Lattice multiplication

Lattice multiplication, also known as gelusia multiplication, sieve multiplication, shabakh, Venetian squares, or the Chinese 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.

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.

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 wrtten top-down). Then each cell of the lattice is filled in with product of its column and row digit.

As an example, let's 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.

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).

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).

The lattice technique can also be used to multiply decimal fractions. For instance, to multiply 5.8 by 2.13, a vertical line could be drawn straight down from the decimal in 5.8, and a horizontal line straight out from the decimal in 2.13. The lines are extended until they intersect, 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.

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:

...
Wikipedia