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Proving -1 = 1 wrong

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  1. Mar 11, 2015 #1

    Rectifier

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    Gold Member

    Hey there!
    These is this falsidical paradox that I cant seem to prove wrong.

    ## -1 = (-1)^1 = (-1)^\frac{1}{1}= (-1)^\frac{2}{2} = (-1)^{\frac{2}{1} \cdot \frac{1}{2}} = (-1)^{2 \cdot \frac{1}{2}} = ((-1)^2)^{\frac{1}{2}} = (-1)^{2 \cdot \frac{1}{2}} = ((-1)^2)^{\frac{1}{2}} = (1)^{\frac{1}{2}} = 1 ##

    Any ideas?
     
  2. jcsd
  3. Mar 11, 2015 #2

    Gil

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    You have (1)1/2, and it can be +1 or -1. The mistake you do is that you sould consider ((-1)2)1/2=|-1| = 1
     
  4. Mar 11, 2015 #3
    No it can't. It's always 1.
     
  5. Mar 11, 2015 #4

    symbolipoint

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    Someone will say it eventually. [tex] i^2=-1[/tex]
    [tex](i^2)^(1/2)=+-(-1)^(1/2)[/tex], the text is there but it does not look right.
    [tex]√(i^2)=+-√(-1)=i[/tex]
     
  6. Mar 11, 2015 #5

    Gil

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    The square root of 1 is ±√1 = ±1, isn't it?
     
  7. Mar 11, 2015 #6
    1 has two square roots: -1 and 1. But ##1^{1/2} = 1##.
     
  8. Mar 11, 2015 #7
    A number raised to the power of 1/2 is the exact same thing as a square root. So [itex](-1)^{1/2} = \pm 1[/itex]. Information is lost upon squaring. That's where I'd say the issue comes from.

    Namely, the OP starts with a number [itex]x[/itex], and takes [tex]x = x^{(2)(1/2)} = (x^2)^{1/2}[/tex]

    but information is lost when we square, because [tex] x = a \implies x^2=a^2[/tex]

    but [tex]x^2 = a^2 \nRightarrow x = a[/tex]



    Though someone else may see farther than me regarding this problem.
     
  9. Mar 11, 2015 #8

    Mark44

    Staff: Mentor

    micromass is correct. ##\sqrt{1} = +1##.
     
  10. Mar 11, 2015 #9

    Stephen Tashi

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    It isn't in general true that [itex] x^{ab} = (x^a)^b [/itex].
     
    Last edited: Mar 11, 2015
  11. Mar 11, 2015 #10

    Mark44

    Staff: Mentor

    No. The sqaure root of -1 is the imaginary unit i.
    That's irrelvant to this question. We're taking the square root, not squaring something.
     
  12. Mar 11, 2015 #11

    Mark44

    Staff: Mentor

    There is a misconception about square roots that shows up here quite often. An expression such as ##\sqrt{4} = 2##, not ##\pm 2##. While it's true that 4 has two square roots, one positive and one negative, the symbol ##\sqrt{4}## represents the positive square root.

    More generally, for any positive real number a, the expression ##\sqrt{a}## represents the positive number b such that b2 = a.
     
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