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Explain the difference between these square roots

  1. Feb 18, 2009 #1
    Hey guys, I was just wondering what the difference between these two statements are:

    V¯(x) = ± 4

    V¯(x) = - 4 ---> does not exist.

    This is the quote from my text, "...we remind you of a very important agreement in mathematics. The square root sign V¯ always means take the positive square root of whatever is under it. For instance, V¯(4) = 2, it is not equal to -2, only 2. Keep this in mind in this section, and always. "

    Maybe I've been staring at the pages too long, but how is (4) different from (-2)²? And why can we right ± 2, but not -2? I know this is basic but I'm embarrassingly confused about this.
     
  2. jcsd
  3. Feb 18, 2009 #2
    [itex]\sqrt{x}[/itex] means the positive square root of x (this way you can refer to [itex] \sqrt{x}[/itex] and only be talking about a single number, not two numbers). If the author says [itex]\sqrt{16}=\pm 4[/itex], he is just making the point that both [itex]4^2[/itex] and [itex](-4)^2[/itex] equal 16, however the correct notation is[itex]\pm \sqrt{16}=\pm 4[/itex].
     
  4. Feb 18, 2009 #3

    HallsofIvy

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    As qntty said, the first, [tex]\sqrt{4}= \pm 2[/itex] is simply wrong. [tex]\sqrt{4}= 2[/tex] because [tex]\sqrt{x}[/tex] is defined as the positive number y such that [tex]y^2= x[/tex]. That is why we must write the solution to [tex]x^2= a[/tex] as [itex]\pm\sqrt{a}[/itex]- because [itex]\sqrt{a}[/itex] does not include "[tex]\pm[/tex]".
     
    Last edited: Feb 20, 2009
  5. Feb 20, 2009 #4
    And also because that we wish that square-root should be a "function", and for being a function it has to be defined like that only. By definition, a function takes a value from a set A and maps it into B, and no two numbers in A can map to the same number in B.
     
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