Mathematical Fallacy: A Classic Proof That 1 Equals 0

Mathematical proofs are one of the most distinguishing beauty of mathematics. But the taken for granted beauty “on the other side” are the fallacies. These are construction of mathematical proofs that conceals some errors and inconsistencies and therefore presents an untrue statement to be true. One example of such is a classic proof that 1 equals 0. I’m sure, you will see more of this if you search the internet.

Below is the proof. Find the step/s that makes the proof wrong.

Theorem: 1 equals 0

Proof:

$a = b$

Multiplying both sides by $a$ gives

$a^2 = ab$

Subtract $b^2$ from both sides results to

$a^2 - b^2 = ab - b^2$.

Factoring both sides we have $(a + b)(a-b) = b(a-b)$.

Dividing both sides by $a - b$, results to

$a + b = b$.

Now, since $a = b$, the preceding equation is equivalent to

$b + b = b$ which is the same as $2b = b$.

Dividing both sides by $b$, we have $2 = 1$.*

For the final step, subtracting 1 from both sides, we have

$1 = 0$ and we are done with the proof.

Do you see the error in the proof?

Mathematical fallacies are often created in a clever way that it is not easy to spot the error in the proof. Some of such fallacies are created for fun and others for pedagogical reasons.

*Note that this also ‘proves’ that 2=1.