One Equals Two? Really?

Mathematical proofs is the heart of mathematics. It is what separates mathematics from other fields. By stating logical deductions, a proof can demonstrate whether a statement is true or false including all possible cases without enumerating all possibilities.

1 = 2? Really?

People who have knowledge in mathematical proofs — not necessarily mathematicians — can read proofs written by others. They can tell whether the proof is valid or not.

Let us test your proof knowledge using the proof below. The proof is quite detailed to be able to accommodate non-mathematics majors. Don’t worry, the proof only requires knowledge of high school algebra.

Proof that 1 = 2

Let $a = b$. Multiplying both sides by $a$ gives us

$a^2 = ab$.

Adding $a^2$ to both sides results to

$2a^2 = a^2 + ab$.

Subtracting both sides by $2ab$,

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

After simplifying the right hand side, the equation is

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

Factoring out $2$ results to

$2(a^2-ab) = a^2 - ab$.

Dividing both sides by $a^2 - ab$,

$2 = 1$

which is what we want to prove.

Can you tell why the proof of one equals two is valid or not?

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Photo Credit: Eneas via Flickr