A binary operation * is defined on the set of real numbers by $a * b = \frac{a+b}{1+ab}$, where $ab \neq -1$. If the operation is commutative, find the identity element $e$ such that $a * e = a$.

For the identity element e, we need a * e = a, so (a + e)/(1 + ae) = a. This gives a + e = a(1 + ae) = a + a²e, so e = a²e, meaning e(1 − a²) = 0 for all a, which holds when e = 0. Verifying: a * 0 = (a + 0)/(1 + 0) = a. ✓ The identity element is 0.