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What's wrong with proofs that say 1 + 1 = 1, or 2 = 1?

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.

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.

And another:

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 values). In this case, the plus sign gives an extraneous result, and the minus sign is the one that gives the right conclusion.

Finally, here are extensive discussions of a number of false proofs:

Classic Fallacies
   - Philip Spencer, for The University of Toronto Mathematics Network

Of course, these aren't really proofs, because they all have some error in them. What's important about these examples is that they show ways you can make a mistake in using math if you aren't careful enough. When you can understand where the error is, then you can look for the same kinds of errors in your own work, whether it's a proof for school or a calculation you make when you're designing a bridge. (It also explains why mathematicians and scientists don't publish their results without first having others check them to make sure there isn't some subtle error in their calculations.)

      [From the Dr. Math archives: 1 + 1 Doesn't Equal 2.]

More from the Dr. Math archives:

And see: "Plausibility Arguments," a thread from the Math Forum's math-teach discussion group archive.

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