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Math Help - is square root of 1 equal 1?

  1. #1
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    is square root of 1 equal 1?

    1) Is \sqrt{1} = 1 ?



    2) And if so, then 1 + \sqrt{x} = \sqrt{1} + \sqrt{x} = \sqrt{1 + x} ?

    3) If 1 and 2 is true, then:


    (1 + \sqrt{x})^2 = (\sqrt{1 + x})^2


    But if you actually square (1 + \sqrt{x})^2 and (\sqrt{1 + x})^2, you'll find out that they are different:

    (1 + \sqrt{x})^2 = (1 + \sqrt{x})(1 + \sqrt{x}) = 1 + 2\sqrt{x} + x

    (\sqrt{1 + x})^2 = 1 + x

    Why is that?
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  2. #2
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    \sqrt{a+b}=\sqrt a+\sqrt b is false, but it's true that \sqrt{a+b}\le\sqrt a+\sqrt b.
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  3. #3
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    That's intresting information. Maybe some links on the topic, where I can have some in-depth study of the situation (c) (all the rights to Jersey Shore) here? Youtube videos would be great, but some text explanation will do too.
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  4. #4
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    Quote Originally Posted by Aero763 View Post
    That's intresting information. Maybe some links on the topic, where I can have some in-depth study of the situation (c) (all the rights to Jersey Shore) here? Youtube videos would be great, but some text explanation will do too.
    Your original post is essentially a proof that \sqrt{a} + \sqrt{b} \ne \sqrt{a + b} for arbitrary a, b. In fact, from the last two lines you already have the inequality that Krizalid posted if you replace 1 with \sqrt{y}.
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  5. #5
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    Thank you guys.

    I got which is the problem step was.

    But what is the difference between \sqrt{1} + \sqrt{x} and \sqrt{1 + x}<br />
?

    I want to get some intuition: why it is different?

    Because on the first glance they not that different: you can say they look basically the same.

    So, let me get it straight:

    What i got from this discussion - is that two numbers under one root - is not two numbers, but one.

    Thank you, gentlemen.
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  6. #6
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    Quote Originally Posted by Aero763 View Post
    But what is the difference between \sqrt{1} + \sqrt{x} and \sqrt{1 + x}<br />
    I want to get some intuition: why it is different?
    Plug x = 4: \;\;\;\;\;\;\;\;\; 1+\sqrt{4} = 1+2 = 3 and \sqrt{1+4} = \sqrt{5}.
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  7. #7
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    Quote Originally Posted by Aero763 View Post
    2) And if so, then 1 + \sqrt{x} = \sqrt{1} + \sqrt{x} = \sqrt{1 + x} ?
    Is 1 + sqrt(4) = sqrt(5)?
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  8. #8
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by Aero763 View Post
    Because on the first glance they not that different: you can say they look basically the same.
    I don't understand this statement. I suppose \frac{1}{a}+\frac{1}{b} looks "basically the same" as \frac{1}{a+b}?
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  9. #9
    Newbie adhiluhur's Avatar
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    2) is wrong.
    therefore, the next statements are wrong.
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