Re:What are the complex numbers, really?
by monochromatic on Tuesday December 19, 2017 @07:14AM
(#15959590)
Doesn’t use the word closure but it does mention this.
> Complex numbers were introduced because they cropped up naturally in the solution of cubic equations. What might we need to introduce to solve higher order polynomial equations? What if we let the coefficients of our polynomials be complex numbers themselves?
> The astonishing answer is: nothing. Once you've allowed yourself complex numbers it turns out that you don't need anything else for higher order polynomials, or even polynomials with complex coefficients. The fundamental theorem of algebra tells us that any polynomial of degree n
with complex coefficients (and that includes real ones, since a real number is a complex number with an imaginary part of zero) has n complex roots.
> Complex numbers were introduced because they cropped up naturally in the solution of cubic equations. What might we need to introduce to solve higher order polynomial equations? What if we let the coefficients of our polynomials be complex numbers themselves?
> The astonishing answer is: nothing. Once you've allowed yourself complex numbers it turns out that you don't need anything else for higher order polynomials, or even polynomials with complex coefficients. The fundamental theorem of algebra tells us that any polynomial of degree n with complex coefficients (and that includes real ones, since a real number is a complex number with an imaginary part of zero) has n complex roots.
> That's a beautiful mathematical result.