I am sort of sad tonight on the passing of my favorite comedian-- Robin Williams. I have never seen a finer comedian than was Robin. Many people are classified as comedians, but to me, there should be two classes -- silly talk and comedian. With silly talk I may not laugh for a full hour of that silly talk, but with Robin Williams a true comedian, I laugh about every minute. Recently we lost the very best singer in the world Pavarotti, and now we lose the very best comedian.
Now there is not much to have to change in the Fundamental Theorem of Algebra with the recent revision of the Infinite-Number, where all Infinite numbers come as at least pairs of two different numbers. Now there is a concept in physics that is almost like the concept of Infinite Number versus Rational number. The concept is the vector, where the vector has magnitude and direction whereas the Rational number has only magnitude.
So, in the Fundamental theorem of Algebra, consider two classes of numbers that makes up all numbers,either the Rationals or the Infinite-numbers (vectors) and given any equation of the four operations of add, subtract, divide, multiply is there a solution from Rationals with Infinite-numbers.
Now the idea of an irrational number as a vector is very tempting so that the right triangle of 1, 1, sqrt2 has a magnitude sides of 1 and a vector side of sqrt2, and depending on the direction of sqrt2 it is either a magnitude of 1.4142 or 1.4143 in pretend 10^3 is infinity.
And, also, it is easy to see that Complex Numbers were really vectors.