I have been enjoying James Tanton’s website. In this video, Tanton explains a foolproof method for creating Egyptian fractions:
See more posts on Egyptian math.
Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer (relatively) soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original post. Figure them out for yourself — and then check the answers just to prove that you got them right.
Alex made a poster of Egyptian-style fractions, from 1/2 to 9/10. Many of the fractions were easy. She knew that…
Therefore, as soon as she figured out one fraction, she had the answer to all of its equivalents.
She had the most trouble with the 7ths and 9ths. She tried converting these to other fractions that were easier to work with. For example, 28 has more factors than 7, making 28ths easier to break up into other fractions with one in the numerator.
Photo from Library of Congress via pingnews.
Archaeology professor Dr. Fibonacci Jones came home from a long day of lecturing and office work. Stepping inside the front door, he held up a shiny silver disk.
“Ta-da!” he said.
“All right!” said his daughter Alexandria. “The photos are here.”
They had to chase Alex’s brother Leon off the computer so they could view the images on the CD, but that wasn’t hard. He wanted to see the artifacts, too. Alex recognized several of the items they had dug up from the Egyptian scribe’s burial plot: the wooden palette, some clay pots, and of course the embalmed body.
Then came several close-up pictures of writing on papyrus.
Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original posts. Figure them out for yourself—and then check the answers just to prove that you got them right.
Would you like to study “the knowledge of all existing things and all obscure secrets”? That is how Scribe Ahmose (also translated Ahmes) described his mathematical papyrus. Ahmose’s masterpiece is now called the Rhind Papyrus, after Alexander Henry Rhind, a Scotsman who was one of the first archaeologists to make meticulous records of his excavations (rather than simply hunting for treasures). Rhind purchased the papyrus from an antiquities dealer in Luxor, Egypt, in 1858.
Ahmose’s writing included a huge table of fractions as well as story problems, geometry, algebra, and accounting. Can you solve any of Scribe Ahmose’s problems?
I have one last puzzle for those of you who are following my Alexandria Jones series on hieroglyphic math and the Egyptian scribe’s method of multiplication by doubling. Here is the “teaser” problem from the cover of the Sept./Oct.1998 issue of my newsletter:
One more Egyptian math puzzle (pdf, 53KB)
What we know about ancient Egyptian mathematics comes primarily from two papyri, the first one written around 1850 BC. This is called the Moscow papyrus, because it now belongs to Moscow’s Pushkin Museum of Fine Arts. The scroll contains 25 problems, mostly practical examples of various calculations. Problem 14, which finds the volume of a frustrum (a pyramid with its top cut off), is often cited by mathematicians as the most impressive Egyptian pyramid of all.
(In the last episode, Dr. Fibonacci Jones discovered a torn scrap of papyrus, covered with hieroglyphic numbers. He promised to teach his daughter, Alexandria, how the ancient Egyptian scribes worked multiplication problems using only the times-two table.)
Click on the image for a larger view. Translate the numbers for yourself before reading on. If you need help, read Egyptian Math in Hieroglyphs.
Hieroglyphs came first. They were carved in the stone walls of temples and tombs, written on monuments, and used to decorate furniture. But they were a nuisance for scribes, who simplified the pictures and slurred some lines together when they wrote in ink on paper-like papyrus. This hieratic writing — like some people’s cursive today — can be hard to read, so we are only using hieroglyphic numbers on this blog.
Download this page from my old newsletter, and try your hand at translating some Egyptian hieroglyphs:
Then try writing some hieroglyphic calculations of your own.
Edited to add: The answers to these puzzles (and more) are now posted here.
Read all the posts from the September/October 1998 issue of my Mathematical Adventures of Alexandria Jones newsletter.
She knelt down to whisper in the ear of her faithful dog Ramus. “In this ring, grad students carefully brush away another layer of sand. In the next ring, the artist sketches every piece as it is found.” She waved her arm. “And over there, our flashiest attraction — drum roll, please — the photographers shoot each shard of pottery from every possible angle. But where is the Master of Ceremonies?”
Alex and Rammy found Professor Jones near the back of the tent, talking to another student. While she waited for her dad, she looked through an assortment of numbered artifacts that were ready to be packed and sent to the museum.
Next time, a new adventure (sort of)…
I confess: I lied — or rather, I helped to propagate a legend. Scholars tell us that the Egyptian rope stretchers did not use a 3-4-5 triangle for right-angled corners. They say it is a myth, like the corny old story of George Washington and the cherry tree, which bounces from one storyteller to the next — as I got it from a book I bought as a library discard.
None of the Egyptian papyri that have been found show any indication that the Egyptians knew of the Pythagorean Theorem, one of the great theorems of mathematics, which is the basis for the 3-4-5 triangle. Unless a real archaeologist finds a rope like Alexandria Jones discovered in my story, or a papyrus describing how to use one, we must assume the 3-4-5 rope triangle is an unfounded rumor.
[In the last episode, Alexandria Jones discovered a mysterious treasure: three wooden sticks, like tent pegs, and a long loop of rope with 12 evenly spaced knots. Her father explained that it was an ancient Egyptian surveyor's tool, used to mark right angles.]
“Geometry,” Fibonacci said.
“Geo means earth, and metry means to measure. So geometry means to measure the earth. That is what the Egyptian rope stretches did.”
Alex thought for a moment. “So in the beginning, math was just surveying?”
[In the last episode, Alexandria Jones, daughter of the world-famous archaeologist, caught her father's arch-enemy trying to uncover the Pharaoh's Treasure.]
…”I can’t believe it!” Simon Skulk threw down the last stone in disgust and walked away. At the mouth of the cave, he turned back and shook his fist. “You haven’t seen the last of me, Alexandria Jones.”
Her muscles aching, Alex sank to the ground and hugged her dog. The she gave him a little push toward the front of the cave. “Rammy, go get Dad.”
Ramus barked once and took off running.
Alex turned back to look at the Pharaoh’s Treasure. Where the last stone had stood was a hole. In the hole lay three wooden sticks, like tent pegs, and a long loop of rope with 12 evenly-spaced knots.
What could it be?
Alexandria Jones stood outside her father’s tent. The glare of the sun on the rocky desert hurt her eyes. Holding up a hand to shield her gaze, she spotted her dad (the world-famous archaeologist) arguing with the foreman.
Poor Dad, she thought. He was sure this was the right site, but so far he’s found nothing.
She looked down at her feet, where her faithful dog Ramus waited, panting. “Well, Rammy, it looks like Dad will be busy for while. What do you say? Shall we go exploring?”
Alexandria ducked into the tent for her backpack and canteen.
Thump! Something bounced against the side of the tent. Ramus barked.
Alex stepped outside and looked quickly around. No one was in sight. She saw a fist-sized rock beside the tent, with a note tied to it. She picked it up and read:
Ha! The real Pharaoh’s Treasure lies under a pyramid of stones, and it’s mine. You can’t stop me this time! —Simon Skulk