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Thread: Square root question/help

  1. #1 Square root question/help 
    Forum Freshman
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    Ok, so I hope I don't sound completely retarded here.
    My question is: Does √(-1)√(-1) = (√(-1))^(2) or √(1)

    I am wondering because : √(x)√(y) = √(xy) doesn't it? √(25)√(30)=√(900)
    So if x=-1 and y=-1 then shouldn't it just be √[(-1)(-1)] ?

    Or does order of operations take precedence and I just don't understand the √(x)√(y)

    I hope that makes sense!
    Any help is appreciated!


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  3. #2  
    Forum Freshman +SCIENCEgirl+'s Avatar
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    you are right about √(x)√(y) = √(xy).

    but the square root of a negative number is not defined, so you actually can't say that √(-1)√(-1) = √(-1*-1) = √(1) = 1.

    to do calculations with the square root of a negative number we introduced a symbol i , where i^2 = −1 (or i =√(-1)) .


    the answer to √(-1)√(-1):

    √(-1)√(-1) = (√(-1))^2 = -1 [because (√(a))^2 = a]
    or:
    √(-1)√(-1) = i * i = i^2 = -1

    so you can only use √(x)√(y) = √(xy) when x and y are positive numbers, because when they're not, the square root is not well defined.


    + Don't walk where the path may lead, walk instead where there is no path, and leave a trail...+
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  4. #3  
    Forum Professor serpicojr's Avatar
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    +SCIENCEgirl+ gave an excellent explanation, but let me just add a little bit about the subtle difference between the meaning of "square root" in the contexts of positive real numbers and of numbers in general.

    In one sense, a square root of a number x is any number y which satisfies y<sup>2</sup> = x. Any nonzero complex number has two square roots in this sense--if y is a square root, then so is -y. In general, there's no good way of choosing one over the other as being "the" square root--for example, is the square root of -2i equal to -1+i or 1-i?

    However, positive real numbers always have a positive square root, so we can define the square root function of positive reals to be the positive root. This square root function then satisfies the nice "multiplicative" property you guys describe above--the product of the square roots is the square root of the product. This property simply does not extend to the previous notion of square roots--the product of two square roots is not necessarily a square root of the product, as you have shown when taking the square root of -1 to be i.

    Things become confused because people write i = √(-1). You just have to remember that this is not defining the function square root on negative numbers (i.e., we're not extending the second sense of square root to negative numbers). Rather, we're saying that i is a complex number which is a square root in the first sense: i is a complex number satisfying i<sup>2</sup> = -1. Just because the same notation is used doesn't mean that the properties of the square root function translate to this expression.
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  5. #4  
    Forum Freshman
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    Oh, alright I see what you mean. Thanks for the responses!
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