Lattice multiplication

Last updated

Lattice multiplication, also known as the Italian method, Chinese method, Chinese lattice, gelosia multiplication, [1] sieve multiplication, shabakh, diagonally or Venetian squares, 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. [2]

Contents

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. [3] [4]

Method

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 (the number written left to right), and one digit per row down the right side for the second multiplicand (the number written top-down). Then each cell of the lattice is filled in with product of its column and row digit.

As an example, consider the multiplication of 58 with 213. After writing the multiplicands on the sides, consider each cell, beginning with the top left cell. In this case, the column digit is 5 and the row digit is 2. Write their product, 10, 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. [2]

Step 1 Example of step 1 of lattice (shabakh) multiplication algorithm.svg
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 Example of step 2 of lattice (shabakh) multiplication algorithm.svg
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). In the example shown, the result of the multiplication of 58 with 213 is 12354.

Step 3 Example of step 3 of lattice (shabakh) multiplication algorithm.svg
Step 3

Multiplication of decimal fractions

The lattice technique can also be used to multiply decimal fractions. For example, to multiply 5.8 by 2.13, the process is the same as to multiply 58 by 213 as described in the preceding section. To find the position of the decimal point in the final answer, one can draw a vertical line from the decimal point in 5.8, and a horizontal line from the decimal point in 2.13. (See picture for Step 4.) The grid diagonal through the intersection of these two lines then determines the position of the decimal point in the result. [2] In the example shown, the result of the multiplication of 5.8 and 2.13 is 12.354.

Step 4 Example of step 4 of lattice (shabakh) multiplication algorithm.svg
Step 4

History

Folios 9v and 10r from the manuscript "Raqaiq a-haqaiq fi hisab ad-daraj wa ad-daqaiq", from the Bibliotheque nationale de France, showing lattice multiplications with old Arab numerals rqy'q lHqy'q fy Hsb ldrj wldqy'q - lSfHtyn 9Z w10w.png
Folios 9v and 10r from the manuscript "Raqâiq a-haqâiq fi hisab ad-daraj wa ad-daqâiq", from the Bibliothèque nationale de France, showing lattice multiplications with old Arab numerals

Though lattice multiplication has been used historically in many cultures, a method called 'Kapat-sandhi' very similar to the lattice method is mentioned in the commentary on 12th century 'Lilavati' a book of Indian mathematics by Bhaskaracharya. It is being researched where it arose first, whether it developed independently within more than one region of the world. [5] The earliest recorded use of lattice multiplication: [6]

The mathematician and educator David Eugene Smith asserted that lattice multiplication was brought to Italy from the Middle East. [7] 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). [8] 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 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 [9] compares and contrasts multiplication by gelosia (lattice), by scacherii (chessboard), and other tableau methods.

Other notable historical uses of lattice multiplication include: [6]

Derivations

Derivations of this method also appeared in the 16th century works Umdet-ul Hisab by Ottoman-Bosnian polymath Matrakçı Nasuh. [10] 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. [11]

Matraki2.jpg

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

See also

Related Research Articles

Arabic numerals are the ten symbols most commonly used to write decimal numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as computer symbols, trademarks, or license plates. The term often implies a decimal number, in particular when contrasted with Roman numerals.

<span class="mw-page-title-main">Arithmetic</span> Elementary branch of mathematics

Arithmetic is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th century, Italian mathematician Giuseppe Peano formalized arithmetic with his Peano axioms, which are highly important to the field of mathematical logic today.

<span class="mw-page-title-main">Fibonacci</span> Italian mathematician (c. 1170–1245)

Fibonacci, also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano, was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".

<span class="mw-page-title-main">Multiplication</span> Arithmetical operation

Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a product.

<span class="mw-page-title-main">Multiplication table</span> Mathematical table

In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system.

A multiplication algorithm is an algorithm to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient than others. Efficient multiplication algorithms have existed since the advent of the decimal system.

<i>Liber Abaci</i> Mathematics book written in 1202 by Fibonacci

Liber Abaci is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci.

<span class="mw-page-title-main">Napier's bones</span> 1617 device for calculating products and quotients

Napier's bones is a manually-operated calculating device created by John Napier of Merchiston, Scotland for the calculation of products and quotients of numbers. The method was based on lattice multiplication, and also called rabdology, a word invented by Napier. Napier published his version in 1617. It was printed in Edinburgh and dedicated to his patron Alexander Seton.

A numerical digit is a single symbol used alone or in combinations, to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits of the hands correspond to the ten symbols of the common base 10 numeral system, i.e. the decimal digits.

The promptuary, also known as the card abacus is a calculating machine invented by the 16th-century Scottish mathematician John Napier and described in his book Rabdologiae in which he also described Napier's bones.

Location arithmetic is the additive (non-positional) binary numeral systems, which John Napier explored as a computation technique in his treatise Rabdology (1617), both symbolically and on a chessboard-like grid.

<i>Līlāvatī</i> Mathematical treatise by Bhāskara II

Līlāvatī is Indian mathematician Bhāskara II's treatise on mathematics, written in 1150 AD. It is the first volume of his main work, the Siddhānta Shiromani, alongside the Bijaganita, the Grahaganita and the Golādhyāya.

<span class="mw-page-title-main">Hindu–Arabic numeral system</span> Most common system for writing numbers

The Hindu–Arabic numeral system or Indo-Arabic numeral system is a positional decimal numeral system, and is the most common system for the symbolic representation of numbers in the world.

In mathematics, ancient Egyptian multiplication, one of two multiplication methods used by scribes, is a systematic method for multiplying two numbers that does not require the multiplication table, only the ability to multiply and divide by 2, and to add. It decomposes one of the multiplicands into a set of numbers of powers of two and then creates a table of doublings of the second multiplicand by every value of the set which is summed up to give result of multiplication.

The grid method of multiplication is an introductory approach to multi-digit multiplication calculations that involve numbers larger than ten. Because it is often taught in mathematics education at the level of primary school or elementary school, this algorithm is sometimes called the grammar school method.

<span class="mw-page-title-main">Rod calculus</span>

Rod calculus or rod calculation was the mechanical method of algorithmic computation with counting rods in China from the Warring States to Ming dynasty before the counting rods were increasingly replaced by the more convenient and faster abacus. Rod calculus played a key role in the development of Chinese mathematics to its height in Song Dynasty and Yuan Dynasty, culminating in the invention of polynomial equations of up to four unknowns in the work of Zhu Shijie.

Nasuh bin Karagöz bin Abdullah el-Visokavi el-Bosnavî, commonly known as Matrakçı Nasuh for his competence in the combat sport of Matrak which was invented by himself, was a 16th-century Turk-Bosniak statesman of the Ottoman Empire, polymath, mathematician, teacher, historian, geographer, cartographer, swordmaster, navigator, inventor, painter, farmer, and miniaturist.

Matrak is an Ottoman combat sport based on sword and shield fighting, Invented by the Ottoman Bosnian statesman, historian and scientist Nasuh Matrakčija Visočak (full name in Turkish: Nasuh bin Karagöz bin Abdullah el-Bosnavî) in the 16th century. It is played with wooden sticks covered with leather simulating a sword, and a wooden leather covered shield. The top of the sticks are rounded and slightly wider than the body resembling bowling pins. The game is a kind of combat simulation, and is played on a lawn. It was used by Ottoman soldiers as practice for melee combat.

<span class="mw-page-title-main">Yupana</span> Incan abacus

A yupana is an abacus used to perform arithmetic operations, dating back to the time of the Incas.

<i>Principles of Hindu Reckoning</i>

Principles of Hindu Reckoning is a mathematics book written by the 10th- and 11th-century Persian mathematician Kushyar ibn Labban. It is the second-oldest book extant in Arabic about Hindu arithmetic using Hindu-Arabic numerals, preceded by Kibab al-Fusul fi al-Hisub al-Hindi by Abul al-Hassan Ahmad ibn Ibrahim al-Uglidis, written in 952.

References

  1. Williams, Michael R. (1997). A history of computing technology (2nd ed.). Los Alamitos, Calif.: IEEE Computer Society Press. ISBN   0-8186-7739-2. OCLC   35723637.
  2. 1 2 3 Thomas, Vicki (2005). "Lattice Multiplication". Learn NC. UNC School of Education. Retrieved 4 July 2014.
  3. Boag, Elizabeth (November 2007). ""Lattice Multiplication"". BSHM Bulletin: Journal of the British Society for the History of Mathematics. 22 (3): 182–184. doi:10.1080/14794800008520169. S2CID   122212455 . Retrieved 25 February 2022.
  4. Nugent, Patricia (2007). ""Lattice Multiplication in a Preservice Classroom"". National Council of Teachers of Mathematics. 13 (2): 110–113. doi:10.5951/MTMS.13.2.0110 . Retrieved 25 February 2022.
  5. Jean-Luc Chabert, ed., A History of Algorithms: From the Pebble to the Microchip (Berlin: Springer, 1999), p. 21.
  6. 1 2 Jean-Luc Chabert, ed., A History of Algorithms: From the Pebble to the Microchip (Berlin: Springer, 1999), pp. 21-26.
  7. Smith, David Eugene, History of Mathematics, Vol. 2, “Special Topics of Elementary Mathematics” (New York: Dover, 1968).
  8. 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).
  9. 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.
  10. 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.
  11. Capraro, Robert (January 2010). "Corlu, M. S., Burlbaw, L. M., Capraro, R. M., Han, S., & Çorlu, M. A. (2010). The Ottoman palace school and the man with multiple talents, Matrakçı Nasuh. Journal of the Korea Society of Mathematical Education Series D: Research in Mathematical Education, 14(1), 19–31". D-수학교육연구.