Where is the error in the proof that 2 = 1?

[Byrgg]

On another forum, someone had this in a sig:

If a=1 and b=1: then a squared - 1 = ab-b: then (a + 1)(a - 1) = b(a - 1):then a + 1 = b: thus 1 + 1 = 1: as a result 2 = 1

I couldn't follow all of it, could someone explain this to me?

 

[neutrino]

In the step, (a+1)(a-1) = (b)(a-1), one "cancels" the a-1, which in this case in the same as cancelling zero on both sides. You can't cancel zero because, a.0 = b.0 doesn't necessarily mean that a = b.

 

[dav2008]

You couldn't follow the work or you don't understand why it works that way?

Here's the work:

a=b=1

a*a-1=a*b-b [since a=b=1]
(a+1)(a-1)=b(a-1) [factoring the right-hand side and factoring a b out of the left-hand side]
a+1=b dividing both sides by a-1[/color]
1+1=1
2=1

The problem is with the line I highlighted in red. You are dividing by a-1, but since a=1 you are actually dividing by zero, which is not defined.

Here's a simpler one to make it easier to see:

5*0=3*0
5=3 ["divide both sides by 0"] - obviously 5 does not equal 3.

 

[Byrgg]

Oh, ok so they're telling lies sorry I forgot about the division by 0, now I have to go tell them how wrong they are. Thanks.

 

[HallsofIvy]

There is a difference between a "lie" and a "joke".