| | Rodney (may I call you that?).
There is nothing mysterious about the imaginary unit. The understanding of i = sqrt(-1) was wrapped up and dealt with by the middle of the 19-th century with the development of the theory of ideals (a kind of multiplicative ring analogous to normal subgroups of groups). It was initially a "mystery" because the Italian mathematicians Cardano and Tartaglia who first used them worked in a restricted context. With the widening of context came the understanding.
Take the real polynomial ring R[x] define and equivalence relation among polynomials modulo (x^2 + 1). This produces a division ring (aka a field) in which x^2 + 1 has a root!
No mystery at all.
Do you like matrices? Identify the comply number a + b*i with the real matrix
[a , b]
[-b , a]
or the matrix M whose elements are m11 = a, m12 = b, m2 = -b,
m22 = a (just in case the square matrix does not print out in a lined up manner).
The algebra of such matrices is isomorphic to the algebra of complex numbers where matrix multiplication corresponds to numerical multiplication and matrix addition corresponds to numerical addition. When a and b are not both zero the above matrix has an inverse which corresponds to the multiplicative inverse of a + b*i which is the product of the number with its conjugate divided by a*a + b*b.
a and b are real in all of the above.
No mystery at all and it does not require Ayn Rand or Objectivism. All of this stuff was worked out well before Ayn Rand was born. It was worked out by European mathematicians steeped in the religious/altruistic culture of Europe at the time. Go figure.
Bob Kolker
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