Multiplication Methods
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Multiplication Methods
A multiplication algorithm is an algorithm (or method) to multiply two numbers. Depending on the
size of the numbers, different algorithms are in use. Efficient multiplication algorithms have existed
since the advent of the decimal system. The grid method (or box method) is an introductory method
for multipledigit multiplication that is often taught to pupils at primary school or elementary school
level. It has been a standard part of the national primaryschool mathematics curriculum in England
and Wales since the late 1990s. Both factors are broken up ("partitioned") into their hundreds, tens and
units parts, and the products of the parts are then calculated explicitly in a relatively simple
multiplicationonly stage, before these contributions are then totalled to give the final answer in a
separate addition stage.
This calculation approach (though not necessarily with the explicit grid arrangement) is also known as
the partial products algorithm. Its essence is the calculation of the simple multiplications separately,
with all addition being left to the final gatheringup stage. The grid method can in principle be applied
to factors of any size, although the number of subproducts becomes cumbersome as the number of
digits increases. Nevertheless it is seen as a usefully explicit method to introduce the idea of multiple
digit multiplications; and, in an age when most multiplication calculations are done using a calculator
or a spreadsheet.
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Long multiplication : If a positional numeral system is used, a natural way of multiplying numbers is
taught in schools as long multiplication, sometimes called gradeschool multiplication, sometimes
called Standard Algorithm: multiply the multiplicand by each digit of the multiplier and then add up all
the properly shifted results. It requires memorization of the multiplication table for single digits.
This is the usual algorithm for multiplying larger numbers by hand in base 10. Computers normally use
a very similar shift and add algorithm in base 2. A person doing long multiplication on paper will write
down all the products and then add them together; an abacususer will sum the products as soon as each
one is computed.
Lattice multiplication : First, set up the grid by marking its rows and columns with the numbers to be
multiplied. Then, fill in the boxes with tens digits in the top triangles and units digits on the bottom.
Finally, sum along the diagonal tracts and carry as needed to get the answer Lattice, or sieve,
multiplication is algorithmically equivalent to long multiplication. It requires the preparation of a lattice
(a grid drawn on paper) which guides the calculation and separates all the multiplications from the
additions. It was introduced to Europe in 1202 in Fibonacci's Liber Abaci. Leonardo described the
operation as mental, using his right and left hands to carry the intermediate calculations. Matrakci
Nasuh presented 6 different variants of this method in this 16thcentury book, Umdetul Hisab. It was
widely used in Enderun schools across the Ottoman Empire.[4] Napier's bones, or Napier's rods also
used this method, as published by Napier in 1617, the year of his death.
Binary Multiplication : In base 2, long multiplication reduces to a nearly trivial operation. For each
'1' bit in the multiplier, shift the multiplicand an appropriate amount and then sum the shifted values.
Depending on computer processor architecture and choice of multiplier, it may be faster to code this
algorithm using hardware bit shifts and adds rather than depend on multiplication instructions, when the
multiplier is fixed and the number of adds required is small. This algorithm is also known as Peasant
multiplication, because it has been widely used among those who are unschooled and thus have not
memorized the multiplication tables required by long multiplication. The algorithm was also in use in
ancient Egypt. On paper, write down in one column the numbers you get when you repeatedly halve the
multiplier, ignoring the remainder; in a column beside it repeatedly double the multiplicand. Cross out
each row in which the last digit of the first number is even, and add the remaining numbers in the
second column to obtain the product.
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