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Math Help - Surds ii

  1. #1
    Member GAdams's Avatar
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    Surds ii

    I got (i) simplified to 12 + sq.rt35 + sq.rt35

    (ii) How would you know if it is irrational without a calculator?
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  2. #2
    MHF Contributor red_dog's Avatar
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    (\sqrt{7}+\sqrt{5})^2=7+5+2\sqrt{7}\cdot\sqrt{5}=1  2+2\sqrt{35}.
    Suppose that 12+2\sqrt{35}\in\mathbf{Q}.
    Let q\in\mathbf{Q} such as 12+2\sqrt{35}=q\Rightarrow \sqrt{35}=\frac{q-12}{2}\in\mathbf{Q}\Rightarrow \sqrt{35}\in\mathbf{Q}, false.
    So 12+2\sqrt{35}\not\in\mathbf{Q}.
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  3. #3
    Member GAdams's Avatar
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    Quote Originally Posted by red_dog View Post
    (\sqrt{7}+\sqrt{5})^2=7+5+2\sqrt{7}\cdot\sqrt{5}=1  2+2\sqrt{35}.
    Suppose that 12+2\sqrt{35}\in\mathbf{Q}.
    Let q\in\mathbf{Q} such as 12+2\sqrt{35}=q\Rightarrow \sqrt{35}=\frac{q-12}{2}\in\mathbf{Q}\Rightarrow \sqrt{35}\in\mathbf{Q}, false.
    So 12+2\sqrt{35}\not\in\mathbf{Q}.

    I don't understand this. What do the signs like the Q mean? And that fancy E means same as right?
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  4. #4
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    Quote Originally Posted by GAdams View Post
    I don't understand this. What do the signs like the Q mean? And that fancy E means same as right?
    The "Q" means "rational" as in a fraction.

    And \in means "in", it means the number is "in" the rationals, i.e. the number is a fraction.
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  5. #5
    Member GAdams's Avatar
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    I'm sorry, I still don't understand how to work out if it is irrational or not without a calculator.
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  6. #6
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by GAdams View Post
    I'm sorry, I still don't understand how to work out if it is irrational or not without a calculator.
    If the number under the square root sign is a perfect square, then it is rational. If it is not a perfect square then it is irrational. So anything with a \sqrt{35} in it is going to be irrational.

    -Dan
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