math - dual number magnitude and normalization -

i need figuring out how calculate magnitude of dual number , how use normalize number.

the closest answer found this: properly normalizing dual quaternion

however, there either error there or (more likely) not understand notation. reply post states magnitude of unit dual quaternion (still form of dual number) follows:

qq' = (r, d)(r*, d*) = (rr*, rd* + dr*) = (1, 0) 

note dual number, not single value. have seen same result in many articles on dual quaternions, none of them explain why dual number 0 (zero).

i assume that, if dual quaternion normalized, real , dual component quaternions normalized. in case, quaternion conjugate equivalent it's inverse, , rr* indeed = 1. if not normalized, rr* not = 1, do then?

additionally, rd* + dr* not 0 (unless i'm reading notation wrong), rather

rd* + dr* = [2(r.scalar)(d*.scalar) + 2dot(r.vector,d.vector), <0,0,0>] 

which quaternion non-zero scalar , "zero vector", speak. quaternion not of magnitude 0.

besides, wiki on dual quaternions, blog post (be careful when reading terms, because uses bold q's , normal q's , q's subscripts dual quaternions, quaternions, , real numbers, terribly confusing, correct: http://simonstechblog.blogspot.com/2011/11/dual-quaternion.html), , own calculations confirmed dual number times it's conjugate dual number, not single value.

do divide dual number it's dual number magnitude normalize? how work? i'm stuck.