If you know only heights of $a$ and $b$, you may estimate heights of $a+b$, $a/b$ and $ab$. Assuming that $h$ is an absolute (Weil) height: $$h(ab)\leq h(a)+h(b)$$ $$h(a/b)\leq h(a)+h(b)$$ $$h(a+b)\leq\log 2 +h(a)+h(b)$$ This bounds are sharp. You may find this, for example, in M. Waldschmidt "Diophantine approximation on linear algebraic groups", Chapter 3.
Oleg Eroshkin
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