Why if $a =b$ then $a = 0$ is not a correct statement

Question

Bogus Claim: If $a$ and $b$ are two equal real numbers, then $a = 0$

$a = b$

$a^2 = ab$

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

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

$a + b = b$

$a = 0$

I found this in my proof handouts, and correct me if I'm wrong,but is it wrong because after line 4 we divide both sides by $(a - b)$ which would be $0$ if $a = b$ ?

This is essentially the same as this question. The conclusion is different ($1=2$ instead of $a=0$) but the argument and flaw are the same. — Noah Schweber, Dec 19 '16 at 19:35
Question :What wrong most of the people do in such fake proofs?? Answer :They divide 0 by 0. — Vidyanshu Mishra, Dec 19 '16 at 19:43
@THELONEWOLF.: Do you actually have a proof of that??? I would tend to guess that even if you do somehow, it is most certainly faked (and most certainly because you've based it on insufficient statistics rather than dividing $0$ by $0$). — barak manos, Dec 19 '16 at 19:46
@barakmanos, honestly speaking, what you are saying is going over my head. Do you want me to prove that in mathematics, dividing 0 by 0 is illegal ?? — Vidyanshu Mishra, Dec 19 '16 at 19:49
@THELONEWOLF.: No, I asked if you have a proof for the fact that you stated. — barak manos, Dec 19 '16 at 19:51
@THELONEWOLF.: Do you actually have a proof of your claim that most of the people who write such faked proofs actually divide $0$ by $0$? — barak manos, Dec 19 '16 at 20:16
To close this discussion , I would say I have no proof. :) :) :) — Vidyanshu Mishra, Dec 19 '16 at 20:26

score 1 · Answer 1 · answered Dec 19 '16 at 19:32

1

You are dividing the 4th line by $a-b$, which is zero by your assumption

answered Dec 19 '16 at 19:32

Canardini

4,147

Thank you,that's what i wanted to know. – August Dec 19 '16 at 19:35

score 1 · Accepted Answer · answered Dec 19 '16 at 19:41

1

You take the equation $(a-b)(a+b)=(a-b)b$
You divide each side by $(a-b)$
You get the equation $a + b = b$

But $a=b\implies(a-b)=0$, which means that step #2 is illegal.

answered Dec 19 '16 at 19:41

barak manos

43,109

Why if $a =b$ then $a = 0$ is not a correct statement

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