relationship between complex numbers and rotational movements?

[igno]

Hi!

I'm new here, and I don't know if this is the right place to ask, but here I go:

I'm studying some maths and physics and I'm taught about the characteristics of the things but not about the reason why things are, this matter is considered up to my self so I speculate...

Is there any relationship between the complex numbers and any rotational movement or something? is the "complex thing" related with any circumference specularly symmetric to the unitary one, but directing its ratio to the inside sense of rotational axes, such vectorial products...? in this situation I'm thinking of I'd understand the negative value of the square of "i"...What do you think about the subject? Am I too imaginative? Do you think I'm missing something important or can you give me a reference where I can find an application of complex numbers?

By the way, forgive me if my explanations are confusing, if you pass I'll understand. I'm just doing all my best!!!

Thanks you all in advance!

Cheers

[swansont]

Well, there's Euler's formula,

[math]e^{i\theta} = cos\theta + isin\theta[/math]

 

http://en.wikipedia.org/wiki/Euler's_formula

[D H]

There are also quaternions, which can be viewed as (and were "created" as) an extension of the complex numbers. Representing a quaterion with scalar part [math]q_s[/math] and imaginary vector part [math]\vec q_v[/math] as [math]\bmatrix q_s \\ \vec q_v \endbmatrix[/math], the quaternion product

 

[math]

\bmatrix \cos \frac{\theta}2 \\ \sin \frac{\theta}2\hat u \endbmatrix

\bmatrix 0 \\ \vec x \endbmatrix

\bmatrix \cos \frac{\theta}2 \\ -\sin \frac{\theta}2\hat u \endbmatrix

[/math]

rotates the three vector [math]\vec x[/math] about the unit vector [math]\hat u[/math] by an angle of [math]\theta[/math].

 

http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

[igno]

I'm sorry that I didn't answer, my Internet contacts weren't fair enough, and besides...there was nothing much to say. I knew Euler's formula but I can't understand it. It seems to me that Euler made his work to represent a power into a circumference, trying to make it periodic and not exponentially growing but I still don't understand how... About quaternions, they seem to me a natural way to make something useful with complex numbers in 3D, using unitary vectors and all this...Ok, I'll read some more about complex numbers and polar coordinates.

Whatever, if you have an idea I'll be glad to read it. I'm particularly fascinated with rotational coordinates. It happens to me that I like to find them everywhere!

Thanks for your answers!

Greetings!

[MrMongoose]

Euler's formula simply converts between polar and cartesian coordinate systems in a complex plane, and can easily be shown by expanding into power series.

 

Even with hindsight, the elegance is mindblowing. Euler is definitely my favourite pure mathematician.

[igno]

maybe you are familiar with this link...Here you can find the work of some pure mathematicians and physicians of the 17th and 18th century. It's a huge job of translation made by Ian Bruce

 

http://www.17centurymaths.com