And here we have practically the poster child for elementary failure of understanding.
First of all, it’s perfectly possible for people to utter absurd and irrational statements, which on its own destroys your facile asserted equality above. Indeed, I was directed recently to a public statement by one Marjorie Taylor Greene, attributing the recent California wildfires to Jewish space lasers paid for by the Rothschilds. Since I observed her uttering this statement on a TV news channel, I know it’s possible for people to treat batshit insane ideas as fact. Indeed, treating batshit insane ideas as fact, is practically a definition of both absurdity and irrationality. But wait, there are plenty of people out there who perform this operation, and do so frequently. Pedlars of ex recto apologetics gatecrashing this site spring to mind at this juncture.
As for impossible, well, it’s impossible for a real number to equal the square root of -1, but that doesn’t mean the concept of the square root of -1 is absurd or irrational, as any competent pure mathematician will tell you. Indeed, an entire branch of pure mathematics, namely complex analysis, arose from the act of defining i to be the square root of -1, and extending the number system via said definition.
Extending the number system in this manner has turned out to be immensely useful from a practical standpoint, courtesy of the existence of conformal mapping theory, which, among other uses, has been used to design aerofoils for commercial jetliners (see “Joukowski aerofoil” for more on this).
In addition, extending the number system via this means, turned up some interesting surprises. In the realm of the complex number field, the French mathematician Augustin-Louis Cauchy alighted upon a startling result, centred upon his work aimed at providing a rigorous definition for the derivative in complex number theory. That startling result consisted of a proof that if the derivative exists, for a function in the complex number field, then that function is infintely differentiable in the complex number field. This is not true for functions restricted to the real number field, a good many of which are only finitely differentiable.
Then there’s another startling outcome from extending the number field to the complex numbers, which centres upon whether or not it is possible to extend the definition of a function to a region outside its original region of definition. Quite simply, it has been proven that this is always possible for functions in the complex number field. The process consists of the following steps:
Step 1: Given a region R1 within which a function f(z) is defined, now define a new region R2, in such a manner that it overlaps R1. In other words, the region R1 ⋂ R2 is non-empty.
Step 2: Define a function g(z) within the region R2, in such a manner that within the region R1 ⋂ R2, f(z) = g(z) for all points in that region .
Step 3: The function g(z) in R2 therefore constitutes the extension of the definition of f(z) into the region R2.
This process is known as analytic continuation, and is a perfectly proper and respectable means of extending a function definition to a new region of the complex plane. Indeed, applying this process to the zeta function allowed Bernhard Riemann to extend the definition thereof to the entire complex plane, with the exception of a singularity at z = 1+0i.
While doing so, he alighted upon an extremely interesting hypothesis about the non-trivial roots of that function, which has significant impact upon our knowledge of the distribution of prime numbers. That hypothesis, incidentally, has tested and destroyed numerous world class mathematicians, and remains unproven to this day, which is why the Clay Mathematical Institute is offering a $1 million prize to the first mathematician to succeed in proving or disproving the hypothesis.
So, while it’s impossible for a purely real number to equal √-1, allowing that number to exist leads not only to an internally consistent new number field, but to a host of illuminating results covering such topics as functional analysis, vector spaces, etc. Indeed, extending tensors to the complex number field allows for a much easier transition to spinors than would otherwise be possible.
As a corollary of the above, if you have problems with this elementary concept, I see no reason why anyone should trust the rest of your apologetics.