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Math Help - Without Calculator

  1. #1
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    Without Calculator

    How do I find the appoximation of square roots that are not perfect squares WITHOUT a calculator?

    SAMPLES:

    (1) sqrt{21} =

    (2) sqrt{79} =

    DO NOT USE A CALCULATOR.

    Thanks!
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  2. #2
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    Quote Originally Posted by symmetry View Post
    How do I find the appoximation of square roots that are not perfect squares WITHOUT a calculator?

    SAMPLES:

    (1) sqrt{21} =
    \sqrt{16}<\sqrt{21}<\sqrt{25}

    Now notice that 21 is almost right inbetween 16 and 25, so we'll say that: \sqrt{21}\approx 4.5

    and a calculator gives: \sqrt{21}\approx 4.58

    So I was off by an eight-hundredth, still pretty close.
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  3. #3
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    ok

    How would you do sqrt{79} THE SAME WAY?
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  4. #4
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    Quote Originally Posted by symmetry View Post
    How would you do sqrt{79} THE SAME WAY?
    \sqrt{64}<\sqrt{79}<\sqrt{81}

    Notice that 79 is a lot closer to 81 than 64, so I'll estimate that \sqrt{79}\approx 8.9


    And the calculator gives: \sqrt{79}\approx 8.89

    So I was off by about 1-hundredth, which is very close.
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  5. #5
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    Quote Originally Posted by symmetry View Post
    How would you do sqrt{79} THE SAME WAY?
    Generally you would use Taylor Polynomials, which involves calculus to find such without a calculator. It's far more accurate, but Quick's method suffices.

    Sqrt(81) = 9

    Sqrt(64) = 8

    Thus, sqrt(79) is between 8 and 9. Note that sqrt(79) is quite close to sqrt(76), which is 2*sqrt(19) (gotten from sqrt(4)*sqrt(19) = sqrt(76).

    And you can do the same deal with sqrt(19) as I did before to try and get a better result. Obviously the result will be in the high 8's, since sqrt(79) is close to sqrt(81).
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  6. #6
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    ok

    My study book for June state math exam takes me through the basics of high school mathematics going through a few calculus 1 and 2 topics.

    I will cross that bridge when I get there.

    In terms of Taylor Polynomials, let's not go there just yet.

    Thanks!
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