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 algorithmically the same as regular long multiplication, but it breaks the process into smaller steps, which some practitioners find easier to use.
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.
Description
A grid is drawn up, and each box is split diagonally. The first and second numbers are positioned along the top and right of the lattice respectively, with each digit being above a column, or next to a row. Simple products are written in each box, corresponding with numbers along the top and to the right of each box. For example, if the number above the box is 3, and the number to the right is 6, [1/8] (for 18) will be written in the box. If the simple product lacks a tens place, simply fill in the tens place with a 0.1
After all the boxes are filled in this manner, the diagonals are added from right to left, bottom to top, with the numbers added and written where the diagonal leads. When the sum contains more than one digit, the value of the tens place is carried over up to the next diagonal. 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).
Multiplication of Decimals
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.
History
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. The earliest recorded use of lattice multiplication:
- in Arab mathematics was by Ibn al-Banna' al-Marrakushi in his Talkhi? a‘mal al-?isab, 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. 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 Mu?ammad ibn Musa al-Khwarizmi (Baghdad, c. 825) or by Fibonacci in his Liber Abaci (Italy, 1202, 1228). 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 compares and contrasts multiplication by gelosia (lattice), by scacherii (chessboard), and other tableau methods.
Other notable historical uses of lattice multiplication include
- Jamshid al-Kashi’s Mifta? al-?isab (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 Ga?esa’s commentary on Bhaskara II’s Lilavati (16th century).
Derivatives
Derivations of this method also appeared in the 16th century in Matrakci Nasuh's Umdet-ul Hisab. Matrakçi 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.
