Okieinexile's Blog

By Bobby Neal Winters

We live in an age of technology. It is all around us. I used the word “technology” there, but that’s just one way of looking at it.  Maybe the word “culture” would be better, maybe “framework of interlocking systems.”

Maybe I will let someone else tell me what the word is and I will give you an example.  

Suppose that you had two sacks of coffee beans–each of the sacks having the same weight–and there were three customers who you wanted to divide them among.  How would you go about it?

Well, in our world of technology, or framework of interlocking systems, we would determine what the total weight of the contents of the two bags was.  We would then divide that weight by 3.  Then we would give each of our customers that weight in coffee beans.  To do this we have used the fact we have a system of weights and measures, we have used the technology of a scale with a numerical (either digital or old-fashioned analog), and we have used the ability to divide numbers by three.  

That last part is not something to be sneezed at.  Unless you are really lucky, you will need to make use of decimals, and the decimal system is a mathematical technology.

Take away the system of weights and measures; take away the scale; take away the decimal system. Then this is a different problem.

However, this sort of problem is not new. You could imagine the ancient Egyptians doing something of the sort, such as in the Biblical story of the division of the grain among Joseph’s brothers.

As an Egyptian grain merchant trying to divide up two bags of grain among, say, three customers in a way that was transparently fair, what would I do?

First thing, let me tell you what we are not going to do.  We are not going to just clump it all in a pile and divide by three.  When you have something separated out into equal piles, this gives you information. You don’t want to lose that information if you don’t have to.  You want to use it in some way.

Okay, back to the problem.  We’ve got two bags and three people.  If we had three bags, the problem would be solved.  

What if we had four?

If we had four, we could give each of the three customers a bag.  

Then it is just a matter of dividing that bag that is leftover by three.

So let’s take each of the two bags and divide it in half.  Then we have four equal quantities.   Then each customer has half a bag, and I have half a bag left.  I take the half a bag that I have left and divide it into three equal piles.  Each of these piles contains one-sixth of what was in one of the original bags.  So I’ve got 1/2 + 1/6 and if you do the math on that you will see that this is 2/3, i.e. two-thirds of a bag.  Just like before.

Modern mathematicians who refer to these sorts of numbers call them “unit fractions” because they are a sum of fractions with a one in the numerator.  I prefer the name Egyptian fractions because my colleague who photographs Egyptian hieroglyphics tells me this is how the Egyptians did fractions.

Notice that I’ve not used any modern-ish technology that the Egyptians wouldn’t have had, such as weighing on numerical scales. I’ve just used the ancient technology of dividing into equal looking piles.  

Children can do this.  

I said the quantities need to be transparently the same.  This technique yields equal piles that can be put into one-to-one correspondence.  That is to say, that each of the customers has a pile that is half a sack and each has a pile that is one sixth of a sack.  They don’t need to weigh it to know they have the same amount; they can see it with their own eyes.

This can be done with other numbers besides 2 and 3, but I will leave that to you.  If you have questions, catch me at the coffee shop sometime with pen and paper, and I will tell you way more than you want to know.

Bobby Winters grew up near Harden City, Oklahoma.  He teaches mathematics and computer science, does woodworking, and blogs at okieinexile.com.