# Where does this problem fail?

• Nov 9th 2009, 06:26 PM
audax
Where does this problem fail?
So, someone was showing me a math problem where he said he could get 1 = 2. Here's what he did:

\$\displaystyle x = 1\$

Thus
\$\displaystyle x^2 = x^2\$

So
\$\displaystyle x^2 - x = x^2 - 1\$
Right?

Then, he factored:
\$\displaystyle x(x -1) = (x - 1)(x + 1)\$

Divided by (x - 1):
\$\displaystyle [x(x-1)]/(x-1) = [(x-1)(x + 1)]/(x - 1)\$
to cancel out (x - 1)

Resulted in:
\$\displaystyle x = x + 1\$
From where he plugged in x = 1:
\$\displaystyle 1 = 1 + 1\$
Therefore:
\$\displaystyle 1 = 2\$

Please explain to me why this fails, because it seems like it does in so many places, but I want to know why exactly! For instance, dividing by \$\displaystyle (x - 1)\$ would be the same as dividing by zero, which would fail, but he thinks he's pretty much found where math has a weakness. I need expert verification!
• Nov 9th 2009, 06:40 PM
Bacterius
There is a division per zero while simplifying, which invalidates all the following, sorry. You cannot divide per zero, because it would mean that \$\displaystyle 2 = 1\$, crumbling the whole theory of mathematics. Look at this :

\$\displaystyle a = b\$

\$\displaystyle a^2 = ab\$ (multiply by \$\displaystyle a\$)

\$\displaystyle 2a^2 - 2ab = a^2 - ab\$ (add \$\displaystyle a^2 - 2ab\$)

\$\displaystyle 2(a^2 - ab) = a^2 - ab\$ (factorize on left side)

\$\displaystyle 2 = 1\$ (simplify by \$\displaystyle a^2 - ab\$)

Which is impossible, because \$\displaystyle a^2 - ab = 0\$ (division per zero).
• Nov 9th 2009, 06:46 PM
audax
Ray, you are awesome! Thanks!

Quick edit: When you say (add \$\displaystyle a^2 - 2ab\$), do you mean \$\displaystyle 2a^2 - 2ab\$?
• Nov 9th 2009, 07:07 PM
Bacterius
Quote:

Originally Posted by audax
Ray, you are awesome! Thanks!

Quick edit: When you say (add \$\displaystyle a^2 - 2ab\$), do you mean \$\displaystyle 2a^2 - 2ab\$?

No, I mean \$\displaystyle a^2 - 2ab\$ (Happy)
• Nov 10th 2009, 06:46 AM
picozzi
can't divide by 0
note that x-1=1-1=0, so you can't divide by 0, as you have done.