That is, it is not true that $2\pi=360$ (in fact, this is obviously false, since $\pi<4$ so $2\pi<8$). Rather, it is true that $$2\pi\text{ radians }=360\text{ degrees.}$$ This is similar to how $1$ foot is $12$ inches, or $1$ mile is $5280$ feet. In this case, however, the ratio between the units "radians" and "degrees" is not just a simple integer ratio like $12$ or $5280$, but an irrational number! In fact, $1$ radian is equal to $\frac{180}{\pi}$ degrees.
(In fact, in advanced mathematics, it is conventional to consider "radians" as not being units at all, but just plain numbers. If you adopt this convention, then the term "degree" is just a shorthand for the number $\frac{\pi}{180}$. That is, "$360$ degrees" means $360\cdot \frac{\pi}{180}=2\pi$.)
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 $360^\circ = 2\pi$.