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Thread: Square Root = Two Answers

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    Square Root = Two Answers

    Taking the square root always yields a positive and negative answer. Why is this the case?

    Example:

    root {9} = -3 and 3.
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    Re: Square Root = Two Answers

    Quote Originally Posted by mathdad1965 View Post
    Taking the square root always yields a positive and negative answer. Why is this the case?

    Example:

    root {9} = -3 and 3.
    this is incorrect.

    The square root function always returns a positive value.

    While it's true $3^2 = 9$ and that $(-3)^2 = 9$

    $\sqrt{9} = 3,~\sqrt{9} \neq -3$
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    Re: Square Root = Two Answers

    The square root function only yields one answer \sqrt{x^2}=|x|.

    But there are two numbers that have a square equal to x^2. Namely x and -x. So the reverse operation to squaring has two possible results.

    In your particular case 3^2=(-3)^2=9.
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    Re: Square Root = Two Answers

    Quote Originally Posted by romsek View Post
    this is incorrect.

    The square root function always returns a positive value.
    positive should of course be "non-negative"

    $\sqrt{0} = 0$
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    Re: Square Root = Two Answers

    Thanks. I always thought that the square root of 9 = -3 and 3.
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    Re: Square Root = Two Answers

    Both -3 and 3 are square roots of 9. The operator/function $\sqrt{.}$ is defined to return the non-negative square root of its argument (or the principle value if complex)
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    Re: Square Root = Two Answers

    Quote Originally Posted by zzephod View Post
    Both -3 and 3 are square roots of 9. The operator/function $\sqrt{.}$ is defined to return the non-negative square root of its argument (or the principle value if complex)
    Thanks.
    Last edited by skeeter; Dec 27th 2016 at 10:17 AM. Reason: removed statement irrelevant to the thread topic.
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    Re: Square Root = Two Answers

    Notice the difference in wording. "Both -3 and 3 are square roots of 9." "The square root of 9 is 3."
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    Re: Square Root = Two Answers

    Quote Originally Posted by HallsofIvy View Post
    Notice the difference in wording. "Both -3 and 3 are square roots of 9." "The square root of 9 is 3."
    I see that -3 is not a square root of 9. We cannot take the square root of negative numbers.
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    Re: Square Root = Two Answers

    Quote Originally Posted by zzephod View Post
    Both -3 and 3 are square roots of 9. The operator/function $\sqrt{.}$ is defined to return the non-negative square root of its argument (or the principle value if complex)
    out of curiosity what do you think are the value(s) of $9^{1/2}$

    i.e. the concept of the square root without the $\sqrt{\text{ }}$ operator
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    Re: Square Root = Two Answers

    Quote Originally Posted by mathdad1965 View Post
    I see that -3 is not a square root of 9. We cannot take the square root of negative numbers.
    Then you are not understanding. You are taking the square root of 9, a positive number. You are NOT taking the square root of -3.
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    Re: Square Root = Two Answers

    Quote Originally Posted by HallsofIvy View Post
    Then you are not understanding. You are taking the square root of 9, a positive number. You are NOT taking the square root of -3.
    I get it. The sqrt {positive number} yields a positive answer.

    root {4} = 2

    root {16} = 4

    root {25} = 5 and so on.........
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    Re: Square Root = Two Answers

    Quote Originally Posted by romsek View Post
    out of curiosity what do you think are the value(s) of $9^{1/2}$

    i.e. the concept of the square root without the $\sqrt{\text{ }}$ operator
    It depends on the context. Without qualification I would take it to denote the (two) distinct values of $3e^{n\pi i};\ n\in \mathbb{Z}$, because that is the interpretation in the contexts that I normally work with.

    .
    Last edited by zzephod; Dec 27th 2016 at 10:04 PM.
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    Re: Square Root = Two Answers

    Quote Originally Posted by zzephod View Post
    It depends on the context. Without qualification I would take it to denote the (two) distinct values of $3e^{n\pi i};\ n\in \mathbb{Z}$, because that is the interpretation in the contexts that I normally work with.

    .
    There is no context to consider. You are being asked straight-up for value(s) of  \ \  9^{1/2}.

    Look at these steps worked out here at:

    www.quickmath.com

    Type: 9^(1/2)

    Then click "Simplify" and look at the steps.
    Last edited by greg1313; Dec 28th 2016 at 10:07 PM.
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    Re: Square Root = Two Answers

    Quote Originally Posted by greg1313 View Post
    There is no context to consider. You are being asked straight-up for value(s) of  \ \  9^{1/2}.

    Look at these steps worked out here at:

    www.quickmath.com

    Type: 9^(1/2)

    Then click "Simplify" and look at the steps.
    that doesn't tell the whole story. In the second step you can replace

    $(3^2)^{1/2}$

    with

    $((-3)^2)^{1/2}$

    and end up with $-3$ as an answer

    However Mathematica says that the only answer is $9^{1/2}=3$ even when expressed as $\left(9 e^{i 2 \pi}\right)^{1/2}$
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