Crook

What's wrong with proofs that say 1 + 1 = 1, or 2 = 1?
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There are lots of "proofs" that claim to prove something that is obviously not true, like 1 + 1 = 1 or 2 = 1. All of these "proofs" contain some error that most people aren't likely to notice. The most common trick is to divide an equation by zero, which is not allowed (in fact, you cannot ever divide by zero.) If a "proof" divides by zero, it can "prove" anything it wants to, including false statements.
It's important to recognize that while these "proofs" may be funny and cute, they always contain some error, and are therefore not real proofs.
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Here's one from the Dr. Math archives:
    • "Proof" that 1 + 1 = 1
      a = 1
      b = 1
      a = b
      a2 = b2
      a2 - b2 = 0
      (a-b)(a+b) = 0
      (a-b)(a+b)/(a-b) = 0/(a-b)
      1(a+b) = 0
      (a+b) = 0
      1 + 1 = 0
      2 = 0
      1 = 0
      1 + 1 = 1
      You'll want to read the detailed discussion in the archives. This false proof relies on dividing both sides of an equation by zero (disguised as a-b), whereas you can divide both sides of an equation by the same thing only as long as you are NOT dividing by zero.
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And another:
    • "Proof": 2 = 1
      a = b
      a2 = ab
      a2 - b2 = ab-b2
      (a-b)(a+b) = b(a-b)
      a+b = b
      b+b = b
      2b = b
      2 = 1
      What is wrong with this proof? Well, the fact that we got 2 = 1 is proof that the method of getting the solution was not mathematically sound. (Read a detailed discussion and find the fallacy in our archives.)
    • Here's a similar example from Eric Weisstein's World of Mathematics.
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Here's a proof that doesn't use division by zero:
    • "Proof": 2 = 1
      -2 = -2
      4 - 6 = 1 - 3
      4 - 6 + 9/4 = 1 - 3 + 9/4
      (2 - 3/2)2 = (1 - 3/2)2
      2 - 3/2 = 1 - 3/2
      2 = 1
      What's wrong with this? Taking square roots requires the use of the double plus-or-minus sign (or absolute...