Elegant 1/(1-x) Identities for Complex Algebraic Manipulations and Logarithms

[ClamShell]

Found out that:

1/(1-x) = the infinite product of (1+x^(2^N)) from N=0 to infinity.

From "The Harper Collins Dictionary of Mathematics" by
Borowski and Borwein.

Makes me wonder whether this identity could be used to make
some complex algebraic manipulations into real manipulations.
And it has nice logarithms too.

Yes, I have been reading the NewScientist article "Reality bits".

 

[maajdl]

This identity is obviously wrong, since the first factor (N=0) in the infinite product is always 2.

 

[ClamShell]

2^0=1; x^1=x; not 1; but anyway...

It's not x^(2*N), it's x^(2^N). 2^N is 1,2,4,8,16,...

 

[maajdl]

Right!
Impressive!
How could one derive this identity?

 

[ClamShell]

It's a "standard" example of an infinite product
found in many advanced textbooks. It's pole
is at 2*2*2*2*2*..., not 1/0 when x=1.

So often I see 1/(1-x) result leads to complex
algebra. Like the squaring of complex strengths
leading to purely real potentials. So it is seductive
to replace it with factors all of which are greater
than or equal to unity and less than(or equal to?) 2.
Have you had the chance to read "Reality bits"
in January 25-31, 2014 of NewScientist?

 

[maajdl]

No, I didn't read it.
I stopped reading NS since a few months by lack of time.
I couldn't find the (full) paper on their web site.
However, I will take the time to read this:

http://www.newscientist.com/article...o-u-searching-for-the-quantum-master-bit.html

http://arxiv.org/abs/1210.4535

http://en.wikipedia.org/wiki/U-bit

Concerning these U-bit idea, I mus say that I don't catch it very well.
After all, complex numbers are used also in electronics and almost everywhere.
Replacing them by a rotating vector is just changing their name, isn't it?

Added:

I like this sentence in the ArXiv paper:

"We are thus led to consider the following model. Every system is to be described as a
quantum object in a real vector space, with the same dimension it would normally have in
the complex theory, and in addition, there is a single auxiliary rebit. We call this auxiliary
rebit the universal rebit, or ubit, because in this model it needs to be able to interact
with every object in the world."

Reminds me of some usual dreams I have and the question of entanglement.
Will read further later to see if this is confirmed.