An apparent contradiction in the simple Lie algebra $E_8$

Question

The following is the Dynkin diagram for simple Lie algebra $E_8$

My question is the following: It is clear that $e_i+e_j$ for $i \neq j$ is a positive root. Let $\alpha _8$ be the fundamental positive root $-\frac{1}{2}\sum_{i=1}^8e_i$. Then $\sum_{i=1}^{8}e_i+\alpha_8=-\alpha_8$ which shows that $-\alpha_8$ is a positive root. This is a contradiction. Where did I go wrong? Thanks for your answer.