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Spivak 2=1 Proof fallacy

  1. Dec 16, 2009 #1
    Hi, I've gotten Spivak's calculus and I have a question on the second proof in the first chapter

    What is wrong with the following "proof"?

    suppose x=y

    1. x² = xy

    2. x² - y²= xy - y²

    3. (x + y)(x - y)=y(x - y)

    4. x + y=y

    5. 2y = y

    6. 2 = 1

    I just want to clarify that the error is in the transition from step three to step four as subtracting both sides by (x - y) is to subtract by zero as if x=y then x -y = 0.

    Step two is also saying that 0 = 0.
  2. jcsd
  3. Dec 16, 2009 #2


    Staff: Mentor

    No, that's not it at all. You have correctly identified the step that is incorrect, but not the reason. What they have done in going from step 3 to step 4 is to divide by x - y, not subtract x - y. There is never a problem subtracting the same amount from both sides of an equation, but you can run into problems by dividing both sides by a quantity that happens to be zero.
  4. Dec 16, 2009 #3
    Sorry I meant divide, misuse of language. Good to hear I got it, Spivak doesn't seem so difficult now :tongue:

    - foolish last words
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