Other
algorithms for long multiplication
Lattice
multiplication
The
lattice method of multiplication appears in the first printed arithmetic
book, printed in Treviso (Italy) in 1478. It shows this exact method
and a variation as well as some variations of the long multiplication
algorithm commonly taught today.
Lattice
multiplication and variations of today's standard long multiplication
were introduced into Europe by Fibonacci (whose correct name is
Leonardo of Pisa). He was an Italian who learnt the use of Arabic
numerals from a Moorish teacher in North Africa. Before the Hindu-Arabic
system was used in Europe, multiplication was often done with counters
because Roman numerals were ill-suited to calculation and very few
people knew how to multiply. The Hindu-Arabic system has made calculation
fairly simple.
In
lattice multiplication, the partial products are laid out in a lattice
and adding along the diagonals gives the answer to the multiplication.
The
lattice method of multiplication is illustrated by the following
examples.
Example
1
28
x 57 = 1596
As
28 and 57 have two digits each, a lattice is set out with two columns
and two rows. The diagonals are drawn in each cell as shown below.
28 is written above the lattice with 2 above the first column and
8 above the second. 57 is written to the right of the lattice with
5 along the first row and 7 along the second.
The
partial products of these digits taken two at a time is set out
in the corresponding cells with the tens above the diagonal and
ones below. e.g. the partial products in this case are 5 x 8 (=
40), 5 x 2 (= 10), 7 x 8 (=56) and 7 x 2 (=14).These are set out
as shown below.
The
sum along each diagonal is then recorded as shown below and these
digits 1, 5, 9 and 6 form the answer to the multiplication.
Thus
28 x 57 = 1596
Example
2
183
x 49 = 8967
The
lattice set out for this multiplication will have 3 columns and
two rows as 183 has 3 digits. As before the numbers are set out
as shown below and the partial products are written down in their
respective positions. The numbers along the diagonals are added
to give the answer.
Note
that in this example adding along the third diagonal gives 19 which
needs 1 to be carried to the diagonal above it. Therefore the addition
should begin with the lowest diagonal ( the product of the ones
from the two numbers).
183
x 49 = 8967
The
Russian Peasant Algorithm
This
interesting algorithm of multiplication, which was used long ago,
is based on the principle of doubling and halving. Its advantage
is that it is only necessary to be able to double and halve numbers:
you do not have to know your multiplication tables.
Of
the two numbers to be multiplied, one is halved and the other is
doubled successively until the number in the halves column is reduced
to 1. Remainders in halving odd numbers are ignored. The numbers
in the doubles column corresponding to the odd numbers in the halves
column are chosen and the rest are ignored. The sum of these selected
numbers in the doubles column gives the product.
The
following examples illustrate the use of the algorithm.
Example
1
28
x 57 = 1596
Two
columns are set up as shown below, one for halves and the other
for doubles. 28 is placed in the halves column and 57 in the doubles
(They can be placed the other way round as well).
In
the next step 28 is halved and 57 is doubled and written below the
respective numbers.
Halves
28
14
|
Doubles
57
114
|
Next
14 is halved to get 7 and 114 is doubled to give 228. This process
is continued as shown below until the number in the halves column
becomes 1. Note that remainders are ignored when halving odd numbers.
Halves
28
14
7
3
1
|
Doubles
57
114
228
456
912
|
To
work out the answer, amazingly the rows containing the even numbers
in the halves column are now ignored. The remaining numbers in the
halves column are 7, 3 and 1. The numbers in the doubles column
corresponding to these, are added to give the product.
228
+ 456 + 912 =1596 which is the product of 28 and 57.
Example
2
183
x 49 = 8967
To
use the Russian peasant algorithm, 49 is placed in the halves column
and 183 in the doubles column. The halving and doubling process
is represented below.
Halves
49
24
12
6
3
1
|
Doubles
183
366
732
1464
2928
5856
|
The
odd numbers in the halves column are 49, 3 and 1. The corresponding
numbers in the doubles column are added to give the product.
183
+ 2928 + 5856 = 8967.
183
x 49 = 8967
Why
does it work?
The
method is simple enough to understand and most people who come across
the method have no difficulty in doing it. However what surprises
most people is why it works at all. While the principle of halving
and doubling is logical and familiar enough. This method is simply
ignores remainders and some of the rows. Here are some observations
about the algorithm which help in seeing why it works.
-
The
method is compensatory in nature as with each subsequent step
a factor 2 is moved from the number in the left column to the
number in the right column.
-
The
numbers in the doubles column contain 1, 2, 4, 8, 16,... groups
of the given number in the right column. In Example 1 the right
hand column contains 1, 2, 4, 8 and 16 groups of 57. This shows
a connection with the binary system of numeration.
-
In
example 1, the retained rows contain 4 groups, 8 groups and
16 groups of 57, giving 28 groups of 57 in all. In fact the
retained lines correspond to the binary expansion of 28. This
is the case with all examples and a little thought explains
how it works.
- To
construct a formal proof, think about the numbers in binary notation.
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