What step is wrong?

Question

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Finding the error in a proof

$a=b$
$ab=a^2$
$ab-b^2=a^2-b^2$
$b(a-b)=(a+b)(a-b)$
$b= a+b$

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

$1=2$

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.

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

score 5 · Answer 1 · answered Oct 21 '12 at 13:24

5

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

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

answered Oct 21 '12 at 13:24

Alex

2,354

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

score 4 · Answer 2 · answered Oct 21 '12 at 13:24

4

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

answered Oct 21 '12 at 13:24

fbg

407

score 4 · Answer 3 · answered Oct 21 '12 at 14:13

4

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!

answered Oct 21 '12 at 14:13

P.K.

7,742

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

What step is wrong?

3 Answers3