positive root

If $R\subset E$ is a root system, with $E$ an Euclidean vector space, then a subset$R^{+}\subset R$ is called a set of positive roots if there is a vector $v\in E$ such that$(\alpha ,v)>0$ if $\alpha \in R^{+}$, and if $\alpha \in R\backslash \backslash R^{+}$.http://planetmath.org/node/3645Roots which are not positive are called negative. Since $-\alpha$ is negative if and only if$\alpha$ is positive, exactly half the must be positive.