>>10967431

What do you mean by "applications". Practical applications are interested in getting rationals.

>And scalars aren't real (in the physical sense), since they lack dimension.

>real

Nothing we have words for "is real" if you ask long enough - certainly not in the sense of materialist realism.

>Then what does 'y' equal? Remember the goal is to avoid using 'i', and deal purely with reals

They were talking about constructing C given R and you can do that by formally* looking at the ring R[X] or finite sequences of polynomials** and taking the quotient w.r.t. the ideal X*X+1.

*in the sense that you can operate with those things in a computably enumerable way

**sequences of X functioning as basis, and sequences of X's being nicely enumerable, i.e. you can write stuff down and compute and even teach a computer how to do it

That is to say, you consider the totality of polynomials with coefficients in R and X*X=-1.

So e.g.

-7 + 2 * X + 5 * X * X + 8 * X * X * X * X * X

and by above ideal rule, this above polynomial is judged to be equal to

-7 + 2 * X + 5 * (-1) + 8 * (-1) * (-1) * X

i.e. this above polynomial is judged to be equal to

-13 + 10 X

Working in the ring of polynomials and with the above ideal rule amounts to just working with what's otherwise called the complex numbers. Executing the rule of replacing all instances of X*X with -1 is formally and thus computationally completely unproblematic.

Sidenote

-7 + 2 * X + 5 * X * X + 8 * X * X * X * X * X

is also

-7 X^0 + 2 * X^1 + 5 * X^2 + 0 X^3 + 0 X^4 + 8 * X^5

so the representation

{-7, 2, 5, 0, 0, 8}

for polynomials suggests itself. This way you're also done with "X" and algebra w.r.t. complex arithmetic is reduced to arithmetic and reduction rules with list:

{-7, 2, 5, 0, 0, 8} becomes the pair (-7+(-1)*5+0, 2-(-1)*0+8), i.e. (-13, 10)

If you already believe in R, then complex arithmetic are just very simple functions on lists of reals (the polynomials)