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1. THE CONTEXT OF THE PROBLEM

This question came to me when I was exploring complex exponents. The key identity to computing expressions with complex exponents is the Euler's identity:

$$e^{i\theta}=\cos\theta+i\sin\theta$$

This enables us to compute, for example what $2^i$ is by some algebraic manipulation. The computation is as follows:

$$2^i=e^{\ln2^i}=e^{i\ln2}=\cos\ln2+i\sin\ln2\;\approx\;0.769+0.639i$$

2. THE PROBLEM

A question arises in my head. We compute sines and cosines with radian values. And since $\ln2\approx0.693$ then $\sin\ln 2$ will be the $y$ coordinate of the unit circle when the angle is $0.693$ radians. But if we use turns instead of radians, then $\sin\ln 2$ will be the $y$ coordinate of the unit circle when the angle is $0.693$ turns. So the value of sine when using turns or radians is different. Similarly, the value of cosine is different.

But that creates a problem. When using radians, $2^i$ computes to about $0.769+0.639i$. But when using turns, $2^i$ computes to about $0.994+0.110i$.

3. POSSIBLE IMPLICATIONS

The problem above illustrates that $2^i$ and any expression with complex exponent is a generalisation that directly depends on what units we use for angles.

There are only two possibilities of "the state of truth" that this fact implies. Either complex exponents are a concept completely made up by humans which would mean that expressions like these are really undefined to the code of the universe (we only make it defined because we think that radians are the true angle units), or there must be a unit that is the truest unit for measuring angles, whether that would be radians or something else.

If the first statement is true, then this would mean that we can accept $2^i$ to be $0.769+0.639i$. After all, this concept would be defined by radians only because we defined it like this.

However, if the second statement is true, then there are even more questions to be asked. If there is a "truest" unit for measuring angles then what is it? Perhaps radians are really the truest unit, meaning that $2^i$ is unambiguously $0.769+0.639i$, but if so, what justifies this fact? What makes radians truer than turns?

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    $\begingroup$ there is no truest angle measure. Recommend Hartshorne, Geometry: Euclid and Beyond $\endgroup$ – Will Jagy May 21 '18 at 1:55
  • $\begingroup$ Note that your step $\ln(2^i) = i \ln(2)$ is generally not correct. If we denote the principal branch of the complex logarithm with $Log(\cdot)$, then $0 = Log(1) = Log((-1)^2) \neq 2 Log(-1) = 2 i \pi$. However, your overall result is still correct because $2^i :=_{\text{per def.}} e^{i Log(2)} = e^{i\, \ln(2)}$ because $Log = \ln$ on the reals. $\endgroup$ – ComFreek May 21 '18 at 11:32
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The actual definition of the complex exponential map is equivalently via an ODE ($f'=f$, $f(0)=1$) or via power series ($f(z)= \sum_{k\geq 0}\frac{z^k}{k!}$).

Similarly, cosine and sine functions are defined via power series above all. If you are to change the way you measure angles and therefore define a new, distorted version of sine and cosine, then the most natural thing to say would be that the Euler formula simply doesn't hold in your new settings. Or you could use it as a new definition for a distorted exponential map, but then what is more important for you: the fact that Euler holds, or the fact that $\exp$ satisfies the above ODE?

All of mathematics rely on the fact that we agree on axioms, language and definitions. How natural these are depend on how beautiful the resulting theorems seem to the trained eye.

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    $\begingroup$ Interpreting $\exp$ and trigonometric functions as a series clears it all at once. Why didn't I think about it before? Thank you for that answer, that's what I was looking for essentially :) $\endgroup$ – KKZiomek May 21 '18 at 3:01
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    $\begingroup$ @KKZiomek You're welcome. Thank you for this question of a very high quality. $\endgroup$ – Arnaud Mortier May 21 '18 at 3:05
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The main issue here is that Euler's formula is correct for radians only. Your questions are arising from using it with turns instead of radians, but you're starting from a false premise.

Often, the order we learn things isn't equivalent to the most precise mathematical definition of things.

For example, at school we first learn exponentiation with natural exponents, then we generalize for integers and even rationals, but how about irrational exponents, for example? The best path to follow if you're looking for precise math (and that's what I suggest you do here) in this case is to consider the definition of the exponential function as the nonzero function that satisfies the property $f(x+y) = f(x)f(y)$. And then if you go further to complex numbers, you'll see that actually one needs to choose what is called a branch of the logarithm function, and so on.

In fact, $2^i$ can have more than one value, depending on the branch choice, but usually we take the so-called principal branch. Regardless, this is all a set-up to reiterate that if you go to the very definition of complex exponentiation / principal complex logarithm, you'll see that Euler's formula only applies to radians in the first place, so your premise is false.

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While Arnaud Mortier provided very good answer, I'd like to make another very simple observation - that you consider blindly changing of the units of only some of the values, but forgot to check&update the formula/constants/etc.

This is a thing you always need to keep track of when dealing with physics, electronics, etc, where it's common to use various units for the same kind of values, and often we keep unit markings all the time (think Ohm: not just 10/0.5=10, but 5V/0.5A => 10Ohm, etc) especially to not forget and mix them up.

I see that you've tried to formalize the question, so forgive me if my answer is too naive or too lax..

But, let's see. For generic typical (planar/Euclid/..) triangle:

$${\alpha}+{\beta}+{\gamma} = {\pi}$$

If I use radians for all input angles, all's fine:

$$\pi/4+\pi+\pi/4 = {\pi}$$

But If I try to use turns or degrees, everything detoriates:

$$45+90+45 = {\pi}$$

That's unless the PI constant adapted to turns/degrees as well.

$$45+90+45 = 180$$

But that's the easy case. Sum is linear operation, so i.e. changing units ("scale", "multiplier", etc) on one side implies need for just changing units on the other side (a+b=c => (kx+ky)=c => x+y=c/k => x+y=z // a=kx,b=ky,c=kz, all units kept same, you may do it "blindly")

In your case, you started to wonder about using turns and degrees in:

$$e^{i\theta}=\cos\theta+i\sin\theta$$

but:

  • first of all, there are some constants (e, i) and you have not even tried to update them to new units
  • more importantly, some of the operations are not linear (exponent), hence units will not "just transfer to the other side"

This just reminds us to keep the formula up-to-date with our assumptions. Now, if we keep the units in mind, we might update the formula:

$$e^{i(\theta/2\pi)}=\cos(\theta/2\pi)+i\sin(\theta/2\pi)$$

which would work on turns.. I suppose that's pretty obvious and simplistic, but that's because I chose the simplest possible adaptation, just because I wanted to show the point why units matter.


Another thing I just noticed, you've said about the sine/cosine:

sin(ln2) will be the y coordinate of the unit circle when the angle is 0.693 radians. But if we use turns instead of radians, then sin(ln2) will be the y coordinate of the unit circle when the angle is 0.693 turns

IMHO, again, dropping units forced you into error.

If we define sin/cos in terms of abstract numerics, we often conventionally use radians. However, at least in my country, in primary school, concept of "pi" and "radian" is a bit too much, so they use degrees (0°-360°) and sin/cos are defined in terms of degrees (a.k.a "deg scale"), so sin(0) = sin(180) = sin(360) = 0, sin(90)=1, etc.

Note how that's ambigous to other readers? If I didn't state it explicitly, the 360 could be as well read as radians? It's easy to tell that 360 is probably degrees, but how about 1.123?

If we now truly keep units in mind, I should have written it unambiguosly, something like: sin(0°) = sin(180°) = sin(360°) = 0, sin(90°)=1, etc - if I assumed sin/cos to be defined in degs.

On the other hand, if I assumed sin/cos to be defined in rads, then I would have to translate all constants accordingly (0,π,2π,π/2), or "update the formula" and write sin(0°/(360°/2π)) = sin(180°/(360°/2π)) = sin(360°/(360°/2π)) = 0, because now passing a degree, or turn, would be out of its defined domain..

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    $\begingroup$ If I'm not mistaken, "grad" scale is a rarely-used scale for angles which go from 0 to 100 instead of 0 to 360. I believe you mean "deg" instead of "grad". In my calculator, I have the three options, "deg", "rad" and "grad" where "deg" is the 0-360 one and "grad" is one that I never use. $\endgroup$ – Pedro A May 21 '18 at 22:21
  • $\begingroup$ @Hamsterrific: aiiiiigh, of course, corrected, thank you, I don't know why I came up with "grad" while deg=degree.. probably some ghost memory from similar calculator :) $\endgroup$ – quetzalcoatl May 21 '18 at 22:22
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    $\begingroup$ Just two notes. 1. Possible typo: 'update then' → 'update them'. 2. You can insert a degree symbol ° instead of an apostrophe by typing Alt+0176 on the numeric keypad (if you are using Windows with a full keyboard) in a plain text or with the LaTeX symbol \circ if you decide to swith to MathJax: \sin(180^\circ) → $\sin(180^\circ)$. $\endgroup$ – CiaPan May 22 '18 at 6:51
  • $\begingroup$ @CiaPan: many thanks! yes, I often use apostrophe instead of degree, exactly because I'm used to reduced keyboard. Thanks for LaTeX tip! $\endgroup$ – quetzalcoatl May 22 '18 at 7:03
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    $\begingroup$ @Hamsterrific. Just for clarity, grad has 400 units in a full circle. And, so far as I can see has no benefits, and several drawbacks over degrees or radians. $\endgroup$ – Alchymist May 22 '18 at 10:48
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In some sense, $2^i$ is uniquely defined. If you accept the definition of $2^i= e^{\log(2)i}$, then we could use another definition of $e^x$ to compute the value. For instance, the Maclaurin expansion of $e^x$.

On the other hand, there is no true unit of measurement, but rather we make a choice to make happy coincidences work.

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What makes radians truer than turns?

Well, if not truer, how about "more convenient"? In simple differential calculaus?

Draw some random angle. The sine of that angle has some fixed value. But the numerical value of the size of the angle itself will vary, depending on whether you're using degrees, or grads, or radians, or revolutions, or any other niche unit.

So the value of the ratio $$\frac{\sin (\theta)}\theta$$will depend on the units used.

If you carry out this exercise for smaller and smaller angles, this ratio approaches $1$ only if radians are used.

Finally, if you try to apply the fundamental definition for the derivative of a function as a limit to the sine function, this same ratio is left messing up the result. So convenient to just hand-wave it away as equal to $1$.

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    $\begingroup$ Radians are more convenient in some applications, but units of 2pi radians are more practical in others. I'm boggled, for example, why the designers of HTML5 graphics decided that the "rotate" transform should use radians rather than degrees, since it means that any kind of rotation--even 90 degrees--is likely to introduce rounding errors which may become visible if the action is repeated enough times. $\endgroup$ – supercat May 21 '18 at 22:46
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The length of an arc of a circle is $r\theta$ when $\theta$ is measured in radians. This conveniently avoids any conversion factors.

If turns, $\tau$, were used, the length of an arc of a circle would be $2\pi r\tau$, which might be useful somewhere.

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