A while back I was making my way through the “hills and hollers” of one of the nearby states on my way to a meeting of mathematics teachers. It is a beautiful state full of friendly people and great scenic beauty, but I will keep its identity a secret for reasons that will become clear in the sequel.
I’d gone to this particular annual meeting a number of times, but this time in order to break the routine I’d decided to take a new route, the road less traveled as it were. As so often happens when you take the road less traveled, at some point I became lost. I don’t know how. I might have been adjusting my radio and missed a turn off or, more likely, I might have just had my mind on something besides driving. This is one of the hazards of being a mathematician.
Regardless, the road I was on gradually deteriorated from paved to gravel and from gravel to dirt. I thought about turning around, but, by the time that notion made its way into my head amongst the theorems I’d been thinking about, the road was too narrow for me to do this.
Eventually, I came to a house and pulled onto its lawn. I put it this way because it had no formal driveway just a lawn with various types of broken-down pieces of equipment scattered across it. The house itself was unpainted and ramshackle, but there was a friendly looking, elderly man on the front porch, with and old hound dog by his side and another at his feet. Instead of just turning around, I decided to stop and ask him directions, thinking maybe there was a way to where I was going that didn’t require a lot of backtracking.
I had gotten out of my car and was walking up to the porch in order to ask directions when an ancient pickup truck pulled into the yard behind me effectively blocking me in.
There were three men in the pickup each of whom looked scarier than the others. They were dressed in overalls without shirts underneath and wore worn out cowboy hats. I swallowed hard when I saw them and became acutely aware there was no means of escape. Having seen the movie Deliverance, I had no desire to live it. I fought to keep the sound of banjo music out of my head.
However, my worries were groundless. Except for a sideward glance from the driver as he approached the porch to talk to the old man, they ignored my presence entirely.
“There was a dog food truck wreck up by the interstate,” the driver said. “We managed to get five hundred-pound sacks off before the highway patrol arrived.”
He went on to explain how, because of certain familial obligations, the dog food had to be divided among a total of seven brothers in the family.
“We got a bunch of sacks,” he said to the old man, “and we come to you because we know you can divide it up amongst us fair.”
The old man was about to speak, but my desire to be helpful came out as I saw an opportunity to put my mathematics to work, and I spoke.
“I am a math teacher,” I said. “What you need to do is this. With five bags to be divided among seven brothers, each brother should get a five-seventh share. Take two-sevenths of out of the top of each bag. This will leave five-sevenths in five bags. Give those bags to five of the brothers and divide the portion taken out between the other two.”
It was at this point, the old man who’d been silent so far began to speak. His voice was kindly and ancient.
“Well,” he said, “I can tell you know your cipherin’ but there are some problems with that. Let me show you one of them.”
He reached into one of the chest pockets of his overalls and removed two new plugs of chewing tobacco.
“Come over here boys,” he said to the other two. “I am going to divide this tobacco amongst you three.”
He proceeded to carefully cut the bottom third from each of the tobacco plugs. He gave two-thirds of a plug to each of the pickup’s passengers and gave the two remaining one-third size pieces to the driver.
“Hey,” said the driver, not seeming happy at all, “you gave them bigger pieces.” He began to snarl and turned toward the other two.
“You gave him more pieces,” said one of the others as he turned to face his brother.
“All right,” said the old man as he spat from his own chaw. “Give me back the t‘backy.”
They did as he said, and he cut the big pieces in half so that everyone had a pair of one-third size pieces.
I saw his problem immediately, and it wasn’t mathematical. It was political. The men with whom he was dealing hadn’t had the arithmetical experience to recognize that quantities that looked different could be the same. On the other hand, they had a highly developed sense of justice, especially when they perceived they were not being treated fairly.
I decided to keep my mouth shut and let them proceed. What I saw was fascinating.
He rocked back and forth in his rocking chair with his eyes closed. His lips were moving and by leaning close I could make out what he said.
“Five among seven. That will be a part of two, a part of seven, and a part of fourteen.”
I wasn’t quite sure what he said, but apparently the men from the pickup did. They went to the back of their pickup and cleared the sacks of dog food and other debris from the bed. They then began dividing the dog food. They divided the contents of four of the five sacks in half and put seven of the eight resulting halves into separate sacks. The eighth half they divided into seven equal piles and put each of those into a separate sack. The remaining hundred-pound sack of dog food was divided into seven equal piles and each of those piles was transferred into a bag.
There were then seven sets of three bags, one containing half a bag, one containing one-seventh of a bag, and one containing a fourteenth of a bag. Everyone would have the same number of bags and each of the brothers would have one bag of a particular size.
They thanked the old man and left, but before they did, I got directions from them to where I was going.
Since I didn’t want to get lost again, I waited until I got to my destination to think about how the old man solved the problem.
He’d used Egyptian Fractions, which are also known as unit fractions because if they are written in standard notation there is a one in the numerator.
Having the advantage of modern measuring devices, we could approach the problem of dividing up that dog food by noting there were five-hundred pounds of dog food altogether and giving each brother one-seventh. This could be done by weighing out approximately 71.4 pounds.
The ancient Egyptians didn’t have the advantage of this technology and had to solve the problem in a different way. One example this can be found is in the Rhind Papyrus which dates from about 3500 years ago.
In modern notation, the old man had used the fact that . Writing it this way is misleading, because it presumes a modern way of thinking. The Egyptians weren’t thinking about writing a fraction as a sum of unit fractions; they were thinking about dividing up quantities equally. While their problems and methods of solving them eventually lead into algebra and the modern way, they were living in a different milieu.
It is interesting to note there may be several ways to resolve a modern fraction into an Egyptian fraction. For instance we can also write . However, the other way is better than this in the sense of having fewer terms and smaller denominators. Having fewer terms and smaller denominators would result in an easier division of the quantities as described in the story. The author of the Rhind Mathematical Papyrus usually had the best way to do it.
No one knows how he did it.