# Why is $\partial\alpha=\overline{\bar{\partial}\bar{\alpha}}$?

I'm reading the book Complex Algebraic Geometry and Hodge Theory, and the definitions of $\partial,\bar\partial$. Given a form $\alpha$ of type $(p,q)$, the differential $d\alpha$ splits into parts of type $(p+1,q)$ and $(p,q+1)$, which we define as $\partial\alpha$ and $\bar\partial\alpha$, respectively. The author then remarks that by definition we have the relation

$$\partial\alpha=\overline{\bar{\partial}\bar{\alpha}}.$$

I fail to see why this holds. For instance, if $f$ is a $(0,0)$-form, i.e. $f$ is a $C^{\infty}$ function $M\to\Bbb C$, then we can write

$$df=\sum_i\frac{\partial f}{\partial z_i}dz_i+\sum_i\frac{\partial f}{\partial\bar z_i}d\bar z_i,$$

and we have that $\partial f$ is the first summand, and $\bar\partial f$ is the second. But then, we have

$$\bar\partial\bar f=\sum_i\frac{\partial\bar f}{\partial\bar z_i}d\bar z_i.$$

However, I fail to see why taking the "complex conjugate" here gives us $\partial f$. I may be misinterpreting definitions, but it's not spelled out all that clearly in the book. Could somebody help me see what is going on here?
