Theory behind long division method

[Anukriti C.]

our maths teacher asked us that we all use the long division method to find square roots or cube roots. The question is, why do we do it that way, i.e. taking one or two nos. from the starting, doubling the divisor and all the steps(i guess everyone knows that). can anyone please help me and tell me the main objective or the actual reasons involved in each step?

 

[SteamKing]

our maths teacher asked us that we all use the long division method to find square roots or cube roots. The question is, why do we do it that way, i.e. taking one or two nos. from the starting, doubling the divisor and all the steps(i guess everyone knows that). can anyone please help me and tell me the main objective or the actual reasons involved in each step?

This has been covered in a recent thread at PF.

Peruse this thread and see if some of the replies don't answer your question:

https://www.physicsforums.com/threa...oot-extraction-at-school.821407/#post-5157020

If there is anything you don't understand about the algorithm, post another question here and we'll try to clear it up for you.

 

[Anukriti C.]

This has been covered in a recent thread at PF.

Peruse this thread and see if some of the replies don't answer your question:

https://www.physicsforums.com/threa...oot-extraction-at-school.821407/#post-5157020

If there is anything you don't understand about the algorithm, post another question here and we'll try to clear it up for you.

thanks btw...actually I wanted to know why do we do it that way... I know how to do and what to do... I want to know how was it first derived?
Was it kinda hit and trial method or there is some logic behind it...

 

[SteamKing]

thanks btw...actually I wanted to know why do we do it that way... I know how to do and what to do... I want to know how was it first derived?
Was it kinda hit and trial method or there is some logic behind it...

This algorithm, and similar ones, have been developed at different times in the distant past.

It is not clear how ancient mathematicians first developed these algorithms, but their work can be translated into modern mathematical notation and studied.

https://en.wikipedia.org/wiki/Methods_of_computing_square_roots

http://faculty.ist.unomaha.edu/aparakh/papers/Aryabhatas_Root_Extraction_Methods_IJHS.pdf

http://arxiv.org/ftp/math/papers/0608/0608793.pdf

With the development of logarithms, these algorithms became mathematical curiosities, at least for extracting roots in daily calculations.

The Greeks might have used geometry to deduce the necessary mathematics, but their number system was not well suited for performing complex arithmetic. The algorithms of Aryabhatas depend to an extent on having a place-value number system in which to do arithmetic. The chronology of the development of the decimal place-value system is still shrouded in mystery to an extent:

https://en.wikipedia.org/wiki/Indian_mathematics

 

[mathman]

With the development of logarithms, these algorithms became mathematical curiosities, at least for extracting roots in daily calculations.

Not completely true. Computer square roots use this method, so square roots take the same amount of time as division.

 

[SteamKing]

Not completely true. Computer square roots use this method, so square roots take the same amount of time as division.

It would be a mistake to assume that the algorithms used by computers for FP division and root extraction are merely hard-coded versions of the pen-and-paper procedures.

It appears that the Goldschmidt algorithm can be adapted for both problems, and it takes advantage of the binary representation of FP numbers to speed calculations:

http://www.informatik.uni-trier.de/Reports/TR-08-2004/rnc6_12_markstein.pdf

https://en.wikipedia.org/wiki/Division_algorithm

For FP-intensive calculations, like with games and whatnot, other algorithms have been developed to reduce calculation time further.