Multiplication algorithm: Difference between revisions

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Revision as of 19:47, 21 July 2023

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====Properties of Quarter square multiplication equation ====

If one of {{math|''x''+''y''}} or {{math|''x''&minus;''y''}} is odd, the other is odd too, thus their squares are 1 mod 4, then taking floor reduces both by a quarter; the subtraction then cancels the quarters out, and discarding the remainders does not introduce any difference comparing with the same expression without the floor functions. Below is a lookup table of quarter squares with the remainder discarded for the digits 0 through 18; this allows for the multiplication of numbers up to {{math|9×9}}.

'''Case <math> x = 2p </math> and <math> y = 2q </math>:'''

<math> x + y = 2p + 2q = 2(p+q) \implies (x+y)^2 = 4(p+q)^2 </math>.

It also implies that <math> (x - y)^2 = 4(p-q)^2 </math>.

This results in <math> xy = \frac{1}{4}((x+y)^2 - (x-y)^2)) = (p+q)^2 - (p-q)^2 </math>.

'''Case <math> x = 2p+1 </math> and <math> y = 2q+1 </math>''':

<math> x + y = 2p+1 + 2q+1 = 2(p+q+1) \implies (x+y)^2 = 4(p+q+1)^2 </math>.

It also implies that <math> (x - y)^2 = 4(p-q)^2 </math>.

This results in <math> xy = \frac{1}{4}((x+y)^2 - (x-y)^2)) = (p+q+1)^2 - (p-q)^2 </math>.

'''Case <math> x = 2p </math> and <math> y = 2q+1 </math>:'''

<math> x + y = 2p + 2q+1 = 2(p+q+\frac{1}{2}) \implies (x+y)^2 = 4(p+q+\frac{1}{2})^2 </math>.

It also implies that <math> (x - y)^2 = 4(p-q-\frac{1}{2})^2 </math>.

This results in <math> xy = \frac{1}{4}((x+y)^2 - (x-y)^2)) = (p+q+\frac{1}{2})^2 - (p-q-\frac{1}{2})^2 </math>.

'''Case <math> x = 2p+1 </math> and <math> y = 2q </math>:'''

<math> x + y = 2p+1 + 2q = 2(p+q+\frac{1}{2}) \implies (x+y)^2 = 4(p+q+\frac{1}{2})^2 </math>.

It also implies that <math> (x - y)^2 = 4(p-q-\frac{1}{2})^2 </math>.

This results in <math> xy = \frac{1}{4}((x+y)^2 - (x-y)^2)) = (p+q+\frac{1}{2})^2 - (p-q+\frac{1}{2})^2 </math>.

====Examples ====

Below is a lookup table of quarter squares with the remainder discarded for the digits 0 through 18; this allows for the multiplication of numbers up to {{math|9×9}}.

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If, for example, you wanted to multiply 9 by 3, you observe that the sum and difference are 12 and 6 respectively. Looking both those values up on the table yields 36 and 9, the difference of which is 27, which is the product of 9 and 3.

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

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>