SHF posts #19 - ‘pythagoras’

Deriving "Pythagoras' theorem" from complex-numbers



e^(iθ) as shown above is really a vector, and the real and imaginary parts represent the "x" {real} and "y" {imag] components.



If we treat the right hand side as just a number then we can say that the separation between two components when laid out on a single dimension is:


s^d = [cosθ]^d - [isinθ]^d

where d = {1 or 2}



If you want to find the separation between orthogonal (at right-angles, like x and y axes) component values across the plane (2-D brane-side actually) then this equation turns into "Pythagoras":


d=2:

s^2 = [cosθ]^2 - [isinθ]^2

s^2 = cos^2(θ) + sin^2(θ)




Again, this is not known generally (at all as far as I am aware).

[original typo:

Mal said: ↑

...

e^(iθ) = cosθ - isinθ

[complex number = real number + imaginary number]
​

Apologies, this should be:

e^(iθ) = cosθ + isinθ
​

of course.





It is the fundamental 'separation', s, between the components that takes the form :

s^d = [cosθ]^d - [isinθ]^d​

where d = {1,2}, depending on the dimension in which you are inspecting.


[Note: the difference inmagnitudeof the complex-vector components is the 1-D form]]