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Thread: 4 = 5 'proof' ... what's wrong?

  1. #1 4 = 5 'proof' ... what's wrong? 
    Forum Ph.D. william's Avatar
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    Howdy everyone,
    Here is a cute little 'proof.'

    Theorem: 4 = 5
    'Proof':
    -20 = -20 (obviously...)
    16 - 36 = 25 - 45 (just two different ways to write -20)
    4^2 - 9*4 = 5^2 - 9*5 (factoring...)
    4^2 - 9*4 + 81/4 = 5^2 - 9*5 + 81/4 (add 81/4 to both sides)
    (4 - 9/2)^2 = (5 - 9/2)^2 ("completing the square")
    4 - 9/2 = 5 - 9/2 (get rid of the squares)
    4 = 5 (cancel the 9/2 from both sides)
    Obviously, this is bunk. So what is the mathematical mistake?

    Have fun,
    william


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  3. #2  
    Forum Ph.D. william's Avatar
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    No... I'm not replying to my own post....

    Just want to say that billco PM'd me the correct answer. He didn't want to spoil the fun so soon so he didn't post the solution.

    Thanks billco.

    And for others, if it is old news for you, or you have seen it elsewhere, maybe it would be nice to follow billco's lead and let someone else toil with it for a while.

    ...Or just post the solution and put it out of its misery....

    Cheers,
    william


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  4. #3 i got it... 
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    the mistake begins here....

    (4 - 9/2)^2 = (5 - 9/2)^2
    can't juz remove the squares to

    4 - 9/2 = 5 - 9/2
    coz

    4-9/2 = -1/2
    5-9/2 = 1/2
    (-1/2)^2 = (1/2)^2

    ..but if the squares r juz removed like that....then it's

    -1/2 = 1/2

    and it's wrong..obviously...

    the same goes for

    (4 - 9/2)^2 = (5 - 9/2)^2
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  5. #4  
    Forum Ph.D. william's Avatar
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    Bravo randytsx!

    Cheers,
    william

    edit:
    Actually,
    (4 - 9/2)^2 = (5 - 9/2)^2
    or
    (-1/2)^2 = (1/2)^2
    is true.

    Remember that sqrt(4) = +/- 2. The square-root of a number can be positive or negative.

    But yes... you can't just cancel the squares.

    cheers
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  6. #5 Certainly a good question. 
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    Fantastic question. Question like this will certainly help us to understand underlying concepts of square roots, division of zero etc which are not taught either in our course lectures or in any of the books. Thanks for posting.
    K. Srinivasa Ramanujam
    M.S (by Research) Scholar,
    http://ramanujamblog.blogspot.com
    India
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  7. #6 Illusion breaker 
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    rationally the formula (a-b)^2 = a^2 +b^2 -2ab is a conditional outcome of mor basic vector formulae which is
    (vec(a) + vec(b))^2 = a^2 +b^2 +2ab cos(x) where x is an angle between vec(a) and vec(b).
    and vectorially a-b not equal to b-a
    hence
    the positive quantitiy (4^2 - 9*4 + 81/4) is not equal to (4 - 9/2)^2 which is a square of -ve quantity as for example
    25 is not equal to (-5)^2
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    8 years old. New record?

    We could have almost waged the second Trojan war in the amount of time between posts.
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    Forum Professor Daecon's Avatar
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    Is there an internet forum equivalent of archaeology?
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    Is there an internet forum equivalent of archaeology?
    I hope so. I would love to study that.
    Has there ever been an instance of this where one of the original contributors of the conversation actually replied?
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  11. #10  
    Malignant Pimple shlunka's Avatar
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    Quote Originally Posted by Daecon View Post
    Is there an internet forum equivalent of archaeology?
    Using Internet Explorer as your browser?
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  12. #11  
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    Very good! Kinetic energy in physics works in much the same way; squaring velocity represents a change from a vector quantity to a scalar (there's no square rooting it, the information about direction really is lost).
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  13. #12  
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    Quote Originally Posted by shlunka View Post
    Quote Originally Posted by Daecon View Post
    Is there an internet forum equivalent of archaeology?
    Using Internet Explorer as your browser?
    We have to use it at work, our purchasing software/staff calendars etc. only seem to work in IE and the IT guys won't give me admin rights to install something better for other stuff. It's amazing, where I work we all have PhDs (or are working towarsd getting them) and are trusted with very, very expensive lab kit but we aren't trusted to administer the software and anti-virus stuff on the PCs in our offices. Madness. [/rant]
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    Quote Originally Posted by Coconut_Sundae View Post
    Very good! Kinetic energy in physics works in much the same way; squaring velocity represents a change from a vector quantity to a scalar (there's no square rooting it, the information about direction really is lost).
    could be done but you would have to accept more than one correct answer. Like "X = 5 and -5".
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  15. #14  
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    Quote Originally Posted by Robittybob1 View Post
    could be done but you would have to accept more than one correct answer. Like "X = 5 and -5".
    Well velocity points (classically) most generally in 3 dimensions. If the problem is confined to a line then there are but two possibilities for any given KE, but there's an infinite number of ways to decompose kinetic energy into components of a vector in 3D (or even 2D). They span the surface of a sphere (or circle, respectively) with the radius given by the square root of the kinetic energy.

    Integrals have a similar phenomena - without boundary conditions there are infinitely many solutions to any integral (just add a constant and the derivative will remain unchanged!), which is how we can include the specifics for a problem whose physics remain identical (two different shapes in a stream of air for example).
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  16. #15  
    Genius Idiot Rajnish Kaushik's Avatar
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    Quote Originally Posted by william View Post
    Howdy everyone,
    Here is a cute little 'proof.'

    Theorem: 4 = 5
    'Proof':
    -20 = -20 (obviously...)
    16 - 36 = 25 - 45 (just two different ways to write -20)
    4^2 - 9*4 = 5^2 - 9*5 (factoring...)
    4^2 - 9*4 + 81/4 = 5^2 - 9*5 + 81/4 (add 81/4 to both sides)
    (4 - 9/2)^2 = (5 - 9/2)^2 ("completing the square")
    4 - 9/2 = 5 - 9/2 (get rid of the squares)
    4 = 5 (cancel the 9/2 from both sides)
    Obviously, this is bunk. So what is the mathematical mistake?

    Have fun,
    william
    sir here (-9/2)^2=(9/2)^2
    but -9/2 is not equal to 9/2
    hope i made sense
    "Think like a man of action,act like a man of action"
    Henry Louis Bergson"
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    In what kind of math does 4 - 9/2 = 5 - 9/2 ? Illogical!!
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  18. #17  
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    rationally the formula (a-b)^2 = a^2 +b^2 -2ab is a conditional outcome of mor basic vector formulae which is
    (vec(a) + vec(b))^2 = a^2 +b^2 +2ab cos(x) where x is an angle between vec(a) and vec(b).
    and vectorially a-b not equal to b-a
    hence
    the positive quantitiy (4^2 - 9*4 + 81/4) is not equal to (4 - 9/2)^2 which is a square of -ve quantity as for example
    25 is not equal to (-5)^2
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  19. #18  
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    Quote Originally Posted by saurabh2b471vg View Post
    rationally the formula (a-b)^2 = a^2 +b^2 -2ab is a conditional outcome of mor basic vector formulae which is
    (vec(a) + vec(b))^2 = a^2 +b^2 +2ab cos(x) where x is an angle between vec(a) and vec(b).
    and vectorially a-b not equal to b-a
    hence
    the positive quantitiy (4^2 - 9*4 + 81/4) is not equal to (4 - 9/2)^2 which is a square of -ve quantity as for example
    25 is not equal to (-5)^2
    Well actually 25 is equal to -5^2.
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