What is a complex root?

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Answer 1 · 2015-11-01T09:26:11

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

#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_-}#.