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Assume that $a=1$ and $b=1$. Then we trivially have

$a=b$

Multiply both sides with $a$

$a^2=ab$

Substract $b^2$ from each side

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

Rewrite by factorizing

$(a+b)(a-b)=b(a-b)$

Cancelling $(a-b)$ from both sides gives

$a+b=b$

Since $a=1$ and $b=1$ we get

$2=1$.

Q.E.D. ?

When dividing by (a-b) you just divided by zero. At that point, all it lost.

@Ansolem

That was obvious of course. Nonetheless, these fallacies remain pretty interesting. I’ll post another one soon where I’ll ‘proof’ that 1 is equal to -1 where, just as in this proof, some basic rules can be overseen.

The line after your sentence is false sine a-b=o. We simplify only when a-b\neq 0.

Ciao