Now define the complex numbers to be the set of all equivale

Now define the complex numbers to be the set of all equivalence classes. The operations of addition and multiplication are inherited from the corresponding operations on polynomials. (In the notation of abstract algebra, this definition says that the complex numbers are the quotient ring R[x]/(x^2 +1).) Exercise 12.10. Confirm that this definition makes sense and is compatible with the other definitions of the complex numbers. Which equivalence class corresponds to the complex number i?

Solution

R[x]/(x^2+1) = { [ax+b]}

it is making sense if we make a isomorphism

[ax+b] -> b+ia

now [ax+b].[cx+d] = [(ad+bc)x + acx^2 +bd] = [(ad+bc)x -ac + bd]

in complex numbers (b+ia)(d+ic) = (ad+bc)i +(bd-ac)

So definition is compatible

equivalence class [x] -> i

as [x].[x] = [x^2] = [-1]

and i.i = -1