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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.

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

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closed as off-topic by Andrés E. Caicedo, Xander Henderson, ahulpke, The Phenotype, kimchi lover Feb 4 '18 at 2:11

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    $\begingroup$ Your teacher's wrong. $\endgroup$ – user137731 May 15 '16 at 22:39
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    $\begingroup$ The irrationality of 360° lies in the unit degree (°), not in the number 360. $\endgroup$ – TastyRomeo May 15 '16 at 22:39
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    $\begingroup$ Your teacher is so so wrong. Please tell me you are joking. $\endgroup$ – Chill2Macht May 15 '16 at 22:39
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    $\begingroup$ Are you sure he wasn't yanking your chain? Or you miss understood him? That shows a profound lack of understanding of units. $\endgroup$ – fleablood May 15 '16 at 22:40
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    $\begingroup$ To say one unit of measurement is irrational and the other isn't is ... wrong. It all depends on a unit. A degree divides the circle into 360 units that are each 1/360 of a circle. We can divide the circle into any other number, rational or otherwise, that we like. To declare a circle to be exactly an irrational 2$\pi$ units is perfectly acceptable. $\endgroup$ – fleablood May 15 '16 at 22:43
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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 $360\cdot \frac{\pi}{180}=2\pi$.)

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    $\begingroup$ This is a gif that helps to understand ru.m.wikipedia.org/wiki/Радиан#/media/Файл%3ACircle_radians.gif $\endgroup$ – Narek Maloyan May 15 '16 at 22:48
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    $\begingroup$ The "the term "degree" is just a shorthand for the number $\frac{\pi}{180}$ really did it for me. One more thing, your last point, that $360*\frac{\pi}{180} = 2\pi,$ shouldn't that be $2\pi^R$ (radians)? Or is it implied? $\endgroup$ – Max Li May 15 '16 at 22:49
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    $\begingroup$ @MaxLi: It is implied, because when you adopt the convention I'm talking about there, you just say that quantities measured in radians are pure numbers with no units. That is, if you see an angle measure without any stated units, that means you are talking about radians. $\endgroup$ – Eric Wofsey May 15 '16 at 22:50
  • $\begingroup$ Max, sometimes I've had a line "deg = pi/180" or similar in code in order to make conversion from radians to degrees or vice versa easier. (You multiply by deg to convert from degrees to radians, and divide by deg to convert from radians to degrees.) $\endgroup$ – Michael Lugo May 17 '16 at 17:39
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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 $360^\circ = 2\pi$.

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

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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.

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