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Lattice multiplication, also known as gelosia multiplication, sieve multiplication, shabakh, Venetian squares, or the Hindu lattice, is a method of multiplication that uses a lattice to multiply two multi-digit numbers. It is mathematically identical to the more commonly used long multiplication algorithm, but it breaks the process into smaller steps, which some practitioners find easier to use.[1]

The method had already arisen by medieval times, and has been used for centuries in many different cultures. It is still being taught in certain curricula today.[2][3]

Description[edit]

A grid is drawn up, and each cell is split diagonally. The two multiplicands of the product to be calculated are written along the top and right side of the lattice, respectively, with one digit per column across the top for the first multiplicand, and one digit per row down the right side for the second multiplicand. Then each cell of the lattice is filled in with product of its column and row digit. For example, if the column digit is 5 and the row digit is 2, then 10 will be written in the cell, with the digit 1 above the diagonal and the digit 0 below the diagonal (see picture for Step 1).

If the simple product lacks a digit in the tens place, simply fill in the tens place with a 0.[1]

Step 1

After all the cells are filled in this manner, the digits in each diagonal are summed, working from the bottom right diagonal to the top left. Each diagonal sum is written where the diagonal ends. If the sum contains more than one digit, the value of the tens place is carried into the next diagonal (see Step 2).

Step 2

Numbers are filled to the left and to the bottom of the grid, and the answer is the numbers read off down (on the left) and across (on the bottom).

Step 3

Multiplication of Decimals[edit]

The lattice technique can also be used to multiply decimal fractions. For instance, to multiply 5.8 by 2.13, a line could be drawn straight down from the decimal in 5.8, and a line straight out from the decimal in 2.13. The lines are extended until they reach each other, at which point they merge and follow the diagonal. The positioning of this diagonal line in the final result is the location of the decimal point.[1]

History[edit]

Lattice multiplication has been used historically in many different cultures. It is not known where it arose first, nor whether it developed independently within more than one region of the world.[4] The earliest recorded use of lattice multiplication:[5]

- in Arab mathematics was by Ibn al-Banna' al-Marrakushi in his Talkhīṣ a‘māl al-ḥisāb, in the Maghreb in the late 13th century

- in European mathematics was by the unknown author of a Latin treatise in England, Tractatus de minutis philosophicis et vulgaribus, c. 1300

- in Chinese mathematics was by Wu Jing in his Jiuzhang suanfa bilei daquan, completed in 1450.

The mathematician and educator David Eugene Smith asserted that lattice multiplication was brought to Italy from the Middle East.[6] This is reinforced by noting that the Arabic term for the method, shabakh, has the same meaning as the Italian term for the method, gelosia, namely, the metal grille or grating (lattice) for a window.

It is sometimes erroneously stated that lattice multiplication was described by Muḥammad ibn Mūsā al-Khwārizmī (Baghdad, c. 825) or by Fibonacci in his Liber Abaci (Italy, 1202, 1228).[7] In fact, however, no use of lattice multiplication by either of these two authors has been found. In Chapter 3 of his Liber Abaci, Fibonacci does describe a related technique of multiplication by what he termed quadrilatero in forma scacherii (“rectangle in the form of a chessboard”). In this technique, the square cells are not subdivided diagonally; only the lowest-order digit is written in each cell, while any higher-order digit must be remembered or recorded elsewhere and then "carried” to be added to the next cell. This is in contrast to lattice multiplication, a distinctive feature of which is that the each cell of the rectangle has its own correct place for the carry digit; this also implies that the cells can be filled in any order desired. Swetz [8] compares and contrasts multiplication by gelosia (lattice), by scacherii (chessboard), and other tableau methods.

Other notable historical uses of lattice multiplication include:[5]

- Jamshīd al-Kāshī’s Miftāḥ al-ḥisāb (Samarqand, 1427), in which the numerals used are sexagesimal (base 60), and the grid is turned 45 degrees to a “diamond” orientation

- the Arte dell’Abbaco, an anonymous text published in the Venetian dialect in 1478, often called the Treviso Arithmetic because it was printed in Treviso, just inland from Venice, Italy

- Luca Pacioli’s Summa de arithmetica (Venice, 1494)

- the Indian astronomer Gaṇeśa’s commentary on Bhāskara II’s Lilāvati (16th century).

Derivatives[edit]

Derivations of this method also appeared in the 16th century in Matrakci Nasuh's Umdet-ul Hisab.[9] Matrakçı Nasuh's triangular version of the multiplication technique is seen in the example showing 155 x 525 on the right, and explained in the example showing 236 x 175 on the left figure.[10]

Matraki2.jpg

The same principle described by Matrakci Nasuh underlay the later development of the calculating rods known as Napier's bones (Scotland, 1617) and Genaille–Lucas rulers (France, late 1800s).

References[edit]

  1. ^ a b c Thomas, Vicki (2005). "Lattice Multiplication". Learn NC. UNC School of Education. Retrieved 4 July 2014. 
  2. ^ Boag, Elizabeth, “Lattice Multiplication,” BSHM Bulletin: Journal of the British Society for the History of Mathematics 22:3 (Nov. 2007), p. 182.
  3. ^ Nugent, Patricia M., “Lattice Multiplication in a Preservice Classroom”, Mathematics Teaching in the Middle School 13:2 (Sept. 2007), pp. 110-113.
  4. ^ Jean-Luc Chabert, ed., A History of Algorithms: From the Pebble to the Microchip (Berlin: Springer, 1999), p. 21.
  5. ^ a b Jean-Luc Chabert, ed., A History of Algorithms: From the Pebble to the Microchip (Berlin: Springer, 1999), pp. 21-26.
  6. ^ Smith, David Eugene, History of Mathematics, Vol. 2, “Special Topics of Elementary Mathematics” (New York: Dover, 1968).
  7. ^ The original 1202 version of Liber Abaci is lost. The 1228 version was later published in its original Latin in Boncompagni, Baldassarre, Scritti di Leonardo Pisano, vol. 1 (Rome: Tipografia delle Scienze Matematiche e Fisiche, 1857); an English translation of the same was published by Sigler, Laurence E., Fibonacci’s Liber Abaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation (New York: Springer Verlag, 2002).
  8. ^ Swetz, Frank J., Capitalism and Arithmetic: The New Math of the 15th Century, Including the Full Text of the Treviso Arithmetic of 1478, Translated by David Eugene Smith (La Salle, IL: Open Court, 1987), pp. 205-209.
  9. ^ Corlu, M.S., Burlbaw, L.M., Capraro, R. M., Corlu, M.A.,& Han, S. (2010). "The Ottoman Palace School Enderun and The Man with Multiple Talents, Matrakçı Nasuh." Journal of the Korea Society of Mathematical Education, Series D: Research in Mathematical Education. 14(1), p 19-31.
  10. ^ http://tamu.academia.edu/SencerCorlu/Papers/471488/The_Ottoman_Palace_School_Enderun_and_the_Man_with_Multiple_Talents_Matrakci_Nasuh

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48 news items

 
WQAD.com -- Quad Cities News & Weather from WQAD
Mon, 03 Feb 2014 20:05:44 -0800

Students in some local elementary schools are learning a completely different way of doing multiplication. It's a technique that many parents can't understand. It's called lattice multiplication. The method has been around for hundreds of years. Some ...

Casper Star-Tribune Online

Casper Star-Tribune Online
Sat, 25 Oct 2014 07:03:45 -0700

“Instead of the standard way that we all learned to multiply, they offer up a strategy of lattice multiplication that gives them another opportunity, and there's another one called partial products,” she said. “I think sometimes that's what's confusing ...

Gizmodo Australia

Gizmodo Australia
Tue, 11 Nov 2014 20:36:09 -0800

With methods previously not seen by many American parents (lattice multiplication anyone?), the new curriculum has some pulling their hair out. As one mum said, “I hate the Common Core… I can't help my kid with his homework and I don't understand the ...
 
Gizmodo India
Tue, 11 Nov 2014 05:41:15 -0800

With methods previously not seen by many American parents (lattice multiplication anyone?), the new curriculum has some pulling their hair out. As one mom said, "I hate the Common Core . . . .I can't help my kid with his homework and I don't understand ...
 
Search Engine People (blog)
Wed, 12 Mar 2014 06:26:02 -0700

Not too long ago I stood in front of 30 eight and nine-year-olds showing them the ever exciting method of Lattice Multiplication. (Seriously, the kids LOVED doing it.) Yep, I was a third grade teacher. Ive talked before about the thought process and ...
 
The Dagger
Tue, 18 Mar 2014 17:47:25 -0700

Part II of The Dagger's Q&A with newly appointed Harford County Public Schools Superintendent Barbara Canavan begins with updates on concerns raised by parents over the years, and ends with Canavan's hopes for the future. Included is a potential ...
 
Chicago Tribune
Mon, 06 Aug 2012 17:28:04 -0700

Declan said he especially liked lattice multiplication, developed about A.D. 1200, and partial quotient division — two of the methods Everyday Math uses that give parents headaches. "They are easier and fun," he said. The parents of children at the ...
 
NewHampshire.com
Fri, 07 Dec 2012 09:21:04 -0800

McGee said students benefit from a consistent math framework across grade levels; something Everyday Math provides. As for the aspects that don't seem to be working, like lattice multiplication, which has been around for centuries, those simply aren't ...
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