Euler's Identity in Degrees

Question

Since we have a simple conversion method for converting from radians to degrees, $\frac{180}{\pi}$ or vice versa, could we apply this to Euler's Identity, $e^{i\pi}=-1$ and traditionally in radians, to produce an equation that is in degrees; one that may or may not be as simplistic or beautiful as the radian version, but yet is still quite mathematically true?

If not, why? If so, how did you derive it?

-K

I suppose the explanation here made it sound more difficult than just replacing pi with 180.

Answer 1

Given that

$$180°=\pi$$

we could write

$$e^{180°i}=-1$$

Answer 2

If you define $n^{\circ} = n \pi/180$, then you could write $$ e^{i\pi} = e^{180^\circ i} = -1.$$

Answer 3

I don't think this is a good idea, since the exponential is usually defined as the power series

$$\exp(x)=\sum_{k=0}^\infty \frac{x^k}{k!}.$$

I would say degrees $°$ is some kind of unit like meters. In the power series, you would get something like $(y°)^k$ and sums of (all) different powers of the unit $°$. This makes as much sense as adding meters to square meters.

Of course you can just write $\exp(i180°) = -1$, but in my opinion it is not well-defined nor useful.

Answer 4

But 180° is NOT equal to π. What is true is that an angle with 180° measure is same size as an angle with π radian measure. That is NOT a statement about the numbers.

Answer 5

If your angles are in degrees,

$$\dot e^{i180}=-1,$$

where $$\dot e=e^{\pi/180}=1.0176064912058515755792228003847\cdots$$ is the constant of Euler-the-Fool.