Express $\dfrac{1 + i}{1 - i}$ in the form $a + bi$.

Multiply numerator and denominator by the conjugate $1 + i$: $\dfrac{(1 + i)^2}{(1 - i)(1 + i)} = \dfrac{1 + 2i + i^2}{1 - i^2} = \dfrac{2i}{2} = i$. Cancelling the $1$'s as if they were common factors wrongly gives 1 or $-1$.