We usually associate the term binary with computers, but it is, by no means, restricted to our PCs. It’s a whole numerical system that can be used for all kinds of applications dating back to ancient Egyptian times when complicated multiplication was performed way before computers were invented. The best part is, we know exactly how they did it, so you can learn, too.

Let’s start with the example of what 19 multiplied by 24 equals.

First, make a table with two columns. At the top of the left column, write in 19, and at the top of the right column, write in 24. Start a new row and in the left column, under the number 19, put a 1, and in the right column, put the number 24 under the first 24. Then we make a new row and double each number. So, 2 on the left, 48 on the right.

We keep doing this until the number we double on the left will exceed the number we wrote at the top of the left column. In our example, the top left number was 19, so we do the above method for 1, 2, 4, 8, 16, and then stop. The next number will be 32, which is more than 19.

Our table looks like this:

Next, we add up the numbers in the left column that equal 19. The only possible way is to add 16, 2, and 1 together. Now, highlight the rows with those numbers in them. Add the numbers in the right column that are highlighted. In this case, it’s 24 + 48 + 384, which is 456. And presto! That’s what 19 multiplied by 24 equals.

So where does binary come into this? Well, the left column counts up in base 2, as binary does. And the way we calculate 19 from the left column is similar to how a computer would do it via binary except that a computer counts in base 2, which only uses 1s and 0s. To calculate numbers higher than 1, it constructs a base 2 number where each position represents a larger number.

In base 2, each numerical position is twice as large as the last, compared to our number system, base 10, where each position is ten times as large. Much like how the number ‘456’ in base 10 represents “four hundreds, five tens, and six ones,” 101 in binary means “one four, no twos, and one one”–add them up, and you get 5.

Sound familiar? You did this above with the table. The leftmost column can be written as 10011 in binary–one sixteen, no eights, no fours, a two, and a one. Add them up, and you get 19. Just like how it’s done in binary.

While it’s not quite so easy to do this mentally, it’s still an awesome display of binary understanding by the ancient Egyptians. Who knows? Next time you’re without a calculator, this may come in handy.

Learn More

Ancient Methods of Multiplication

https://mathcurious.com/2019/12/17/ancient-methods-of-multiplication-the-egyptian-form-of-multiplication/