Multiplication algorithm: Difference between revisions

Revision as of 19:47, 21 July 2023

Revision as of 20:18, 21 July 2023

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===Quarter square multiplication===

Two quantities can be multiplied using quarter squares by employing the following identity involving the [[Floor and ceiling functions|floor function]] that some sources<ref>{{citation |title= Quarter Tables Revisited: Earlier Tables, Division of Labor in Table Construction, and Later Implementations in Analog Computers |last=McFarland |first=David|url=http://escholarship.org/uc/item/5n31064n |page=1 |year=2007}}</ref><ref>{{cite book| title=Mathematics in Ancient Iraq: A Social History |last=Robson |first=Eleanor |page=227 |year=2008 |isbn= 978-0691091822 }}</ref> attribute to [[Babylonian mathematics]] (2000–1600 BC).

This formula can in some cases be used, to make multiplication tasks easier to complete:

: <math>

\left\lfloor \frac{\left(x+y\right)^2}{4} \right\rfloor - \left\lfloor \frac{\left(x-y\right)^2}{4} \right\rfloor =

\frac{1}{4}\left(\left(x^2+2xy+y^2\right) - \left(x^2-2xy+y^2\right)\right) =

\frac{1}{4}\left(4xy\right) = xy.

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====History of quarter square multiplication====

TwoIn quantitiesbabylonian can be multiplied usingtime, quarter squaressquare bymultiplication employing the following identity involving theinvolved [[Floor and ceiling functions|floor function]]; that some sources<ref>{{citation |title= Quarter Tables Revisited: Earlier Tables, Division of Labor in Table Construction, and Later Implementations in Analog Computers |last=McFarland |first=David|url=http://escholarship.org/uc/item/5n31064n |page=1 |year=2007}}</ref><ref>{{cite book| title=Mathematics in Ancient Iraq: A Social History |last=Robson |first=Eleanor |page=227 |year=2008 |isbn= 978-0691091822 }}</ref> attribute to [[Babylonian mathematics]] (2000–1600 BC).

Antoine Voisin published a table of quarter squares from 1 to 1000 in 1817 as an aid in multiplication. A larger table of quarter squares from 1 to 100000 was published by Samuel Laundy in 1856,<ref>{{Citation |title=Reviews |journal=The Civil Engineer and Architect's Journal |year=1857 |pages=54–55 |url=https://books.google.com/books?id=gcNAAAAAcAAJ&pg=PA54 |postscript=.}}</ref> and a table from 1 to 200000 by Joseph Blater in 1888.<ref>{{Citation|title=Multiplying with quarter squares |first=Neville |last=Holmes| journal=The Mathematical Gazette |volume=87 |issue=509 |year=2003 |pages=296–299 |jstor=3621048|postscript=.|doi=10.1017/S0025557200172778 |s2cid=125040256 }}</ref>