What is a complex root?

1 Answer
Nov 1, 2015

Answer:

#{z in \mathbb{C} ; f(z) = 0} #

Explanation:

#f(x) = x^3 - 1#

The roots of #f# under a domain #A# are #{x in A ; f(x) = 0}#.

What are the real roots of #f#? #{x in \mathbb{R} ; x^3 = 1} = {1}#.

But there are two other complex roots:

#\frac{x^3 - 1}{x - 1} = x^2 + x + 1 = 0#

#x_± = \frac{-1 ± i sqrt {3}}{2}#

Therefore, three are complex roots of #f#.

#{x in \mathbb{C} ; x^3 = 1} = {1, x_+, x_-}#.