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### Proof that 1 + 1 = 1?

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Date: 09/04/97 at 20:43:37
From: Ben Mayer
Subject: 1 + 1 = 1?

My programming teacher said that there was a proof that proved that
1 + 1 = 1. He also said in this proof that one of the steps was a
little bit "in the gray area." I was wondering if you could shed any
light on this subject.

Thank you,

Benjamin W. Mayer
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Date: 09/08/97 at 11:36:34
From: Doctor Guy
Subject: Re: 1 + 1 = 1?

Sure, I can do that. Here goes:

Let a = 1 and b = 1.

Therefore a = b, by substitution.

If two numbers are equal, then their squares are equal, too:

a^2 = b^2.

Now subtract b^2 from both sides (if an equation is true, then if
you subtract the same thing from both sides, the result is also
a true equation) so

a^2 - b^2 = 0.

Now the lefthand side of the equation is a form known as "the
difference of two squares" and can be factored into (a-b)*(a+b).
If you don't believe me, then try multiplying it out carefully,
and you will see that it's correct. So:

(a-b)*(a+b) = 0.

Now if you have an equation, you can divide both sides by the same
thing, right? Let's divide by (a-b), so we get:

(a-b)*(a+b) / (a-b) = 0/(a-b).

On the lefthand side, the (a-b)/(a-b) simplifies to 1, right?
and the righthand side simplifies to 0, right?  So we get:

1*(a+b) = 0,

and since 1* anything = that same anything, then we have:

(a+b) = 0.

But a = 1 and b = 1, so:

1 + 1 = 0, or 2 = 0.

Now let's divide both sides by 2, and we get:

1 = 0.

Then we add 1 to both sides, and we get what your programming
teacher said, namely:

1 + 1 = 1.

In fact, you can prove that 47 = -3 or anything else you want.
But of course you know that is wrong.

Do you know what I did that was not correct?

Shall I tell you? If you want to work it out for yourself before
viewing my answer, I will space down a few lines so you can hide my
response and work it out for yourself.

hmmm...

not yet...

Okay, here's the bad thing I did. You can divide both sides of an
equation by the same thing ONLY AS LONG AS YOU ARE NOT DIVIDING BY
ZERO. In fact, you cannot ever divide by zero. When I divided by
(a-b), that means a somewhat disguised form of 0, since a = b = 1.
That's where I went wrong. Did you figure that out by yourself, or did
you need the hint?

-Doctor Guy,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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