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Finding the square root

  1. May 18, 2009 #1
    This has been bothering me for a while now. Without a calculator and without using "guess and check" how can find the square root of a number? For example: 6[tex]^{2}[/tex] = 36 but how can you reverse it and figure out what [tex]\sqrt{36}[/tex] equals? Is there some equation to it or can it only be done using guess and check?

    Thanks in advance :smile:
     
  2. jcsd
  3. May 18, 2009 #2
  4. May 18, 2009 #3
    Thats a bit complex for me at my level of math right now. Are there any simpler, more straight forward equations?
     
  5. May 18, 2009 #4
    There is a long process similar to long division. Is that what you're looking for?
     
  6. May 18, 2009 #5
  7. May 19, 2009 #6
    Find the Talor polynomial to whatever degree of accuracy you want, and plug in.

    For example...

    f(x) = sqrt(x)

    f(1) = 1
    f '(1) = 1/2
    f ''(1) = -1/4

    So f(x) ~ 1 + 1/2(x-1) - 1/8(x-1)^2

    Check my arithmetic... ;p

    A special case of this is linear approximation. In this case, and in all cases, it's good to choose a point "close to" the thing you're trying to evaluate to start from. The closer you start, the better your final answer will be.
     
  8. May 19, 2009 #7

    Hurkyl

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    What's wrong with guess and check? It's the best method to solve lots of problems. (Assuming you don't use it in a stupid way)
     
  9. May 19, 2009 #8

    Hurkyl

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    For example, starting from 1 will only work if you want to find the square root of a number less than 2....
     
  10. May 19, 2009 #9

    HallsofIvy

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    This turns out to be equivalent to Newton's method but does not require the general formula:

    To find [itex]x= \sqrt{a}[/itex] is equivalent to solving [itex]x^2= a[/itex] which, for non-zero x, is equivalent to x= a/x. Notice also that is x< [itex]\sqrt{a}[/itex], a/x will be larger than x and vice-versa. In either case, the correct square root is between x and a/x.

    Therefore: Choose x as some reasonable "guess" (or "starting value" if you don't like "guess"!) for [itex]sqrt{a}[/itex]. Calculate a/x. If that is equal to x, we are done. Otherwise we know the correct square root is between x and a/x. We don't know where, but 1/2 way is likely to be close. So use that as a new value and repeat.

    For your example, a= 6, x= 2 is a good starting value. a/x= 6/2= 3 and half way between 2 and 3 is 2.5. Now a/x= 6/2.5= 2.4. Halfway between 2.5 and 2.4 is 2.45. a/x= 6/2.45= 2.449 to 3 decimal places. (2.450+ 2.449)/2= 2.4495 which is still 2.450 to three decimal places to three decimal places which is exactly what a calculator would give. If you want more decimal places, you will need to continue the calculation. This method recovers, generally, one more decimal place for every iteration.
     
  11. May 19, 2009 #10
    you don't even have to start off with a reasonable guess for x. of course, a better initial guess leads to quicker convergence on the solution.

    even working out something like [itex]\sqrt{3844}[/itex], taking a starting guess to be 50 takes about 6 iterations to get the correct answer. taking a starting guess to be 1 only takes 12 iterations to get the result.
     
    Last edited: May 19, 2009
  12. May 19, 2009 #11

    HallsofIvy

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    To what degree of accuracy?
     
  13. May 19, 2009 #12
    using a=3844, x=1, i got (using pl/sql mind)

    using x=50, i get

    as i did mention though, a closer initial estimate leaders to quicker convergence. the only reason that i mention this is that sometimes, a bad initial guess can lead to chaotic results. but that doesn't seem to be the case with the method you mentioned above.
     
    Last edited: May 19, 2009
  14. May 19, 2009 #13

    jtbell

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    This method (the decimal version) is probably the one I learned in high school in the late 1960s. Yes, they actually used to teach this stuff in high school! Maybe it was even in middle school, I don't remember for sure. This was before personal electronic calculators existed, of course. If we wanted to find square roots quickly, we used a slide rule or a table of logarithms.
     
  15. May 19, 2009 #14
    Ah, those were the days, weren't they?

    No, really, I'm asking, weren't they?? I can't remember any more!!! :eek:
     
  16. May 19, 2009 #15
    I learned it in high school in the late 60s too.
     
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