Lattice Multiplication Explained

Date: 10/20/1999 at 09:55:54
From: Julie Durham a
Subject: Lattice Multiplication

I know how to figure out the Lattice Multiplication procedure, but I 
don't understand why it works. We learned the traditional way of 
multiplying, and it works with this but I don't see how it works when 
you put the numbers on a square. Why and how does it work? I asked my 
teacher and she said, "Magic."

I was never really great at math but I like this and I want to 
understand why it works. How many algorithms for multiplication are 
there in this world? We only learned it one way and now I know two and 
I like this one better than the other way. Can you help, please? Can 
you give me other examples of multiplication algorithms?

Date: 10/20/1999 at 12:18:09
From: Doctor Peterson
Subject: Re: Lattice Multiplication

Hi, Julie.

I'm glad you find lattice multiplication useful. In case you want to 
see more of it, here's a page in our archives on it:

   Lattice Multiplication
   http://mathforum.org/dr.math/problems/susan.8.340.96.html   

You might also find Napier's Rods (or "bones") interesting:

   Napier's Rods
   http://mathforum.org/dr.math/problems/phyllis03.05.99.html   

There were many methods of multiplication in earlier times (say the 
16th and 17th centuries), but they were generally similar to either 
our usual method or the lattice method. You might want to check a good 
book on the history of math to see how they worked, but I think only 
these two had lasting value.

As for why the method works, let's try inventing the method ourselves, 
and see why we'd do it the way we do. Math is never magic - it's just 
that the reasons are sometimes pretty well hidden. We'll use this 
example, as in the archived answer I referred you to:

       469
     x  37
     -----

The basis of any method of multiplying is the distributive property:

     a x (b + c) = a x b + a x c

In this case,

     469 x 37 = (400 + 60 + 9) x (30 + 7)
              = (400 + 60 + 9) x 30 + (400 + 60 + 9) x 7
              = 400 x 30 + 60 x 30 + 9 x 30 + 400 x 7 + 60 x 7 + 9 x 7

In other words, we can break each number up into a sum of terms, one 
term for each digit, and the product will be the sum of all possible 
products of a term from one number and a term from the other. How can 
we do that easily? A multiplication table does the same thing - a 
table of all products. So let's make a multiplication table for these 
terms:

                      400     60     9
                   +-------+------+-----+
                   |       |      |     |
                   | 12000 | 1800 | 270 | 30
                   |       |      |     |
                   +-------+------+-----+
                   |       |      |     |
                   |  2800 |  420 |  63 | 7
                   |       |      |     |
                   +-------+------+-----+

(I've written the labels for the rows on the right rather than the 
left, just because I know that they would be in the way on the left.)

Notice that the products along a diagonal from top right to bottom 
left have the same number of zeroes (the same exponent of 10). That's 
because when you go left a column and down a row, you add a zero and 
take it back off. So we can add the numbers along each diagonal line, 
then sum the results:

                      400     60     9
                   +-------+------+-----+
                   |       |      |     |
                   | 12000 | 1800 | 270 | 30
                   |       |      |     |
                   +-------+------+-----+
                 / |       |      |     |
               /   | 2800  |  420 |  63 | 7
             /     |       |      |     |
           /       +-------+------+-----+
         /       /       /      /
     12000 +  4600  +  690  +  63 = 17353

We can drop the zeroes and just write the product of the non-zero 
digits, and put the zeroes back in when we add:

                     4      6      9
                 +------+------+------+
                 |      |      |      |
                 |  12  |  18  |  27  | 3
                 |      |      |      |
                 +------+------+------+
               / |      |      |      |
             /   |  28  |  42  |  63  | 7
           /     |      |      |      |
         /       +------+------+------+
       /       /      /      /
     12      46     69     63

     12
      46
       69
        63
     -----
     17353

But we can make it a little easier by noticing that the first digit of 
each product will add into the second digit of the diagonal to its 
left. So we can split each product into two digits and draw diagonals 
to show this:

                       4      6      9
                   +------+------+------+
                   | 1   /| 1   /| 2   /|
                   |   /  |   /  |   /  | 3
                   | /  2 | /  8 | /  7 |
                   +------+------+------+
                 / | 2   /| 4   /| 6  / |
               /   |   /  |   /  |   /  | 7
             /     | /  8 | /  2 | /  3 |
           /       +------+------+------+
         /       /      /      /      /
        1      5      22     15      3

       1
        5
        22
         15
           3
       -----
       17353

Now if you just do the carries as you write the sum of each diagonal, 
you'll have the lattice method, with as little writing as possible:

                       4      6      9
                   +------+------+------+
                   | 1   /| 1   /| 2   /|
                   |   /  |   /  |   /  | 3
                   | /  2 | /  8 | /  7 |
                   +------+------+------+
                 / | 2   /| 4   /| 6  / |
               /   |   /  |   /  |   /  | 7
             /     | /  8 | /  2 | /  3 |
           /       +------+------+------+
         /       /      /      /      /
     1      7       3      5      3

You can see the same process at work in our normal method, which we 
can spread out as:

       469
     x  37
     -----
        63  <-- 7x9
       42   <-- 7x60
      28    <-- 7x400
       27   <-- 30x9
      18    <-- 30x60
     12     <-- 30x400
     -----
     17353  <-- (30+7) x (400+60+9)

(notice the same six products we had before), or write more compactly 
as:

       469
     x  37
     -----
      3283  <-- 7x(400+60+9)
     1407   <-- 30x(400+60+9)
     -----
     17353  <-- (30+7)x(400+60+9)

Have fun using the lattice!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/