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Finding the error in a proof

  1. $a=b$
  2. $ab=a^2$
  3. $ab-b^2=a^2-b^2$
  4. $b(a-b)=(a+b)(a-b)$
  5. $b= a+b$

Reminder the first step where $b = 2b$ So,


In this case in my opinion the wrong step is that in the third, because you can´t subtract $b^2$ from both sides.

This question it was taken from a math exam.

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marked as duplicate by Douglas S. Stones, Martin Sleziak, Norbert, Belgi, Pedro Tamaroff Oct 21 '12 at 14:51

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

I wonder why the most common questions can't be researched on [a search engine] before being asked. –  Parth Kohli Oct 21 '12 at 14:24

3 Answers 3

You divided by $a-b$ in step 5), which is zero by assumption 1).

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$a=b \implies a-b=0$

from step 4 to 5 you are dividing through by $a-b=0$.

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Where did you see question like this Alex? –  Vinicius L. Beserra Oct 21 '12 at 14:08
@ViniciusL.Beserra a friend has asked me this before. –  Alex Oct 21 '12 at 14:17

The transition from step 4 to step 5 is wrong.$$\rm a = b\qquad\Rightarrow \qquad a - b = 0$$If $\rm \,a - b = 0\, $, then $\rm (a + b)(a - b) = b(a - b) $ can be written as $\rm 0(a + b) = 0b$

While what we do in the transition is, we divide both sides by $0$, hence breaking the so-called “fundamental rule”: never divide by zero!

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Thanks for you commentary. –  Vinicius L. Beserra Oct 21 '12 at 19:01

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