>> Alright,
I'm going to use algebra
to show that two equals win.
Now, we know that's not really
true, so see
if you can find my mistake.
Alright, we're going
to start off with just saying
that A is equal to B.
So let's say we have two
variables, A and B,
and that they're equal
to each other.
The first thing I'm going
to do is multiply both sides
by A. So for instance I'm
going to do A on the left side
and I'm going to multiply by A
on the right side,
and that's going
to give me A squared
equals AB.
And now I'm going
to add A squared
to both sides.
So, I have A squared--
let's see,
I'm going to add A squared
to both sides.
So I'm going to add A squared
to both sides.
And I'll just combine
like terms over here
on the left hand side.
A squared plus A squared
equals 2A squared.
So I have 2A squared equals
AB, plus A squared.
Next up I'm going
to subtract 2AB
from both sides.
So I have 2A squared minus 2AB
on the left,
and on the right hand side I
have AB plus A squared,
and I'm also going
to subtract 2AB
on the right hand side,
and then I'm just going
to simplify the right hand
side by combining like terms
over here.
So we have 2A squared minus
2AB on the left,
on the right hand side I have
A squared and then I have 1AB
minus 2AB, that's just going
to be minus AB.
Now I'm going
to factor both sides
in a certain way.
So on the left hand side I'm
going to take
out the common factor of 2,
so that gives you 2 times A
squared minus AB.
And the right side I'm going
to just write
that as 1 times A squared
minus AB.
So I did not take
out the greatest
common factor.
But notice I have a factor
of A squared minus AB
on both sides of the equation.
So I'm just going
to divide both sides
by that factor.
So if I divide both sides
by A squared minus AB--
Then I can cancel this.
So I can cancel this,
and over here I can cancel
this, and so I have, oh,
I have 2 equals 1.
So what do you think
about that?
See if you can find
my mistake.
I encourage you to try
to figure that out
on your own, but if not,
I'm going to continue
and show you the mistake.
So if you don't want to know,
turn off the video now.
Okay, over on the left is the,
quote, proof that 2 equals 1.
And let's see what went wrong.
Well, if we look
at the very beginning,
we have that A equals B,
right?
That means
that A minus B is equal
to zero.
If we multiplied both sides
by A, you would have A times A
minus B equals A times zero,
which is the same thing
as A squared minus AB is equal
to zero.
So, here comes the problem,
look right here.
I divided by--
A squared minus AB.
Well, that means I divided
by zero, and that is
the problem.
That's an illegal thing to do.
Alright, so the problem is
right here
when I divided by zero.
You're going
to get a false statement.
So remember,
you can never divide by zero,
that is undefined.
So on the very last step here
when I divided by zero,
I get a false statement.
So, therefore,
two does not really equal one.