## Archive for June 10th, 2009

### The First Bunch of Ways to Multiply

June 10, 2009

I blithely mentioned in yesterday’s post that I only knew about 13 ways to multiply [“only” because it would be great to write a book called Twenty-five ways to Multiply], and then Jason asked me to list them.  I was originally going to list them all, but then I started describing them which is taking a lot longer and I won’t have time to watch the next episode of Heroes on Netflix prepare for a committee meeting if I do that all today so I’ll do it in steps and you can read all about multiplication for a few days!   I’ll see if I can get Godzilla do to some demonstrations of the more complicated methods tomorrow.

These are carefully ordering according to The Order In Which I Thought of Them.

(1) Doubling and Halving, as described yesterday (used in Egypt, Ethiopia, and presumably Russia).  This is one of my favorite methods, because it’s surprising that it works.

(2) Duplation, which is a variant on the above method, in which you start with 1 and one of the the numbers and double both.  For example, to multiply 14 and 12 you’d start with 1 & 12, and double both until the left-hand column was going to be bigger than the other number, in this case 14.  That’s not at all clear, is it?  Here’s what I mean for 14×12:

1 & 12
2 & 24
4 & 48
8 & 96
16 & <— Oh, I can stop because 16 is bigger than 14.

Since 14 can be written as 8+4+2, put a little mark by those rows:
1 & 12
*2 & 24
*4 & 48
*8 & 96

and add up the corresponding numbers on the right:  96+48+24=168.  And there’s your product!  What you’re really doing is adding the appropriate doubles (96 is 8 12s, 48 is 4 12s, and 24 is 2 12s so when you add them you get 14 12s, as you wanted).  This was also used by the Ancient Egyptians, and it’s referred to on the video from yesterday.

One thing that neat about these methods is that all they use is adding, subtracting, doubling, and/or halving.   The folk who used them had to remember the process, but they didn’t have to memorize 45 separate single digit multiplication facts.

(3) and (4) Mesopotamian Multiplication.  Like the Egyptians, the Babylonians broke down multiplication into addition, subtraction, and halving.  They had one more trick, though:  they had tables that contained the squares of all the numbers from 1 to 60 [which was all they really needed, since they used a Base 60 system].   So to multiply a and b they used one of two formulas:

$\frac{\left( a+b \right) ^2 - a^2-b^2}{2}$ or $\frac{\left( a + b \right) ^2 - \left( a - b \right) ^2}{4}$

In other words, to find 14×12, to use the first formula you’d look up 262, 142, and 122 in your table, subtract the last two numbers (196 and 144)  from the first one (676), and then cut that answer in half.  With the second formula you’d look up 262 and 22 in your table, take the difference, and cut that in half twice.

This method totally amuses me because although you can check the algebra to make sure it works (indeed, I usually introduce this by asking students to come up with a formula for ab that uses only addition, subtraction, doubling, halving, and a2, b2, (a+b)2, and/or (ab)2), it’s really an ancient plug-and-chug method.  You don’t have to think at all about why it works, you just do it.

Incidentally, if you walpha a number, Wolfram Alpha will give it to you in Babylonian symbols.  Here’s  34.  But watch out: it’s sometimes wrong.

Edited 7/8 to add: Although I read in one book (I’m not sure which now; it’s been several years) that this was done by the Babylonians, Victor J. Katz’s A History of Mathematics doesn’t match that.   I’m not quite sure what to believe now.

(5) Grid or lattice multiplication.  A lot of schools are teaching this today because, although it takes a little while to draw the boxes, you do all your single digit multiplication first and then all your addition.  I’ve heard people complain about it being a new-fangled method, and I like to point out that it’s actually about a thousand years old (Doesn’t that sound snarky?  I try to say it in a friendly way.  I probably fail.).

It’s possible that it originated in India (I’ve seen sources claim this, but I think it’s not considered absolutely certain); it certainly appeared in Arabic books before it made its way to Western Europe.

Here’s how it works.  Suppose you want to multiply 345 and 12, which are conveniently the numbers that I found in this Wikipedia example.  You have a 3-digit number and a 2-digit number, so you make a 3×2 array of boxes.  Put the first number (345) on the top, and the second number (12) on the side.  One of those is the multiplier and one is the multiplicand, but I can never remember which is which.

The picture above also shows the next step:  divide each little square in half, and then multiply each pair of digits.  For example, 5×2 is 10, so you write the 1 above the diagonal and the 0 below.

For the next step, you add along the diagonals, carrying as necessary.

The reason this works is that the digaonals automatially take care of place value.  For example, look at the third diagonal, with the purple arrow (that ends up in the hundreds place).  It has the 4 of 4×1, which was really 40×10=400.  It has the 0 in the tens place of 4×2, which really would stand for the hundreds place of 40×2.  It has the 6 of 3×2, which really comes from 300×2=600.  Plus it has anything that was carried from the previous diagonal.

Writing the total flat along the bottom makes more sense if you’re teaching it to kids, but in some older books people wrote the answer on the bottom and left-hand side, like on this example from page 23 of The Treviso Arithmetic (Arte dell’Abbaco), an Italian textbook [published in the town of Treviso in Northeastern Italy] from 1478.  This example shows that 934×314=293,276.

When I do this in class I often decorate the sides when I’m done, because I’ve seen that done before.  Then it looks like this:

(5½) There’s a variation of the above method in which the diagonals go in the other direction but you write the number on the right “upside down”.   Here’s an example from the same page of the Trevisio, again showing 934×314=293,276.

I didn’t want to count this as a separate method, because it has all of the same principles as the previous grid.  But I wanted to mention it because if you look at it, it’s a small step to getting the next version of 934×314, shown below, in which the boxes above have had the diagonals removed but you still add along the diagonal (although the total is written at the bottom):

This is actually a little bit confusing because with the diagonals removed, the numbers don’t quite line up the way they’re supposed to.   It’d be clearer if the rows were shifted a bit, like this:

And THAT looks an awful lot like “traditional” multiplication, where in this case “traditional” means “the way I learned multiplication in in 1978”.  But this isn’t a new-fangled method  either:  this “traditional” multiplication ALSO appeared in the Trevisio!  And that leads us to….

(6) Multiplication like I first learned it, like this:

Phew!  That was more lattice multiplication than you ever wanted to see, wasn’t it?  But isn’t it neat how all those pictures (all on the same two pages) just lead into one another, even though they actually were in the reverse order in the original book?

Incidentally, while we’re on the subject of The Trevisio it might amuse you to learn that this 500 year old book also contains the problem “If 17 men build 2 houses in 9 days, how many days will it take 20 men to build 5 houses?”.  If you’re wanting to get yourself a full copy, here’s a link to all 24 Mb of it!  Or, if you don’t read Italian, you can learn a little more about it in this 1996 column by Ivars Peterson.

Next up, some different ways that they don’t teach in school.

Edited to add: Here is The Second Bunch of Ways!