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Math Help - Algebra showing 2 = 1. Where's the error?

  1. #1
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    Algebra showing 2 = 1. Where's the error?

    Can anyone explain how this is possible?



    1. Let a and b be equal non-zero quantities

    a = b



    2. Multiply through by a (^ : is raised to)

    a^2 = ab



    3. Subtract b^2

    a^2 - b^2 = ab - b^2



    4. Factor both sides

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



    5. Divide out (a - b)

    a + b = b



    6. Observing that a = b

    b + b = b



    7. Combine like terms on the left

    2b = b



    8. Divide by the non-zero b

    2 = 1
    Last edited by mr fantastic; June 4th 2010 at 06:43 AM. Reason: Re-titled.
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  2. #2
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by chinchu View Post
    Can anyone explain how this is possible?



    1. Let a and b be equal non-zero quantities

    a = b



    2. Multiply through by a (^ : is raised to)

    a^2 = ab



    3. Subtract b^2

    a^2 - b^2 = ab - b^2



    4. Factor both sides

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



    5. Divide out (a - b)

    a + b = b



    6. Observing that a = b

    b + b = b



    7. Combine like terms on the left

    2b = b



    8. Divide by the non-zero b

    2 = 1
    Very old trick which would amuse idiots only.

    look at step 4.. If a=b, then (a-b) = 0, and in step 5, you are dividing by 0. What does your algebra class tell you about divding by 0?
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  3. #3
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    Quote Originally Posted by chinchu View Post
    Can anyone explain how this is possible?



    1. Let a and b be equal non-zero quantities

    a = b



    2. Multiply through by a (^ : is raised to)

    a^2 = ab



    3. Subtract b^2

    a^2 - b^2 = ab - b^2



    4. Factor both sides

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



    5. Divide out (a - b)

    a + b = b



    6. Observing that a = b

    b + b = b



    7. Combine like terms on the left

    2b = b



    8. Divide by the non-zero b

    2 = 1
    If a = b then a - b = 0.

    You can't divide by 0.
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  4. #4
    MHF Contributor undefined's Avatar
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    With division by 0 allowed, I can offer a much shorter proof.

    1 * 0 = 2 * 0

    1 = 2

    Q.E.D.
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    Quote Originally Posted by undefined View Post
    With division by 0 allowed, I can offer a much shorter proof.

    1 * 0 = 2 * 0

    1 = 2

    Q.E.D.
    Except it's NEVER allowed.
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    Thanks harish....

    Thanks harish.......

    Quote Originally Posted by harish21 View Post
    Very old trick which would amuse idiots only.

    look at step 4.. If a=b, then (a-b) = 0, and in step 5, you are dividing by 0. What does your algebra class tell you about divding by 0?
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  7. #7
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    Quote Originally Posted by chinchu View Post
    Thanks harish.......
    And what am I? Chopped liver?
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  8. #8
    Senior Member Mukilab's Avatar
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    Heh this brings back memories of 'you just divided by zero' posters, often featuring someone's head exploding behind a book or a giant whirlpool in the ground :P
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  9. #9
    MHF Contributor ebaines's Avatar
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    This reminds me of a math teacher I had who had dozens of "proofs" that 1 = -1, or 0 = 2, or some such. Whenever he made a mistake working a problem on the board he would try to show why his mistake wasn't really a mistake using another such "proof." Here are two - enjoy!

    1. 1 ^2 = -1 ^2
    2. Take the square root of both sides: sqrt (1^2) = sqrt (-1^2)
    3. since sqrt(a^2)= sqrt(a): 1 = -1
    QED

    And another:
    1. 1 = -1^2
    2. log(1) = log (-1^2) = 2 x log(-1), from log(a^n) = n log(a)
    3. log(1) = 0, so 0 = 2 x log(-1)
    4. Divide through by 2: 0 = log(-1)
    5. Since 0 = log(1): log(1) = log(-1)
    6. If log(a) = log(b) then a = b, so: 1 = -1. QED
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  10. #10
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by ebaines View Post
    This reminds me of a math teacher I had who had dozens of "proofs" that 1 = -1, or 0 = 2, or some such. Whenever he made a mistake working a problem on the board he would try to show why his mistake wasn't really a mistake using another such "proof." Here are two - enjoy!

    1. 1 ^2 = -1 ^2
    2. Take the square root of both sides: sqrt (1^2) = sqrt (-1^2)
    3. since sqrt(a^2)= sqrt(a): 1 = -1
    QED

    And another:
    1. 1 = -1^2
    2. log(1) = log (-1^2) = 2 x log(-1), from log(a^n) = n log(a)
    3. log(1) = 0, so 0 = 2 x log(-1)
    4. Divide through by 2: 0 = log(-1)
    5. Since 0 = log(1): log(1) = log(-1)
    6. If log(a) = log(b) then a = b, so: 1 = -1. QED

    Might as well use expressions like this then.

    \frac{\log (-(\sqrt{-1})^2-1)}{0}
    Last edited by undefined; June 4th 2010 at 01:26 PM.
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