My teacher said that $2\pi$ radians is not exactly $360^{\circ}$?

Question

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.

1. $\pi$ is irrational (and transcendental).
2. $360$ is a whole number.
3. 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 $2\times3.14 \ne 2\pi^R.$

Answer 1

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 $360\cdot \frac{\pi}{180}=2\pi$.)

Answer 2

The unit degree is defined as two pi divided by 360 hence Steamyroot's comment. The unit degree is an irrational number in radians.

Answer 3

Your teacher reasoned incorrectly.

If his/her argument was right, you could similarly argue as follows:

1. $\pi$ is irrational.
2. $2$ halves of the circle makes a circle.
3. 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 $360^\circ = 2\pi$.

Answer 4

Your teachers claim that the irrationality of the number $2\pi$ means $360$ degrees and $2\pi$ radians is false. They are the same because of the way we define the radian. Just because a number is irrational, does not mean it does not perfectly exists. The claim is like saying $\pi$ is not perfectly the ratio of a circles circumfrence to diameter because it is irrational and thus can not be represented with decimals or fractions. It just contradicts the definition.