Lattice Multiplication ExplainedDate: 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/ |
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