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Thread: Question about square roots!

  1. #1 Question about square roots! 
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    Is it mathematically correct to say √16=4, or would it be correct to say that √16=4, unless specified √16? This isn't for any specific problem or anything, but I'm just curious if is correct to have , because I know every number has two square roots.
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    Quote Originally Posted by muon321 View Post
    Is it mathematically correct to say √16=4, or would it be correct to say that √16=4, unless specified √16? This isn't for any specific problem or anything, but I'm just curious if is correct to have , because I know every number has two square roots.
    Yes. Every real number number (except zero) has two unique square roots. However, (almost always) denotes only the principal square root, which is the positive one. So it's somewhat of a misuse of notation to state . The correct use would be and also .

    Nothing will change the fact that 16, or any other real number, has two square roots, but we use the principal value instead of both where it is applicable. One will often see it in basic geometry, since negative values for lengths, area, etc. are meaningless (though you might use in the process of solving certain problems). On the other hand, one will probably use more in other areas, especially when encountering some equation in the form .

    1) Given a square with an area of 16 square meters, find its side length.

    2) Solve for roots: .

    In the first, one would use the principal value. In the second, one would typically use the square root, since both values are certainly roots of the polynomial.
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  3. #3  
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    Quote Originally Posted by muon321 View Post
    Is it mathematically correct to say √16=4, or would it be correct to say that √16=4, unless specified √16? This isn't for any specific problem or anything, but I'm just curious if is correct to have , because I know every number has two square roots.
    by convention. It is a matter of notation that the square root of a real number is taken to be the non-negative square root. This is mathematically unimportant. What is important is that you recognize that there are two numbers the square of which is 16.

    As you note any complex number, except 0 has two distinct square roots. When one is dealing with other than real numbers there is notational convention that selects between the two.
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