Complex number confusion

[jonel]

Hi All,
Just as I thought I was really understanding the geometrical interpretation of the Euler Formula I ran straight into an issue that, quite frankly, has thrown a spanner into this understanding.

Below is some text that I was following, and up to this point I was fully with the explanation.

The magical thing about the exponential (e) being here, is that if we think of elevating a number to an imaginary exponent as turning α radians around this circumference of radius 1, if we take ANY other number except e and we do this, we will have travelled a certain distance around the circumference like shown in the following figure:




Let me explain. When we take the number 2 and elevate it to iα, if α = 1 radian, we would be on the orange point, thus having travelled a distance along the circumference from the grey ‘start’ point to the orange point, along the yellow line of 0,693.
If we do this for the number 5, we would have travelled a distance from the grey point to the green point of 1,609. However, if we use the magical number e the distance we would have travelled is 1, exactly the same as the value we have given α. Woah.

Now it would be truly remarkable to me if I understood this with regard to bases other than e (for which the Euler Formula DOES work and for which a series can also prove this fact., but sadly I don't understand this. In fact I don't really believe it. But I've thought that many many times before and have usually been wrong. I would be really happy if someone could point out that this IS the case and perhaps a bit better explanation. I have to say the author of this article has done a fantastic job for me right up to this point.

Thanks

Jonel

 

[skeeter]

let’s use the base 2 example ...

[MATH]e = 2^{\log_2{e}} = 2^{1/\ln{2}}[/MATH]
[MATH]e^{i \theta} = 2^{i \theta/\ln{2}}[/MATH]
Euler ...

[MATH]e^{i \theta} = \cos{\theta} + i \sin{\theta}[/MATH]
[MATH]2^{i \theta/ \ln{2}} = \cos{\theta} + i \sin{\theta}[/math]
[MATH]2^{i \theta} = \left( \cos{\theta} + i \sin{\theta} \right)^{\ln{2}}[/MATH]
DeMoivre ...

[MATH]2^{i \theta} = \cos(\ln2 \cdot \theta) + i \sin(\ln{2} \cdot \theta)[/MATH]
on the unit circle in the complex plane, arc length [MATH]s = r (\ln{2} \cdot \theta)[/MATH]
now let [MATH]\theta = 1[/MATH] radian ...

 

[jonel]

Hi Skeeter,
Thank you very your detailed and excellent explanation. This has shone a stronger light on my geometrical interpretation of what Euler means with regard to this formula and I hope I am on the way to moving forward with this.

So, clarifying something in my own mind:

[MATH]\displaystyle s = r (\ln{2} \cdot \theta) [/MATH]
With angle of 1 radian then the only time the length around the unit circle is 1 is when [MATH] \displaystyle e [/MATH] replaces [MATH]\displaystyle \ln{2}[/MATH] and the angle is 1 radian.

That being the case, if the base is switched back to [MATH]\displaystyle \ln{2}[/MATH] the the distance around the circumference is 0.6931... with a smaller angle, meaning the angle subtended is diiferent from the 1 radian used to calculate the result.

Thanks

 

[skeeter]

correct ...

[MATH]b^{i \theta}[/MATH] subtends an arclength on the unit circle of [MATH]s = \ln{b} \cdot \theta[/MATH], which equals 1 when [MATH]b=e[/MATH]

 

[pka]

Suppose that each of \(z~\&~w\) is a complex number. Then \(z^w=\exp(w\cdot\log(z)\).
To use that one need to understand the complex logarithm: \(\log(z)=\log(|z|+i\cdot\arg(z)\)
Example: \({\large (-1)^i}=\exp\left( i\cdot(\log(|i|)+i\cdot(\pi)\right)=e^{-\pi}\)