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 $(\alpha ,v)<0$ if $\alpha\in R\backslash R^+$ . Roots which are not positive are called negative. Since $-\alpha$ is negative if and only if $\alpha$ is positive, exactly half the roots must be positive.