I recently saw a video demonstration of “Egyptian Multiplication” in which the presenter described how both Egyptians and modern computers multiply using binary. It was presented like this:

- Write the problem down horizontally.
- Below the first factor, list the value of each place in binary (1, 2, 4, 8… etc).
- Circle the ones that compose your first number. If your number is 22, for example, you would circle 16, 4 and 2. All numbers can be written this way – or in any other base – but the neat thing about base 2 is that each place can only count once (or none) – so you never need to double-circle any number).
- Below the second factor, you create a doubling list, starting with the number itself. (If the number were 5, your list would read 5, 10, 20, 40… etc).
- Circle the numbers in the second list that are across from numbers that were circled in the first (below).
- The sum of those circled numbers is the product.

This is how computers multiply? This video was made in the 80’s. I am certain that now, if not then, there is more to it. Regardless, I like this because I talk a lot about multiplication strategies in my 5th grade classroom, modeling how multiplication works and what it means. There is so much that is mathematically powerful in asking and answering the questions, *Why did you do it that way? How do you know it works?* Ideally, this helps kids flexibly choose the method that’s most appropriate rather than always resorting to the standard American algorithm, which is most adult’s go-to but often the least efficient way. Take the case of 98 x 26, for example.

Of course, kids get creative with multiplying in this way if they understand what multiplication means – that it’s shorthand for making groups.

I asked myself about this binary multiplication the same questions I’m constantly asking my students: *Why does it work?* The video wasn’t meant to be transparent, opting instead for the ‘wow’ factor of a cool math trick.*

So what’s the relationship of this doubling method to more familiar ways of multiplying? Are there numbers this doesn’t work for? This seems ridiculously efficient and easy – is there some reason we don’t all multiply this way instead? Maybe it gets unwieldy if you use certain factors.

I saw that the connection between numbers that are across from each other is the key to the question of why this method works. In 22 x 5, for example, the pairs are 1-5, 2-10, 4-20… Looking at it, the second column is just groups of 5. First one, then two, then four, etc… doubling. Since every number can be composed in binary, then you can figure out how much 22 groups of 5 is by taking the sum of 16 groups, 4 groups, and 2 groups.

Isn’t decomposing numbers into smaller groups and then multiplying just what we do when we multiply in our own number system? How is this any different? I started to experiment by using this method with base 10 instead of base 2. I tried it (below), listing our own place values (1, 10, 100…) below the first factor and below the second, multiplying the factor by ten as opposed to 2.

There are two problems. First of all, unlike binary, each place can be occupied many times over – you can have 2 ones and 2 tens, for example, rather than just one of each. This muddies the method and adds steps. Above, I had to circle some numbers multiple times. Second, the size of the places are so far from each other, it sometimes doesn’t help at all to use the method.

If you want to multiply something by 67, for example, you are only using two of the places – the ones and the tens – and then you have to go through all the trouble of multiple copies of each value. Binary is so elegant because even small numbers are spread across many places. You could use base 3 for this method without too much trouble, but most of us are not as facile with tripling as we are with doubling, so it’s harder.

You can use this same binary method for dividing, incidentally, with a slight reversal of the steps. I’m wondering if I can or should present this 5th graders.

Wow, this is so interesting! Thanks for sharing this, and your thinking about it. I had to work out 67 x 12 myself to believe that it could easier than the standard American algorithm. I’m not sure it beats the distributive property here, though. 67 x 10 + 67 x 2 is pretty quick. I wonder how binary multiplication would play out with 3- or 4-digit numbers or larger. I imagine the doubling could become tedious, to the point where you’re more-or-less using the SAA to figure out the numbers in the second column.

I teach high school math, not elementary, but I am very interested in the idea of what is fundamental about the basic arithmetic operations. (And I *love* that an elementary teacher is thinking about math as much and as deeply as you do!)

I have been thinking a lot lately about the 2009 essay by Hung-Hei Wu http://www.aft.org/pdfs/americaneducator/fall2009/wu.pdf

in which he suggests that the primary thing to understand about all of the operations is that any of them can be broken down into a series of single-digit-by-single-digit operations. I would be curious to hear your take on this.

(And not to throw too many things in the stew, but this also leads me to think about what our goal is in teaching large-digit multiplication in the age of omnipresent calculators.)

Thanks for the though-provoking post. Keep up the good work!

[…] @AnaFoxC has a blog named Make Math. The third post for the Blogging Initiation is titled “Computer Multiplication” and the author sums it up as follows: “I recently saw a video demonstration of […]