Infinite tetration and superroot of infinitesimal

[andydude]

1) What would be its orientation relative to I ? ( most likely 90 degrees )

If your "imaginary infinitesimal" is equivalent to \( i\epsilon \), then it would be parallel, but that depends on which interpretation you use for infinitesimals. Since infinitesimals could be added to any real number, you could consider the hyper-real field to be 2D, or 1D, its up to you, there is no right or wrong way. I think its useless to think about things like this, since you can visualize it however you want.

2) In which space we would see this orientation?

*sigh*, in whatever space you want.

3) If multiplying by I means rotation by 90 degress in complex plane, would multiplying by dI mean infinitesimal rotation in what plane?

This is only in the standard interpretation of complex numbers, you could construct a skew coordinate system for visualizing complex number, in which case the amount of rotation would depend on each point.

4) What is self root of dI? Or what tetration would result in dI?

Whatever it is, the conversion of this number to the reals will be zero. Since any possible solution would be "zero" in the reals, the distinction of different scales in hyper-reals wouldn't make any difference. Even if it is a number like \( 2^{\epsilon} - 1 \), the distinction of scales only holds for hyper-real numbers, not real numbers, which would make this value convert to zero. So basically, anyone who doesn't use hyper-reals would say its just 0.

Andrew Robbins