One Plus One isn't Two

Date: 10 Jan 1995 11:18:54 -0500
From: Stephanie Huang
Subject: 1+1=?

Hiya Dr. Math,

Hello, I'm a Senior at Monta Vista HS in Cupertino, California. In our 
Internet class we are doing a project where we ask you a question and 
hopefully receive an answer. I was once shown that 1+1 isn't 2 and I 
don't remember how it was done. Could you please e-mail me with an 
answer? Thanks for your time!

Stephanie 

Date: 16 Jan 1995 11:53:47 -0500
From: Dr. Ethan
Subject: 1+1=?

Well the first thing to say is that this cannot be done. You cannot use
CORRECT mathematics to prove something that is untrue. Secondly, yes I 
have constructed a proof. My challenge to you is to find the flaw. (I 
have seen more sophisticated proofs where the flaw is harder to find, 
but each one essentially has the same flaw that mine has.)

Given a = b then this implies

     a - b + b  =    b

Now divide both sides by (a-b) and we have

     a - b + b       b
     ---------  =  -----
       (a-b)       (a-b)

Then reduce the left side to be
              
           b         b
     1 + -----  =  -----
         (a-b)     (a-b) 

Then subtract b/(a-b) from both sides and you have

             1  =  0      

Then add one to both sides and you have

          1 + 1 = 1

Wow, pretty neat huh?

Remember. This is a flawed proof and no correct proof exists. Untrue 
things cannot be proved through correct mathematics.

Ethan, Doctor On Call

Date: 9 Sep 2000 01:31:38 -0400
From: Graeme McRae
Subject: 1+1=?

I read your joke proof that 1 = 0, and would like to share a couple 
I've found over the years. I collected them on my web page,

   http://mcraeclan.com/graeme/math/MathJokes.htm   

The first one uses the square root incorrectly, neglecting that there 
are two square roots of every real number. The second one is -- if I do 
say so myself -- even simpler than yours, yet harder to spot the flaw. 
Here it is, I love it -- the flaw doesn't happen until the last step:

               let a  =  b

Multiply both sides by a

                 a^2  =  ab

Add (a^2 - 2ab) to both sides

     a^2 + a^2 - 2ab  =  ab + a^2 - 2ab

Factor the left, and collect like terms on the right

         2(a^2 - ab)  =  a^2 - ab

Divide both sides by (a^2 - ab)

                   2  =  1

A corollary -- and what proof is complete without a corollary -- is 
this: a proof by contradiction that a can never equal b! How funny is 
that?

The third "proof" on my web page is an alleged proof that there are no 
uninteresting numbers. It uses the old technique of making a set of 
numbers, and using the Well Ordering Principle to pick out the lowest 
one, and then reaching a contradiction about it.

Please feel free to copy or link to this web page, and above all, 
ENJOY!