A math puzzle

Discussion in 'Miscellaneous' started by Tiberius, Nov 20, 2012.

  1. Tiberius

    Tiberius Commodore Commodore

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    I'm familiar with the erroneous proof that 1=2, but then I found this on Futility Closet and I can't find where the error is...

    [​IMG]
     
  2. Roger Wilco

    Roger Wilco Admiral Admiral

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    I literally haven't thought about such things since high school, but how exactly do you get from step 2 to step 3?
     
  3. Scroogourner

    Scroogourner Admiral Admiral

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    Admitting that I suck at math.....

    With the top being level 1, the transition from level 2 to level 3 is incorrect. that transition does not take into account 24 on the left and 40 on the right.


    ETA: Beaten to the answer while trying to come up with the right words.
     
  4. rhubarbodendron

    rhubarbodendron Rear Admiral Rear Admiral

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    the error is in line#1:
    9-24 = -15
    25-40 = 15

    While techically (meaning the way the equation is being treated with completing the square, radicating [correct word??] etc.) the procedure is correct, you start the "proof" under the false premise that -15 = +15
    This triggers a cascade of errors with the rest of the calculation, much like a domino-effect:
    In Line #2 you get 1 = 31
    In Lines #3 and #4 the equation reads as -1 = +1


    The important thing - both in mathematics and in life - is to never take anything for granted but always to check the supposed facts yourself, without blindly relying on others.
     
    Last edited: Nov 20, 2012
  5. Zulu Romeo

    Zulu Romeo World Famous Starship Captain Admiral

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    Actually, the first line is correct.

    Given the leap of adding 16 to each side in statement 2, the transition from level 2 to level 3 is also correct.

    I think the main mistake is in the 4th statement - both sides should be divided by the same common multiplier to keep things equal, but instead the left is divided by (3-4) and the right by (5-4).

    Taking the rest of the sequence beyond that error, you still end up with a difference of 2 instead of equality, whether you physically subtract 4 from each side or just let them cancel each other out.

    Do correct me if I am wrong - I've just woken up.
     
    Last edited: Nov 20, 2012
  6. rhubarbodendron

    rhubarbodendron Rear Admiral Rear Admiral

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    oops, how embarassing! You are right, of course, Zion! Thanks for pointing it out. My only excuse is that I got only 4 hours sleep last night, due to a somewhat loud party in the neighbourhood.
    At any rate you proved my point of never taking anything for granted and always checking the facts :D

    The trick with such equations is usually that you discreetly multiply with zero and then cancel down. In this case, however, it works differently:

    Line #1 reads as -15 = -15 which is correct
    Line #2 says 1 = 1 still correct
    Line #3 states -1² = +1² which is basically correct, as both is 1 but
    then at line #4 you radicate again which you shouldn't, because -1 is not +1.
    So line #4 is definitely wrong, and then the last line automatically gets wrong as well.
     
  7. Roger Wilco

    Roger Wilco Admiral Admiral

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    Can you explain? What did he do to get from #2 to #3? I'm stumped.
     
  8. Shaytan

    Shaytan Vice Admiral Admiral

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    #2 to #3 is what we call in French identité remarquable, must have a name in English too, it's supposed to be known by heart to use it to solve simple equations:

    (a+b)²= a²+2ab+b²
    (a-b)²=a²-2ab+b²
    (a-b)(a+b)=a²-b²
     
  9. Roger Wilco

    Roger Wilco Admiral Admiral

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    Ah, thank you, now I get it. Wow, that was a long time ago for me that I dealt with that. :D
     
  10. stj

    stj Rear Admiral Rear Admiral

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    It's kind of interesting to think about the best way to explain the error.

    In line 3, when the expanded terms are reduced to a binomial, neither 3-5 nor 5-4 are binomials in the usual sense of unknown variables that cannot be summed because they are unknown. The usual procedure is to consolidate the constants, which would have given us -1 squared and 1 squared in line 4. Which are indeed equal.

    But then, it would have been much more obvious that square roots are commonly limited to the positive roots even though they are both positive and negative. For example, the square roots of 4 are 2 and -2, but only 2 is commonly written down. But here it is arbitrarily written, in effect, that one side the root is only -1 while on the other it is only +1. It should have been +1 and -1 on both sides.

    PS Another way of putting it: It is obvious that the square root of a^2 is both plus and minus a. But if we write (a)^2, the parentheses appear to exclude the negative root. The moral here is that parentheses are conveniences, not genuine mathematical operations.
     
    Last edited: Nov 20, 2012
  11. Lumi

    Lumi Captain Captain

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    I love math. I'm terrible at it, but it's always fun trying to figure things out. More topics like this! ;)
     
  12. Ood Sigma

    Ood Sigma Commander Red Shirt

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    The real problem is in the transition from lines 3 to 4, in which it appears that they are taking the square root of both sides. You cannot do this, as there are two solutions to any square root (for example, the square root of 9 is both 3 and -3).

    The third line is correct:
    (3-4)^2 = (-1)^2 = 1
    (5-4)^2 = (1)^2 = 1

    But it is dependent on the fact that 1 has square roots of both 1 and -1.
     
  13. Asbo Zaprudder

    Asbo Zaprudder Rear Admiral Rear Admiral

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    Yeah, a square root has two values. The error arises by choosing the wrong sign leading to the difference of 2 on each side. Write line 3 as (3-4)(3-4) = (5-4)(5-4) and it becomes obvious that you can properly divide through by either (3-4) or (5-4), for example:

    (3-4) = (5-4)(5-4)/(3-4) = 1x1/-1 = -1
     
  14. rhubarbodendron

    rhubarbodendron Rear Admiral Rear Admiral

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    yep, that's what I pointed out in my last post, only less elaborate. It's quite difficult to explain mathematics in a foreign language. There are so many special terms you never learned at school.

    Totally off-topic, but I just have to ask: what kind of spider is the one in your avatar, Asbo Zaprudder? It's pretty!
     
  15. stj

    stj Rear Admiral Rear Admiral

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    A language note: The math term for taking a root is exponentiation. The exponent for taking a square root is 1/2.
     
  16. SmoothieX

    SmoothieX Vice Admiral Admiral

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    I think the problem is going from line 3 to line 4.

    Line 3 is saying that -1^2 and 1^2 are both equal to 1. Which is true.

    Line 4 is telling you 1 = -1. Someone conveniently forgot order of operations (PEMDAS) here.
     
    Last edited: Nov 20, 2012
  17. MacLeod

    MacLeod Admiral Admiral

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    BODMAS on this side of the pond.
     
  18. Lindley

    Lindley Moderator with a Soul Premium Member

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    They are assuming that x^2=y^2 implies x=y, which is false.
     
  19. SmoothieX

    SmoothieX Vice Admiral Admiral

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    I have the same math opinion as the two previous posters. 18 years later, I finally got something out of freshman HS algebra.
     
  20. Asbo Zaprudder

    Asbo Zaprudder Rear Admiral Rear Admiral

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    A jumping spider - family Salticidae - although I'm not sure which one of the approx 5000 species.