3 ms·

Why not e^{2 pi i x} = cos x + i sin x then? We already handle e^{2 pi/360 i x} = cos x + i sin x for x in degrees just fine. It's not that euler no longer hold

Why not e^{2 pi i x} = cos x + i sin x then? We already handle e^{2 pi/360 i x} = cos x + i sin x for x in degrees just fine. It's not that euler no longer holds, it's that you just have to be clear about what units[0] you use when comparing the explicitly angular/geometric cos and sin with the numeric exponential, and then deciding on a default numeric cos and sin/a default "unit" for angles. If we want to get real interesting with it, this could also motivate an explicitly geometric "unit" aware exp operation, and depending on the defaults we use for the angular scale and the linear scale, 2 pi could be the conversion factor, making exp(2 pi i x) = cos x + i sin x actually interesting and useful and clarifying. [0] pedantically, they're not units, or at least not dimensional units, so whatever the word for dimensionless units are, as in degrees vs radians vs turns.

kazinator 2mo ago
The problem is that angles in the complex plane are related to multiplication, which is related to exponentiation. When you multiply two complex numbers z1 and z2, their angles add: Arg(z1 z2) = Arg(z1) + Arg(z2). That carries into exponentiation: Arg(z^2) = 2 Arg(z). The exponent 2 has an interpretation as doubling the angle. In other words, e^2πix has an interpretation as working with angles. When you have that 2π in there, but not in the sin and cos expressions, you're using different angles for multiplication/exponentiation and for sin/cos. Your left side shows that you are sticking with Arg(z) being in radians! But on your right side, you have turns: the expression cos x + i sin x is literally saying that the point whose angle is x on the unit circle in the complex plane is the complex number <cos x, sin x>. But your left side essentially says that the Arg of this point: Arg(cos x + i sin x) is not x, but 2πx! When we have a point on a unit circle whose Arg is x, then if we raise e to the power of ix, we get that point. That's what the original left hand says, without the pi. You have a "trigonometric angle" and "Arg" that are separate, right in a fundamental equation.
- 6gvONxR4sf7o 2mo ago
  If we add types or a geometric abstract manifold or something, the issue is that we do want cos and sin to take numbers with a scale (e.g. cos(90 degrees) vs cos pi/2 rads), but we generally don't give the same thing to e or exponentiation (no e^i(pi/2 radians) vs e^i(90 degrees). > But on your right side, you have turns: the expression cos x + i sin x is literally saying that the point whose angle is x on the unit circle in the complex plane is the complex number <cos x, sin x>. I totally agree here, and that's purely geometric, regardless of what we express x in. We can talk in terms of abstract points without specific coordinates/embeddings in R. > When you have that 2π in there, but not in the sin and cos expressions, you're using different angles for multiplication/exponentiation and for sin/cos. This part I'm not following. When we talk about a scale, any purely universal identity like Arg(z^2) = 2 Arg(z) is going to hold regardless of the scale. I agree that the Arg stuff nicely motivates interpreting it as an angle, but don't see how it says anything at all about the scale in question. Like, we get an interpretation of e^2πix as working in angles from the Arg reasoning, but we don't get a scale for those angles from it, do we? We'd only get Arg if we impose a scale on Arg itself, right? So if we take e^2πix at x=1/2 turn=1/2, we get e^πi=-1, which gets us Arg(-1)=1/2 turn=pi rads=180 degrees, and we can work from there, but I still don't see how it imposes a unique scale that we can say is still radians and thus incompatible with the RHS's scale of turns.

kazinator 2mo ago

The problem is that angles in the complex plane are related to multiplication, which is related to exponentiation. When you multiply two complex numbers z1 and z2, their angles add: Arg(z1 z2) = Arg(z1) + Arg(z2). That carries into exponentiation: Arg(z^2) = 2 Arg(z). The exponent 2 has an interpretation as doubling the angle. In other words, e^2πix has an interpretation as working with angles. When you have that 2π in there, but not in the sin and cos expressions, you're using different angles for multiplication/exponentiation and for sin/cos. Your left side shows that you are sticking with Arg(z) being in radians! But on your right side, you have turns: the expression cos x + i sin x is literally saying that the point whose angle is x on the unit circle in the complex plane is the complex number <cos x, sin x>. But your left side essentially says that the Arg of this point: Arg(cos x + i sin x) is not x, but 2πx! When we have a point on a unit circle whose Arg is x, then if we raise e to the power of ix, we get that point. That's what the original left hand says, without the pi. You have a "trigonometric angle" and "Arg" that are separate, right in a fundamental equation.

6gvONxR4sf7o 2mo ago
If we add types or a geometric abstract manifold or something, the issue is that we do want cos and sin to take numbers with a scale (e.g. cos(90 degrees) vs cos pi/2 rads), but we generally don't give the same thing to e or exponentiation (no e^i(pi/2 radians) vs e^i(90 degrees). > But on your right side, you have turns: the expression cos x + i sin x is literally saying that the point whose angle is x on the unit circle in the complex plane is the complex number <cos x, sin x>. I totally agree here, and that's purely geometric, regardless of what we express x in. We can talk in terms of abstract points without specific coordinates/embeddings in R. > When you have that 2π in there, but not in the sin and cos expressions, you're using different angles for multiplication/exponentiation and for sin/cos. This part I'm not following. When we talk about a scale, any purely universal identity like Arg(z^2) = 2 Arg(z) is going to hold regardless of the scale. I agree that the Arg stuff nicely motivates interpreting it as an angle, but don't see how it says anything at all about the scale in question. Like, we get an interpretation of e^2πix as working in angles from the Arg reasoning, but we don't get a scale for those angles from it, do we? We'd only get Arg if we impose a scale on Arg itself, right? So if we take e^2πix at x=1/2 turn=1/2, we get e^πi=-1, which gets us Arg(-1)=1/2 turn=pi rads=180 degrees, and we can work from there, but I still don't see how it imposes a unique scale that we can say is still radians and thus incompatible with the RHS's scale of turns.