Wikipedia:Reference desk/Archives/Mathematics/2014 December 9

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December 9

Square Root methods

How many ways does anyone know to obtain the square root of a number? I will list those that I know. I would like to know if there are any others. I don’t know of any that were introduced in the twenty-first century. Can anyone list any other methods?

In approximate order of date of introduction:

a. Use a pencil-and-paper procedure that resembles a more complicated version of long division.

b. Use logarithms.

c. Use an iterative method, such as Newton’s method. These are methods for successive numerical approximation of the solution to any of various equations that cannot be exactly solved by algebra. (This probably would not have been actually done in the eighteenth century, because for square root, it involves more steps than long division.)

d. Use a table of square roots. Interpolate if necessary. The generation of the table of square roots was done using either long division or logarithms. (It wouldn’t have been done by iteration, because for square root, that takes longer than long division.)

e. Use a slide rule.

f. Write a computer program that computes square roots. (The program will probably either use logarithms or implement an iterative method.)

g. Use a calculator. (The calculator may rely on logarithms, but that is only my guess, because I haven’t seen the hardware logic of a calculator.)

h. Use Excel. (Excel probably relies on logarithms. That is only my guess, because I haven’t seen the source code of Excel).

Any other ideas?

Robert McClenon (talk) 15:52, 9 December 2014 (UTC)

There's a Euclidean construction, where you construct a right triangle with hypotenuse 1+x, where the altitude divides the hypotenuse into a segment of length 1 and one of length x. Simliar triangles gives that the altitude has length sqrt(x). Sławomir Biały (talk) 17:12, 9 December 2014 (UTC)

That method is the most ancient, since the paper-and-pencil method relies on Arabic numerals.

You could also use a binomial series. Sławomir Biały (talk) 17:44, 9 December 2014 (UTC)

Thank you. And the computer program could also be using a binomial series. The calculation of the logarithms is also done using a Taylor series. Robert McClenon (talk) 18:08, 9 December 2014 (UTC)

I did some cleanup work on Methods of computing square roots back in the day; not sure what's there now, but you'll likely find some additional information. -- Meni Rosenfeld (talk) 18:14, 9 December 2014 (UTC)

There was an interesting approximate versions the Fast inverse square root used in the Doom video game.--Salix alba (talk): 09:50, 10 December 2014 (UTC)

From the description of sector (instrument), it sounds as if for roots it works similarly to a slide rule but without the slide and instead used with dividers. Various analog computers may implement roots in different ways. I assume that hydraulic computers typically use pressure and flow rates more than volumes, but you can easily imagine a student demonstrator or experiment composed of a plexiglas tank in the form of a triangular prism with vertex down, thick enough that when filled with 1cc of water the level is 1cm, so that when filled with x cc of water the level is √x cm. (Not sure you are interested in that.) -- ToE 23:35, 10 December 2014 (UTC)