Proof that 2=1


1) $a = b$. Given.

2) $a^2 = ab$. Multiply both sides by a.

3) $a^2-b^2 = ab-b^2$. Subtract $b^2$ from both sides.

4) $(a+b)(a-b) = b(a-b)$. Factor both sides.

5) $(a+b) = b$. Divide both sides by $(a-b)$

6) $a+a = a$. Substitute $a$ for $b$.

7) $2a = a$. Addition.

8) $2 = 1$. Divide both sides by $a$.


Usually, if a proof proves a statement that is clearly false, the proof has probably divided by zero in some way.

In this case, the quantity of $a-b$ is $0$ as $a = b$, since one cannot divide by zero, the proof is incorrect from that point on.

Thus, this proof is false.


If this proof were somehow true all of mathematics would collapse. Simple arithmetic would yield infinite answers. This is why one cannot divide by zero.

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