Common Errors in Proofs

Dividing or multiplying by zero

Dividing both sides of an expression by a second expression that is equal to zero can cause problems. Generally, we cannot divide by zero as it can generate nonsense. For instance, we know $7 \times 0 = 5 \times 0$ but we cannot divide both sides by 0 to give $7 = 5$. This issue extends to examples that contain algebra. Here is a classic proof that commits this error [can you spot exactly where the error occurs?]:

Let $x$ and $y$ be non-zero numbers such that $x = y$.

1. Then we can write $x^2 = xy$
2. Subtract $y^2$ from both sides: $x^2 - y^2 = xy - y^2$
3. So $(x + y)(x - y) = y(x - y)$
4. Dividing by $(x - y)$: $x + y = y$
5. As $x = y$, we have: $2y = y$
6. Then dividing by the non-zero number $y$: $2 = 1$
7. Subtracting 1 from both sides: $1 = 0$

Therefore $1 = 0$.