# What is a complex root?

Nov 1, 2015

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

#### Explanation:

$f \left(x\right) = {x}^{3} - 1$

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:

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

x_± = \frac{-1 ± i sqrt {3}}{2}

Therefore, three are complex roots of $f$.

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