Find the mistake a=b a*a=a*b a^(2)=a*b a^(2)-b^(2)=ab-b^(2) (a-b)*(a+b)=b(a-b) a+b=b 2b=b 2=1?

OK, here you go sweetheart :)


We assume a, b ∈ R and a, b ≠ 0


Also assume for arguments' sake that a = b :


a = b
a × a = a × b
a² = ab
a² - b² = ab - b²
(a - b)(a + b) = b (a - b)

Here is when things go terribly wrong!

Dividing through by '(a - b)' means we are losing other solutions which can also exist!

Also, it doesn't make sense to divide through by (a - b) especially if a = b.

Since (a - b) would equal 0 and division by 0 is undefined!

Hence (a - b) ≠ 0

(a + b) = b
2b = b
2b = 1

Again, divding through by 'b' means we are losing other solutions which can exist.

The correct way to proceed would be as follows :

2b = b
2b - b = 0
b (2 - 1) = 0
b = 0


But this would mean that a = b = 0 and that is NOT possible as we have mentioned (a - b) ≠ 0 for the original equation to be valid.


Hence we have a contradiction.



Hope this helps !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Consider the step :
(a - b) * (a + b) = b * (a - b)

As a - b = 0 for a = b, it is actually :
0 * (a + b) = b * 0
implies that 0 = 0

It is INCORRECT to divide the both sides by 0 to give the following step : (a + b) = b


Simply speaking, the mistake is similar to the following :
0 * 2 = 1 * 0
Hence, 2 = 1

Division by zero as with so many of these trick questions.