As a second postscript, the simplest way to explain Euler's version of the equation is to make four observations. They're a *bit* hand-wavy, but should be fairly easy to follow:
1) Exponentiation that uses one constant base can be always rewritten in another base by including an appropriate scaling factor in the exponent. For example, 8^x can be written as 2^(3*x). There's a specific way to find the scaling factor, but it's not important for the discussion here.
2) A bit less obviously, the same principle extends to the imaginary unit: with the right scaling factor, it can be moved from the base to the exponent. For example, if we want to go from base i to base 10, the solution is i^x = 10^(~0.682 i * x). The important point is that both forms -- imaginary^real and real^imaginary -- work the same: they give us continuous rotation in the complex plane.
3) As discussed in the article, the expression i^x implements rotation. In essence, as x goes from 0 to 4, the resulting point, starting at (1,0), travels the distance of 2*pi along the unit circle (radius 1, circumference 2*pi).
Note that this is different from how we usually define sin(x) / cos(x): when using radians, there is a 1:1 correspondence of "input" and "output" distances: 2*pi radians means a full rotation long the circumference of 2*pi. This is why in the formula that equated two methods of rotating a point, we needed a scaling factor on one side for the equality to hold:
i^(2/pi*m) = cos(m) + i * sin(m)
The 2/pi factor means that when the parameter of sin() / cos() changes from 0 to 2*pi, the exponent on the left side changes from 0 to 4 (2/pi*pi*2).
It would be nice to find a different base where the rate of change on the left side naturally matches the right side, so that we could lose the 2/pi bit!
4) There is only base with an initial 1:1 correspondence between an increment in the exponent and the resulting increment in the value of the expression. It's, by definition, the mathematical constant e. If you were introduced to e in a different way and are suspicious, you can run a simple numerical test with a small exponent delta of, say, 0.0001:
Rate of change for base 2: (2^0.0001 − 2^0) / 0.0001 = ~0.69
Rate for base e: (e^0.0001 − e^0) / 0.0001 = ~1.00
Rate for base 4: (4^0.0001 − 4^0) / 0.0001 = ~1.39
This rate changes as we get away from real zero: that's the nature of exponential growth. For example, if the exponent is in the vicinity of 2, the rate jumps to (e^2.0001 − e^2) / 0.0001 = ~7.39. That said, only the real part of the exponent changes the growth rate. In the absence of a real part, the rate of change for e^imaginary will remain 1:1 at all times.
From #1 + #2, if we want to switch to base e, it follows that we can rewrite our scaled, 1:1 rate-of-change expression -- i^(2/pi*m) -- as e^(<sth>*i*m). At first blush, it's not clear what <sth> ought to be. But from #3 + #4, we know that the rate of change for e^(i*m) is always 1, so we don't actually need any further scaling -- <sth> = 1. This gives us Euler's formula:
Whenever I think about complex numbers, I mostly think of them as an X/Y coordinate in 2d space, so I need to convert them to a vector (angle+magnitude), add the angle, and convert them back. I'm currently unable to grasp how the rotation equivalences map into the whole thing.
I have always been a fan of describing the complex numbers as 2d scaling and rotation matrices. If someone has enough familiarity with linear algebra, it is much more concrete and constructive. There is no 'we introduce this variable i with no motivation'.
You quickly get the understanding that there is a square root of the identity matrix. It then just takes a little work to show that the sum of two suchs matrices remains such a matrix.
It might make field extensions a bit more jarring for some students. But most students never need to get there.
I understand Gauss wanted to call the "imaginary" axis the "lateral" axis, which would have been a lot less confusing for math students. There is also the nice fact that understanding multiplication as rotation explains something else that students can find confusing: It makes sense that multiplying positive (real) numbers results in a positive real number, and it sort of makes sense that multiplying a positive real number times a negative real number gives a negative number, but much harder to understand why multiplying two negative numbers results in a positive number. But it's just a matter of 180° rotations.
As a second postscript, the simplest way to explain Euler's version of the equation is to make four observations. They're a *bit* hand-wavy, but should be fairly easy to follow:
1) Exponentiation that uses one constant base can be always rewritten in another base by including an appropriate scaling factor in the exponent. For example, 8^x can be written as 2^(3*x). There's a specific way to find the scaling factor, but it's not important for the discussion here.
2) A bit less obviously, the same principle extends to the imaginary unit: with the right scaling factor, it can be moved from the base to the exponent. For example, if we want to go from base i to base 10, the solution is i^x = 10^(~0.682 i * x). The important point is that both forms -- imaginary^real and real^imaginary -- work the same: they give us continuous rotation in the complex plane.
3) As discussed in the article, the expression i^x implements rotation. In essence, as x goes from 0 to 4, the resulting point, starting at (1,0), travels the distance of 2*pi along the unit circle (radius 1, circumference 2*pi).
Note that this is different from how we usually define sin(x) / cos(x): when using radians, there is a 1:1 correspondence of "input" and "output" distances: 2*pi radians means a full rotation long the circumference of 2*pi. This is why in the formula that equated two methods of rotating a point, we needed a scaling factor on one side for the equality to hold:
i^(2/pi*m) = cos(m) + i * sin(m)
The 2/pi factor means that when the parameter of sin() / cos() changes from 0 to 2*pi, the exponent on the left side changes from 0 to 4 (2/pi*pi*2).
It would be nice to find a different base where the rate of change on the left side naturally matches the right side, so that we could lose the 2/pi bit!
4) There is only base with an initial 1:1 correspondence between an increment in the exponent and the resulting increment in the value of the expression. It's, by definition, the mathematical constant e. If you were introduced to e in a different way and are suspicious, you can run a simple numerical test with a small exponent delta of, say, 0.0001:
Rate of change for base 2: (2^0.0001 − 2^0) / 0.0001 = ~0.69
Rate for base e: (e^0.0001 − e^0) / 0.0001 = ~1.00
Rate for base 4: (4^0.0001 − 4^0) / 0.0001 = ~1.39
This rate changes as we get away from real zero: that's the nature of exponential growth. For example, if the exponent is in the vicinity of 2, the rate jumps to (e^2.0001 − e^2) / 0.0001 = ~7.39. That said, only the real part of the exponent changes the growth rate. In the absence of a real part, the rate of change for e^imaginary will remain 1:1 at all times.
From #1 + #2, if we want to switch to base e, it follows that we can rewrite our scaled, 1:1 rate-of-change expression -- i^(2/pi*m) -- as e^(<sth>*i*m). At first blush, it's not clear what <sth> ought to be. But from #3 + #4, we know that the rate of change for e^(i*m) is always 1, so we don't actually need any further scaling -- <sth> = 1. This gives us Euler's formula:
e^(i*m) = cos(m) + i*sin(m)
If you want a more visual explanation of Euler’s formula, I wrote one a long while back:
https://docs.google.com/presentation/d/1oMNjkDp-LieSGnZEwNpceG8KTIvbnus9olu3KqnM5bg/
Whenever I think about complex numbers, I mostly think of them as an X/Y coordinate in 2d space, so I need to convert them to a vector (angle+magnitude), add the angle, and convert them back. I'm currently unable to grasp how the rotation equivalences map into the whole thing.
I have always been a fan of describing the complex numbers as 2d scaling and rotation matrices. If someone has enough familiarity with linear algebra, it is much more concrete and constructive. There is no 'we introduce this variable i with no motivation'.
You quickly get the understanding that there is a square root of the identity matrix. It then just takes a little work to show that the sum of two suchs matrices remains such a matrix.
It might make field extensions a bit more jarring for some students. But most students never need to get there.
I understand Gauss wanted to call the "imaginary" axis the "lateral" axis, which would have been a lot less confusing for math students. There is also the nice fact that understanding multiplication as rotation explains something else that students can find confusing: It makes sense that multiplying positive (real) numbers results in a positive real number, and it sort of makes sense that multiplying a positive real number times a negative real number gives a negative number, but much harder to understand why multiplying two negative numbers results in a positive number. But it's just a matter of 180° rotations.
This clarified the most fundamental question, that no one has ever answered to me till now. The shortest summary of this entire article!! though
Thank you, that’s gold.