Lattice Method THINGS
TO TRY:
divide
The lattice method is an alternative to long multiplication for
numbers. In this approach, a lattice is first constructed, sized
to fit the numbers being multiplied. If we are multiplying an -
digit number by an -digit number, the size of the lattice is .
The multiplicand is placed along the top of the lattice so that
each digit is the header for one column of cells (the most
significant digit is put at the left). The multiplier is placed along
the right side of the lattice so that each digit is a (trailing)
header for one row of cells (the most significant digit is put at
the top). Illustrated above is the lattice configuration for
computing .
Before the actual multiplication can begin, lines must be
drawn for every diagonal path in the lattice from upper right to
lower left to bisect each cell. There will be 5 diagonals for
our lattice array.
Now we calculate a product for each cell by multiplying the
digit at the top of the column and the digit at the right of the
row. The tens digit of the product is placed above the diagonal
that passes through the cell, and the units digit is put below
that diagonal. If the product is less than 10, we enter a zero
above the diagonal.
Now we are ready to calculate the digits of the product. We
sum the numbers between every pair of diagonals and also
between the first (and last) diagonal and the corresponding
corner of the lattice. We start at the bottom half of the lower
right corner cell (6). This number is bounded by the corner of
the lattice and the first diagonal. Since this is the only number
below this diagonal, the first sum is 6. We place the sum along
the bottom of the lattice below the rightmost column.
Next we sum the numbers between the previous diagonal and
the next higher diagonal: . We place the 9 just below
the bottom of the lattice and carry the 1 into the sum for the
next diagonal group. (The diagonals are extended for clarity.)
We continue summing the groups of numbers between
adjacent diagonals, and also between the top diagonal and
the upper left corner. The final product is composed of the
digits outside the lattice which were just calculated. We read
the digits down the left side and then towards the right on the
bottom to generate the final answer: 783996.
Although the process at first glance appears quite different
from long multiplication, the lattice method is actually
algorithmically equivalent.
SEE ALSO:Long Division, Long Multiplication, Multiplication
This entry contributed by Len Goodman
CITE THIS AS:
Goodman, Len. "Lattice Method." From MathWorld--A Wolfram Web
Resource, created by Eric W.
Weisstein. http://mathworld.wolfram.com/LatticeMethod.html
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