The Big 6 Civilizations (pt 2: Egypt)

1024px-Egypt.Giza.Sphinx.02

2. Egypt

I know I don’t have to tell you about Egyptian civilization, but did you know the Egyptians had maths?

Problem number 56 from the Rhind Mathematics Papyrus (dated to around 1650 BC):

Egyptian seked
Seked of the Great Pyramid

If you construct a pyramid with base side 12 [cubits] and with a seked of 5 palms 1 finger; what is its altitude?[1]

Most Egyptian geometry questions appear to deal with more mundane matters, like the dimensions of rectangular fields and round granaries, rather than pyramids. (The Egyptians had not yet worked out an exact formula for the area of a circle, but used octagons to approximate it.)

 

Picture 4A “pefsu” problem involves a measure of the strength of the beer made from a heqat of grain, called a pefsu.

pefsu = (the number of  loaves of bread [or jugs of beer]) / (number of heqats of grain used to make them.)

For example, problem number 8 from the Moscow Mathematical Papyrus (most likely written between 1803 BC and 1649 BC, but based on an earlier manuscript thought to have been written around 1850 BC):

Example of calculating 100 loaves of bread of pefsu 20:
If someone says to you: “You have 100 loaves of bread of pefsu 20 to be exchanged for beer of pefsu 4, like 1/2 1/4 malt-date beer,”
First calculate the grain required for the 100 loaves of the bread of pefsu 20. The result is 5 heqat. Then reckon what you need for a des-jug of beer like the beer called 1/2 1/4 malt-date beer. The result is 1/2 of the heqat measure needed for des-jug of beer made from Upper-Egyptian grain.
Calculate 1/2 of 5 heqat, the result will be 212. Take this 212 four times.
The result is 10. Then you say to him:
Behold! The beer quantity is found to be correct.[1]

“Behold! The beer quantity is found to be correct,” is one of the most amusing answers to a math problem I’ve seen.

Picture 5The Egyptians also used fractions and solved algebraic equations that we would write as linear equations, eg, 3/2 * x + 4 = 10.

But their multiplication and division was really weird, probably as a side effect of not yet having invented a place value system.

A. Let’s suppose you wished to multiply 9 * 19.

B. First we want to turn 9 into powers of 2.

C. The powers of 2 = 1, 2, 4, 8, 16, 32, 64, etc.

D. The closest of these to 9 is 8, and 9-8=1, so we turn 9 into 8 and 1.

E. Now we’re going to make a table using 1, 8, and 19 (from line A), like so:

1        19
2        ?
4         ?
8         ?

F. We fill in our table by doubling 19 each time:

1        19
2        38
4         76
8         152

E. Since we turned 9 into 1 and 8 (step D), we add together the numbers in our table that correspond to 1 and 8: 19 + 152 = 171.

Or to put it more simply, using more familiar methods:

9 * 19 = (1 +8) * 19 = (19 * 1) +(19 * 8) = (19 * 1) + (19 * 2 * 2 * 2) = 171

Slab stela of Old Kingdom princess Neferetiabet (dated 2590–2565 BC), with number hieroglyphs
Slab stela of Old Kingdom princess Neferetiabet (dated 2590–2565 BC), with number hieroglyphs

Now let’s do 247 * 250:

The closest power of 2 (without going over) is 128. 247 -128 = 119. 119 – 64 = 55. 55 – 32 = 23. 23 – 16 = 7. 7 – 4 = 3. 3 – 2 = 1. Whew! So we’re going to need 128, 64, 32, 16, 4, 2, and 1, and 250.

Let’s arrange our table, with the important numbers in bold (in this case, it’s :

1       250
2        ?
4         ?
8         ?
16       ?
32       ?
64       ?
128      ?

So, doubling 250 each time, we get:

1       250
2       500
4       1000
8        2000
16     4000
32     8000
64     16,000
128    32,000

Adding together the bold numbers in the second column gets us 61,750–and I probably don’t need to tell you that plugging 247 * 250 into your calculator (or doing it longhand) also gives you 61,750.

The advantage of this system is that the Egyptians only had to memorize their 2s table. The disadvantages are pretty obvious.

Berlin Papyrus
Berlin Papyrus

See also the Lahun Mathematical Papyri, the Egyptian Mathematical Leather Roll, the Akhmim wooden tablets, the Reisner Papyrus, and finally the Papyrus Anastasi I, which is believed to be a fictional, satirical tale for teaching scribes–basically, a funny textbook, and the Berlin Papyrus 6619:

The Berlin Papyrus contains two problems, the first stated as “the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other.”[6] The interest in the question may suggest some knowledge of the Pythagorean theorem, though the papyrus only shows a straightforward solution to a single second degree equation in one unknown. In modern terms, the simultaneous equations x2 + y2 = 100 and x = (3/4)y reduce to the single equation in y: ((3/4)y)2 + y2 = 100, giving the solution y = 8 and x = 6.

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20 thoughts on “The Big 6 Civilizations (pt 2: Egypt)

  1. Once one has read any amount of history about the history of mathematics (assuming one has a halfway decent understanding of the time scales involved), the notion in some education theories that kids can “discover” math (beyond the most basic arithmetic and geometry) seems quite absurd. I’ve read and listened to people who seemed to think that any random kid could be the next Gauss or Ramanujan, and yet somehow, even though I was one of the top girls in my state in math, the most I was able to figure out was the multiplication tables at age six, and only after my mom showed me how to arrange buttons for the first few products. (No, I realize this was pretty clever, but if I believed the education professors, I’d think there was something wrong with me for needing that much introduction, or not going farther on my own… or maybe it was just because I’m a girl. Wait…)

    (That said, there’s a lot I probably could’ve done if the teachers had stepped back and let me, rather than making me cut and paste stupid posters all those years… I would’ve loved the mythical classrooms where everyone sat in rows memorizing facts…)

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    • Most people can’t figure out how to accurately draw faces without explicit instruction. People who can rediscover the calculus or set theory from scratch are exceedingly few and far between.
      Folks want to come up with an education method that will transform everyone into geniuses. I can’t blame them. I’d like to be a genius, too. 🙂
      When it comes to education and reform, I tend to trust the already established teachers, who have far more practical experience trying to teach things to kids than I do.

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      • re: “When it comes to education and reform, I tend to trust the already established teachers, who have far more practical experience trying to teach things to kids than I do.” evolutiontheorist

        You dream. Brief story follows:

        My son in last year of college was taking Linear Algebra which was an elective for engineering students of which he was one and now has a MS and Ph.D from a top 20 in the world University. The Linear Algebra course was a hard course only taken by dedicated STEM students. Early in the course the first exam was returned to the students at the beginning of class by the professor. He gave them back in the order from the highest score to the lowest, perhaps to see who the students were by their earned ranking. When he came to the last six exams he did not call their names and give the exams back but said that at the end of class for the students who had not received their exam to come forward and meet with him.

        At the end of class, most of the students left the room but my son stayed behind to ask a question of the professor. He was accompanied by a few others. But before those students could meet with the professor the professor had to deal with the six students who had not received their exams. The professor addressed them by saying:

        Why are you here taking this course? Out of the six a voice said that they were Education Majors who were studying how to be High School science and math teachers. Their advisor told them that they should take one math course from the University Math Department during this their senior year. (All of their math and science had been taught in the School of Education by professors of education.) The advisor checked the catalogue of available courses and found Linear Algebra and said that since the students were already trained in algebra that Linear Algebra would be a good course to take so the group enrolled in this LINEAR ALGEBRA course with the blessings of their advisor.

        The professor replied by saying that of the group of education majors the highest score on the test was a 6 out of 100. He continued and said that they should immediately drop the course as there was no way any of them could pass it as the subject matter was only going to get more and more difficult from here on out.

        One of the students piped up that if the math professor would SIMPLIFY and EXPLAIN the subject they would be able to learn the concepts. The professor replied that they had demonstrated to him that they lacked the ability to understand the concepts being covered and that they could stay but that a LOW Failing grade was the best that they could hope for and that they should heed his advice. The group never returned.

        Teachers like those in that group are the type of “science & math” teachers that American students encounter in the Teacher’s Union run High Schools of the USA. It is nearly hopeless unless the student has a parent or tutor that can supplement what is NOT being taught.

        Dan Kurt

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      • The people creating Common Core math standards haven’t passed linear algebra, either, so this does not refute the original point: I trust someone with multiple decades’ experience teaching a subject to know how to teach it effectively more than I trust someone with no experience teaching that subject.

        For that matter, the vast majority of American students will never need nor be smart enough to take linear algebra.

        I learned math by buying textbooks and reading them. The fact that anybody with access to a library, paper, and pencils can learn math is one of the reasons I love it.

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      • Sigh, yeah… “Funny” story… When I was in high school, I got to take classes at the nearby flagship state university. The summer between my junior and senior years of high school, I took linguistics 101, because I’d already spent so much time reading linguistics books from the library, it seemed time. The class was easy for me (One graduate student TA pulled me aside and made sure I was planning going somewhere better for college.) Also in the class were a few education majors a few credits away from graduating. At this time, I had just finished pre-calc and was about to start AP calc back at my high school in the fall, so nothing special, but then I heard them talking about finishing up pre-calc in order to graduate college…

        My husband and I happened to be rearranging our bookshelves last night, and discovered that we had two different editions of the same Linear Algebra textbook. That was the math class I took before graduating college. That said, I took the “easy” one, but the lowest math class offered where I went was introductory calculus. Incidentally, a lot of Education math people think that it counts as linear algebra if you have a few simultaneous equations or a matrix of something. Likewise, in elementary school levels, they have “algebra” or “algebraic thinking” which they use to make naive parents think that they’re oh-so-advanced (kind of like having kindergartners write “reports”…) and some of it might be actual baby-algebra like “2 + __ = 5”, or manipulative games that can be analyzed as having algebraic concepts, but are age-appropriate, but to the extent things are age-appropriate for an average-to-bright elementary student, it’s not, in my opinion, worth calling algebra, and to the extent that it’s worth calling algebra, it’s better to focus on getting basic math facts solid (whether with games or traditional drills, the latter of which many kids do actually find fun if the teacher doesn’t completely ruin it with their attitude… aside: girls who are promising engineers/scientists/stem-whatever find math drills fun unless the teacher ruins it for them… Second aside: “traditional” math hasn’t happened in many elementary schools for at least 30 or 40 years. Today’s teachers were likely taught by teachers who were likely taught in ed school that traditional math was “bad”…)

        (That said, I know quite a few high school level math teachers who originally trained as engineers but wanted something a bit more laid back… Middle school is a bit of a mix of people who actually like math teaching it and education majors looking for a job, and elementary is where you really need to watch out. There are some good teachers who will actually teach your elementary kids math, and some average teachers who will have materials that won’t trip up average-to-bright students learning math… And, yeah, the most gifted kids can figure out enough on their own to do ok, but the “just gifted” might not figure it out soon enough to, say, be a physics major in a strong program, and we still need technicians and engineers from the “average to just bright” category who don’t have to relearn algebra in college…)

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      • re: E’s comment August 1, 2016 at 1:05 pm

        Nice to see you got the gist my post. My own story in High School took place just after Sputnik. The Feds sent out much money to promote science eduction. In the Eastern city I was raised in a program was instituted that funded for free HS students to take in the evening an electrical engineering course Tuesdays and Thursdays for a semester titled Circuits 1 and if a student passed gain 3 college credits. One had to get permission from a student’s High School and self transport to the class taught by a professor from the city’s premiere world renowned college of engineering. I was the only student who took the course from my Catholic HS. In all there were a bit over 180 of us that started the course. I was a second semester junior at the time. The professor took no prisoners and only 14 or 15 of us were present when the course ended. I received a letter later that summer with my first three college credits which I never used as I went to college and earned a chemistry degree not one in engineering. the real lesson I learned taking the course was that there was no hand holding in college by the professors unlike the hand holding there was in HS as it was truly sink or swim. Ironically in the decade to come the Viet Nam war arrived and the draft could be avoided by going to college so standards invariably dropped as entering students came in numbers that necessitated falling standards because of the tyranny of the IQ curve, that is, accepting more and more from the leftward direction of the curve. In the decades since, more and more students attend college and standards continue to be lowered and content diluted with the curse of grade inflation reaching preposterous levels.

        Dan Kurt

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    • Common core is apparently trying to teach kids that you can take numbers apart and put them back together in different arrangements, eg, 45 + 55 = 40 + 5 + 50 + 5, etc. I understand where they’re coming from on a theoretical level (yes, it is good to understand that big numbers are made up of smaller numbers,) but as a practical matter, this approach confuses the kids.

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      • When I was taking a course in arithmetic for math teachers (in a program aimed at actual math teachers which required taking a calculus class as well, not a “math for elementary school teachers” class, and this needs to be clarified…) “common core” wasn’t yet “a thing” but the theoretical and political issues with math education were already decades old… anyhow, we did have some exercises which were basically recreating ancient methods, which was useful in this context because (a) if you’re teaching math, it’s useful to know each level inside out, upside down, and sideways, and (b) with arithmetic in particular, it’s easy to forget what it’s like not knowing what you know… however, I think what may have happened is that some teachers and quite a lot of education professors forgot these points, and just thought these exercises would make good things to teach kids. Add in lab schools full of professors’ kids who can be subjected to these fads and still do just fine, and you end up with the past few decades…

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  2. Is agriculture a prerequisite for math, and if so, is it because of the excess calories allow for specialists, because agriculture is inherently mathematical, or for some other reason/combination of reasons?

    Also, OT: I picked up a copy of “In the Shadow of Man” last week.

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    • I think trade is basically a prerequisite for math. In general, you need to have enough goods to make trade viable, and someone to trade with. (Thus a herding society could, I assume, also develop some math.) Trading with family members isn’t enough because most people don’t keep exact records of what they gave to their families, but just expect a similar trade in the future. But once you start having to do things like calculate how many bushels of wheat in taxes to expect out of a province based on the number of acres under cultivation in order to predict how many workers you’ll be able to hire next season to work on your pyramid, then you’re really in the math.

      Great!

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      • That’s a great point about trade. In prohibiting writing on the Sabbath, the Talmud prohibits trade (and measuring and weighing things) since it inevitably leads to writing. It’s not stated outright, but the mechanism by which that local iteration occurs probably matches the longer evolution of mathematical thought and writing. Did the North American Indians have any sort of math? If I recall correctly, they had significant trade, but no agriculture (maybe horticulture?). I’m thinking mostly of the northeastern tribes (Iroquois and Algonquin), but I don’t think any of the tribes had a written language.

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      • Aztecs had agriculture, monumental architecture, trade, calendar, writing, and math. Farming of corn, beans, and squash was IIRC widespread everywhere the climate was suitable or could be modified (ie small dams for holding water in the southwest.) Hopi/Pueblo peoples built cities; Mississippi culture IIRC had monumental art (snake mounds). I think hunter-gathering persisted in the Pacific northwest due to the abundance of wild food like salmon; I have a favorable impression of the level of development among the Iroquois/other similar tribes of the northeast, but no knowledge of their having writing. The Cherokee developed writing almost immediately after being exposed to the idea (one of tribesmen invented his own syllabary) and soon had their own publishing industry. I have read that the Cherokee sent relief aid to the Irish during the potato famine.

        But North America lacked good domesticated draft animals for pulling carts (and I don’t recall anything like a paper analogue,) which is definitely a problem for doing much trade/math.

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  3. Interesting facts about the “secrets of the pyramids”. How the pyramids were made is no longer a secret. Joseph Davidovits has come up with a material called geopolymers. They’re the equivalent of plastic polymers but they’re made of minerals. They look just like stone blocks when they’re hardened. If you look at the outside blocks of the pyramids they fit together perfectly. Inside the pyramid they are rough except for the hallways and galleries. So only the outside and covering were cast in geopolymer. Its also been speculated that the whole upper section was cast in order to make it easier as the geopolymer material can be carried in baskets then put into molds. Fair warning Egyptologists don’t like this explanation but to me it’s the only explanation that makes sense. It also explains the delicate vases and sculpture made of Dorite an immensely hard stone. They were cast just like clay. This would also account for the perfect edges on the various caskets, tool marks(they are actually wood mold marks). Where people claim that advanced machining has taken place is actually holes drilled while the casting stone was not completely hardened. One very strong piece of evidence that the stones were cast is the fossils in the stones. In pyramid blocks the fossils are all mixed up in terms of orientation. In a normal block the fossils are laying flat like they landed when they died on the sea floor.

    http://www.geopolymer.org/archaeology/pyramids/are-pyramids-made-out-of-concrete-1

    There’s a video on the above link that shows him making limestone blocks.

    There’s intense interest and research in geopolymers. It’s not there yet but they will have a huge influence on building and material science some day. The cement made this way takes a LOT less energy, last as long as the pyramids and being a polymer can be tuned chemically in an almost infinitely number of ways just like plastic.

    Here’s some pictures of Diorite statues and vases.

    And the probably cast Colossi of Memnon made of super hard quartzite.

    https://en.wikipedia.org/wiki/Colossi_of_Memnon

    http://www.geopolymer.org/archaeology/civilization/colosses-of-memnon-masterpiece-by-amenophis-son-of-hapu/

    All this stuff would be almost impossible to carve delicate writing as it’s so hard. How would you do vast work like this and make no mistakes? If it were cast you would only have to make raised inserts then cast the statue and remove the inserts. Simple.

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