I hate stuff like this but not for the reason that most people think: http://www.youtube.com/watch?v=ICv_0ln_yzw I hate it because people fail at critical thinking. You give some one a demonstration like this which is called a proof and they start picking apart the intermediate steps used to solve the equation. The argument is that the proof is invalid because the intermediate steps are invalid, but they are completely missing the big picture. Here is an example to illustrate my point: Proof that 7 = 5 7 = 1 + (6/2) 7 = 1 + 4 7 = 5 This is proof that 7 = 5. People look at this and say, "wait a minute... 6/2 is 3 not 4. You made a mistake therefore 7 does not equal 5." SEE WHAT I DID THAR This is such epic fail yet so many people do it. Of course my example made no attempt to disguise the glaring gap in my so-called "proof" which is of course that I proved nothing. People use this kind of shallow thinking everywhere in life and it makes me sad.

Yep. He started with this: 1 + 1 = 1 + sqrt(1) This is not equal to begin with, so it's no wonder he was able to equate two different numbers. He only rationalized one side with some bad math. He didn't prove the two sides are equal.

Every positive number has two roots. The square root of 1 is both positive 1 and negative 1. Some people would argue that square root notation dictates the use of a negative sign outside of the square root if you intend to use the negative root instead of the principal root: http://en.wikipedia.org/wiki/Square_root But even this is irrelevant. He still didn't prove anything. Most proofs of this kind start with an identity. See this wiki for examples: http://en.wikipedia.org/wiki/Invalid_proof But the so-called "proof" in this Youtube video doesn't start with anything but an arbitrary and unproven assertion that 1 + 1 = 1 + sqrt(1). You can't say something is proven just because it seems obvious, and that is effectively the claim that is implied by his starting point. If you start with something that is proven (like an identity) then you are clear to start manipulating the equation to draw other conclusions. But this Youtube video has no such starting point and the process he follows proves nothing. http://en.wikipedia.org/wiki/Mathematical_proof

Here is another angle... If I derive a contradiction from a proposition that I claim to be true, then the proposition must necessarily be false. What he did was basically an informal proof by contradiction where he proved that he was wrong. Clearly 1 + 1 = 0 is a contradiction, therefore the original proposition must be false. But he framed it as a direct proof and fooled everyone. The Youtube comments are all trying to fix the intermediate steps which clearly means they have all accepted the original proposition as being true despite the fact that the author of the video demonstrated a contradiction. I guess logic and deduction don't come naturally to most people. When I watch that video I immediately think, "ok, your proposition was never proven and you have demonstrated it to be false. What is your point?" Then I see that he claims the contradiction to be a proven statement and I'm like wtf?! Then I see that everyone else is going along with that claim and I weep bitter tears.

I was wrong. After doing some research I found that square root notation actually requires a negative sign in front of the square root if you wish to specify the negative root, so that is not ambiguous as I originally thought. I no longer have any complaints about the original equation. The problem is indeed in the intermediate steps. Specifically, you need to be careful splitting square roots of negative factors. I am mad at myself. I invested my ego in this debate and now it is betrayed.