The Tachyonics Operator is a Relativistic Imaginary
Reconsider the Relativity Operator, R, defined;
R = 1/[(1 - [(v/c)^2])^(1/2)] = (1 - [(v/c)^2])^(-1/2) ,
and let M denote a moving mass, with m the rest-mass, so
M = Rm = m/[(1 - [(v/c)^2])^(1/2)] .
If v > c, the case for tachyons, then R is an imaginary number; making M imaginary.
A commonly-used definition of a tachyonic mass, tM, has it that tM = -iM .
This would, for instance, be the kind of definition physics professors first give to undergraduate students. And that is fine, considering the way tachyons are presented in the literature. [See "Tachyons" listing by physicist Gerald Feinberg in the Encyclopedia of Physics, Lerner and Trigg (e.g., 2nd Edition, 1991), from VCH (p. 1246 in my copy).]
But this kind of definition leaves room for confusion when standard complex quantities and tachyonic complex quantities are discussed in the same context.
We can explain the situation as follows.
Let Q indicate the absolute-value of the difference "1 - (v/c)^2", denoted;
Q = | 1 - (v/c)^2 | ,
and let the following notation convention be observed;
Q+ = (+1)Q whenever v < c ,
Q0 = (0)Q = 0 whenever v = c ,
Q- = (-1)Q whenever v > c .
Then, because R = (1 - [(v/c)^2])^(-1/2) , we indicate the three cases of R;
R = R+ = (Q+)^(-1/2) if, and only if, v < c .
R = R0 = 0 if, and only if, v = c
(assuming 1/0 is undefined; not infinity).
R = R- = (Q-)^(-1/2) if, and only if, v > c .
In the last case, for tachyons, where v > c, we have;
R- = (Q-)^(-1/2) = [(-1)Q]^(-1/2)] = ... = 1/(i[Q^(1/2)]) = (1/i)[Q^(-1/2)] .
Note, however, that 1/i = -i ,
according to the following proof.
1/i = 1/[(-1)^(1/2)] = (-1)^(-1/2) = (-1)^[(1/2) - 1]
= [(-1)^(1/2)][(-1)^(-1)] = [(-1)^(1/2)][(-1)^(1 - 2)]
= [(-1)^(1/2)][(-1)^(1)][(-1)^(-2)] = [(-1)^(1/2)](-1)/[(-1)^2]
= (-1)[(-1)^(1/2)]/1 = -[(-1)^(1/2)] = -i .
Consequently, if v > c , then R is;
R = R- = (Q-)^(-1/2) = (1/i)[Q^(-1/2)] = -i[Q^(-1/2)] .
Thus, the relativistic tachyonic mass tM is properly defined;
tM = (R-)m = -i[Q^(-1/2)]m , where v > c ,
while the corresponding tardyonic mass M continues to be defined as usual, but also;
M = (R+)m = [Q^(-1/2)]m , where v < c .
Hence, we can legitimately write
tM = -iM ,
when deriving tM using the Relativity Operator R,
but we cannot write tM = iM , in such cases, because the sign is wrong.
We see that, because of the importance of keeping track of the sign on the imaginary-unit in the rigorous derivation of tachyonic mass, tM, we must adopt special rules on the symbols we employ (i.e., we must use a notation convention), and which allow us to represent tachyonic mass in terms of bradyonic mass, while maintaining sufficient rigor to assure accurate conceptualization.
With that established, I address the main source of confusion in the next section.