Can a Binary Number be Viewed as an Orthogonal Basis?

[entropy1]

Could you view a discrete number, for instance a binary number, as a sort of orthogonal basis, where each digit position represents a new dimension? I see similarities between a binary number and for instance Fourier Transform, with each digit being a discrete function.

 

[Grinkle]

How is that different from writing out a vector?

[1 0 1 1 1 0]

vs

101110

Isn't it just a matter of notation and interpretation? The algebra for manipulation doesn't change, I think? There may be a deeper question than I am seeing, I am far from a mathematician.

 

[fresh_42]

Could you view a discrete number, for instance a binary number, as a sort of orthogonal basis, where each digit position represents a new dimension? I see similarities between a binary number and for instance Fourier Transform, with each digit being a discrete function.

Yes. Since $\mathbb{Z}_2$ is a field everything is fine in $\mathbb{Z}_2^n$, the $n-$dimensional unit cube.
You might want to have a look on the following page https://en.wikipedia.org/wiki/Discrete_Fourier_transform about discrete Fourier transformations which are an important tool in information theory.