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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).
When done by hand, this may also be reframed as grid method multiplication or lattice multiplication. In software, this may be called "shift and add" due to bitshifts and addition being the only two operations needed. In 1960, Anatoly Karatsuba discovered Karatsuba multiplication, unleashing a flood of research into fast multiplication ...
The grid method (also known as the box 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.
The method was based on lattice multiplication, and also called rabdology, a word invented by Napier. Napier published his version in 1617. [1] It was printed in Edinburgh and dedicated to his patron Alexander Seton. Using the multiplication tables embedded in the rods, multiplication can be reduced to addition operations and division to ...
It is an extension of Napier's Bones, using two sets of rods to achieve multi-digit multiplication without the need to write down intermediate results, although some mental addition is still needed to calculate the result. The rods for the multiplicand are similar to Napier's Bones, with repetitions of the values.
The Egyptian method of multiplication of integers and fractions, which is documented in the Rhind Mathematical Papyrus, was by successive additions and doubling. For instance, to find the product of 13 and 21 one had to double 21 three times, obtaining 2 × 21 = 42 , 4 × 21 = 2 × 42 = 84 , 8 × 21 = 2 × 84 = 168 .
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implies that the fractions a / b , a / c , c / d , b / d are all irreducible, that is having no common factors (provided the denominators are non-zero, of course). More generally, if p / q is an irreducible fraction, then + + is also irreducible (again, provided the denominator be non-zero).