EGYPTIAN MATHEMATICS - Brigham Young Universitywilliams/Classes/300W2012/PDFs/PPTs...EGYPTIAN MATHEMATICS ... Notice the intermediate numerators: 7>4>1. Fibonacci proved these always

EGYPTIAN MATHEMATICS

TIMELINE

Archaic Period (3100 - 2650 BCE) Old Kingdom (2650 - 2134 BCE) – Large pyramids

built; rich and productive period 1st Intermediate Period (2200 - 2050) BCE – Chaotic Middle Kingdom (2050 - 1640 BCE) – “Golden Age”

Moscow and Ahmes Papyri. 2nd Intermediate Period (1640 – 1550 BCE) New Kingdom (1550 – 1070 BCE) – temple building,

empire building. Ramses, Tutankamon, Moses.

TIMELINE

MESOPOTAM IA

EGYPTInt

Int

1000 BCE1500 BCE2000 BCE2500 BCE3000 BCE

New KingdomMiddle KingdomIntOld KingdomArchaic

AssyriaOld BabylonAkkadiaSumaria

TIMELINE

MESOPOTAM IA

EGYPTInt

Int

1000 BCE1500 BCE2000 BCE2500 BCE3000 BCE

New KingdomMiddle KingdomIntOld KingdomArchaic

AssyriaOld BabylonAkkadiaSumaria

EGYPT’S GEOGRAPHY

Relatively isolated, hard to attack, and therefore stable.

Populated along the Nile, which makes life in Egypt possible.

The Nile floods predictably every July (when Sirius rises), which provides silt and nutrients to the rich soil along the banks.

EGYPT’S GEOGRAPHY

Egypt subsisted on organized and centralized farming in the area flooded annually by the Nile.

Tracking and managing the allocation of land required extensive record-keeping, measuring, and written language.

And, in particular, a calendar.

A SIDE NOTE ABOUT ASTRONOMY

Ancient peoples of both Mesopotamia and Egypt could easily track the movement of the celestial sphere as it revolved around the earth every year. They could also track the movement of the sun in the ecliptic against the celestial sphere. Finally, they could plot the changes in the moon. This gave them four cycles to keep track of: years, seasons, months, and (of course) days.

EGYPTIAN CALENDARS

The beginning of the Egyptian calendar year was when the Nile was predicted to flood, July on our calendars. When Sirius was visible just above the horizon at dawn, the flood was imminent.

Like most calendars, there was some coordination of the cycle of the sun and the moon.

EGYPTIAN CALENDARS

The earliest Egyptian calendar had 12 months, alternating 29 days and 30 days. The actual cycle of the moon is about 29 ½ days.

The “year” was therefore 354 days. So, every 2 or 3 years, an additional month was

added.

EGYPTIAN CALENDARS

The second Egyptian calendar had a 365-day year.

All 12 months were 30 days long. Then an extra 5 days was added at the end. This calendar worked better for tracking the

solar year, but the coordination with the moon cycle was lost.

THE EGYPTIAN SEASONS

The year was divided into three seasons, as suited what was important: Inundation (the flooding of the Nile) Emergence (of the crops)Harvest

BACK TO RECORD-KEEPING, MEASURING, AND WRITTEN LANGUAGE

We all know that Egypt developed a pictorial writing system known as hieroglyphics.

HIERATIC TEXT

A more cursive form of hieroglyphic, the result of quickly drawing signs by hand on a sheet of papyrus with a reed brush. Here the original hieroglyphic signs were reduced to their simplest form.

Used from around 3000 BCE.

DEMOTIC TEXT

Evolved from an even more cursive form of hieratic and became the standard for the administration from the 25th or 26th Dynasty on (around 600 BCE – 450 CE).

WRITTEN LANGUAGE

The Rosetta stone was discovered in 1799 as part of the building material used in the construction of Fort Julien near the town of Rashid (Rosetta) in the Nile Delta. It provided one key to our understanding hieroglyphics.

Jean-François Champollion did the first successful translation, building on the work of Thomas Young.

PAPYRUS

Egyptians developed a sort of paper made from the pith of the papyrus reeds growing on the side of the Nile.

These were made into long strips and then rolled and unrolled for use.


And thus we come to:

“How do we know about Egyptian Mathematics?”

Mostly from various papyrus scrolls that have been found and translated.


Two major sources:Moscow Papyrus (1850 BCE): volume of a

frustrum, other methods for finding areas, volumes.Rhind Mathematical (Ahmes) Papyrus (1650 <

2000 < 2650 BCE): multiplication and division by doubling and halving, use of unit fractions, fraction doubling tables, approximation of π, linear equations, use of cotangent, attempt to square the circle.


A few minor sources:Berlin Papyrus (1300 BCE): a quadratic equation

and its solution; simultaneous equations.Reisner Papyri (1800 BCE): volumes of temples.Kahun (Lahun) Papyrus (1800 BCE): 6 fragments;

much is still not translated. Egyptian Mathematical Leather Roll (1850 BCE):

Table of decompositions into unit fractions.

EGYPTIAN NUMERATION

This is covered pretty well in the text…

DON’T KNOW MUCH ABOUT HISTORY…..

A word about history – There are lots of versions of things.

The internet is a dangerous and wonderful thing.

“Wikipedia is the best thing ever. Anyone in the world can write anything they want about any subject. So you know you are getting the best possible information.”

–Michael Scott

DON’T KNOW MUCH ABOUT HISTORY…..

Books are also wonderful and dangerous things, for that matter.

Scholarship is about differences in opinion. Frog, tadpole, bird, burbot fish Scroll, snare, coil of rope. Astonished man, kneeling figure, scribe, prisoner,

Heh, the god of the unending.Heel, rope, hobble for cattle.

EGYPTIAN NUMERATION - FRACTIONS

For reasons unknown, the ancient Egyptians worked only with unit fractions, that is, fractions with a numerator of 1. The single exception was having a symbol for 2/3*. All other fractions were written as a sum of unit fractions: . So tables were provided

to help with this task.*(Maybe ¾, too. That’s history for you.)


Fractions were symbolized by putting the hieroglyph above the denominator. This may stand for “r” meaning an “open mouth.” No, I don’t know why that has anything to do with fractions.

The direction of the little man’s feet determines whether we add or subtract. (Sometimes just the feet were used for this symbol.)


Eventually, as hieroglyphic moved to hieratic, the open mouth became a line. Some suggest this was the beginning of our fraction bar.

EGYPTIAN NUMERATION – OUR VERSION

When we write Egyptian numbers, we’ll use our regular notation. For fractions, we’ll follow the Egyptian practice of writing them as sums of unit fractions.

is how we will write .

EGYPTIAN ARITHMETIC: ADDITION

We’re not sure how it was done. Tables? Memorization? Counting? Perhaps something like regrouping, based on the base-10 nature of the number system.

But it got done, and done pretty accurately, in the problems found in the Papyri.

Same for subtraction.

DIVISION USING FRACTIONS

“Reckon with 8 so as to get 19.” In other words, “What, times 8, will give you 19?”

8 1

Next double > 19 16 2

Half of 8 4 12

Half ... 2 14

… until you get to 1 1 18

DIVISION USING FRACTIONS

“Reckon with 8 so as to get 19.” In other words, “What, times 8, will give you 19?”

8 1

Next double > 19 16 2 x

Half of 8 4 12

Half ... 2 14

x

… until you get to 1 1 18

x

19 214

18

Or, 2

Find the numbers in the left column that add to 19; add the corresponding numbers in the right column to get the quotient.

MORE DIVISION WITH FRACTIONS

“Reckon with 12 so as to obtain 31”

12 1

The usual doubling 24 2

Half of 12 6 12

Half of 6 – now what?

3 14



12 1


Half of 12 6 12

Half of 6 – now what?

3 14

Dag, yo

.



12 1


Two-thirds of 12 8 23

Half of 8 4 13

Half 2 16

Done 1 112



12 1

The usual doubling 24 2 x

Two-thirds of 12 8 23

Half of 8 4 13

x

Half 2 16

x

Done 1 112

x

31 213

16

112

A LITTLE BIT ABOUT THESE FRACTIONS

The Ahmes Papyrus 2/n table:2/3 = 1/2 + 1/6 2/5 = 1/3 + 1/15 2/7 = 1/4 + 1/28

2/9 = 1/6 + 1/18 2/11 = 1/6 + 1/66 2/13 = 1/8 + 1/52 + 1/104

2/15 = 1/10 + 1/30 2/17 = 1/12 + 1/51 + 1/68 2/19 = 1/12 + 1/76 + 1/114

2/21= 1/14 + 1/42 2/23 = 1/12 + 1/276 2/25 = 1/15 + 1/75

2/27 = 1/18 + 1/54 2/29 = 1/24 + 1/58 + 1/174 + 1/232 2/31 = 1/20 + 1/124 + 1/155

2/33 = 1/22 + 1/66 2/35 = 1/25 + 1/30 + 1/42 2/37 = 1/24 + 1/111 + 1/296

2/39 = 1/26 + 1/78 2/41 = 1/24 + 1/246 + 1/328 2/43 = 1/42 + 1/86 + 1/129 + 1/301

2/45 = 1/30 + 1/90 2/47 = 1/30 + 1/141 + 1/470 2/49 = 1/28 + 1/196

2/51 = 1/34 + 1/102 2/53 = 1/30 + 1/318 + 1/795 2/55 = 1/30 + 1/330

2/57 = 1/38 + 1/114 2/59 = 1/36 + 1/236 + 1/531 2/61 = 1/40 + 1/244 + 1/488 + 1/610

2/63 = 1/42 + 1/126 2/65 = 1/39 + 1/195 2/67 = 1/40 + 1/335 + 1/536

2/69 = 1/46 + 1/138 2/71 = 1/40 + 1/568 + 1/710 2/73 = 1/60 + 1/219 + 1/292 + 1/365

2/75 = 1/50 + 1/150 2/77 = 1/44 + 1/308 2/79 = 1/60 + 1/237 + 1/316 + 1/790

2/81 = 1/54 + 1/162 2/83 = 1/60 + 1/332 + 1/415 + 1/498 2/85 = 1/51 + 1/255

2/87 = 1/58 + 1/174 2/89 = 1/60 + 1/356 + 1/534 + 1/890 2/91 = 1/70 + 1/130

2/93 = 1/62 + 1/186 2/95 = 1/60 + 1/380 + 1/570 2/97 = 1/56 + 1/679 + 1/776

2/99 = 1/66 + 1/198 2/101 = 1/101 + 1/202 + 1/303 + 1/606

WHY THIS PARTICULAR DECOMPOSITION?

Can you express every proper fraction as a sum of unit fractions?

Yes – Proved by Leonardo of Pisa (Fibonacci) in 1202.

The Greedy Algorithm works every time.

THE GREEDY ALGORITHM ILLUSTRATED

Turn into a sum of unit fractions (UFs):

Find the largest UF that less than ,

so we want . Subtract: .

Do it again with . .

So, .

Notice the intermediate numerators: 7>4>1. Fibonacci proved these always form a decreasing sequence, ending with 1.

JUST ONE PROBLEM:

The Egyptians didn’t use the greedy algorithm. What did they use? We don’t know. Some

algorithms have been found that explain many, but not all, of the choices for how fractions are broken up.

So if you need a topic for your PhD in Egyptology or History of Mathematics. . . .

AHMES (RHIND) PAPYRUS

The papyrus is in the form of a scroll about 12-13 inches wide and 18 feet long, written from right to left in hieratic script on both sides of the sheet, in black and red inks.


Giving “Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries...all secrets”.

"This book was copied in regnal year 33, month 4 of Akhet, under the majesty of the King of Upper and Lower Egypt, Awserre, given life, from an ancient copy made in the time of the King of Upper and Lower Egypt Nimaatre (?). The scribe Ahmes writes this copy."

BY THE WAY, THIS IS AN EXAMPLE OF A COLOPHON

A brief description of publication or production notes relevant to the edition.

“An account of Lehi and his wife Sariah, and his four sons, being called, (beginning at the eldest) Laman, Lemuel, Sam, and Nephi. The Lord . . . . They cross the large waters into the promised land, and so forth. This is according to the account of Nephi; or in other words, I, Nephi, wrote this record.”

COLPHONS IN THE BOOK OF MORMON

1 Nephi 1:1-3, "I, Nephi, . . . make a record of my proceedings in my days. . . . And I know that the record which I make is true; and I make it with mine own hand; and I make it according to my knowledge.“

“The account of Alma, who was the son of Alma, the first and chief judge over the people of Nephi, and also the high priest over the Church. An account of the reign of the judges, and the wars and contentions among the people. And also an account of a war between the Nephites and the Lamanites, according to the record of Alma, the first and chief judge.”


2/n and n/10 fraction decomposition tables Division of loaves (1 – 6) Multiplication of fractions (7 – 20) Completion & Aha (heap) problems (21 – 29) Division of fractions (30 – 34) More division of loaves and of hekats of grain (and beer)

arithmetic sequences (35 – 40) Geometry problems; volumes, areas, pyramids, seked (41

– 60) Other Miscellaneous problems, including pesu problems

and one on geometric progressions (61 – 84)

N/10 TABLE

1 divided by 10 yields 1/10 2 divided by 10 yields 1/5 3 divided by 10 yields 1/5 +1/10 4 divided by 10 yields 1/3 +1/15 5 divided by 10 yields 1/2 6 divided by 10 yields 1/2 +1/10 7 divided by 10 yields 2/3 +1/30 8 divided by 10 yields 2/3 +1/10 + 1/30 9 divided by 10 yields 2/3 +1/5 + 1/30

DIVIDING LOAVES AMONG 10 MEN (#3)

The answers to these are the same as the answers in the n/10 table, and the computations show that it is correct.

Divide 6 loaves among 10 men.

1 1/2 + 1/10

2 x 1 + 1/5

4 2 + 1/3 + 1/15

8 x 4 + 2/3 + 1/10 + 1/30

10 6 (1/5 + 1/10 + 1/30 = 1/3, from a table)

MULTIPLICATION OF FRACTIONS (#7)

Problems 7–20 show how to multiply the expressions 1 + 1/2 + 1/4 and 1 + 2/3 + 1/3 by different fractions.


Multiply by .

1 12

114

12

14

128

14

18

156

112

14

12

14

18

114

128

156

1 Answer


The question is, how does this long string of fractions turn into 1?

The fact that was actually used quite

frequently in the records we have.

From there, wouldn’t be too hard.

So: addition facts for unit fractions were probably partly memorized, partly taken from tables, partly determined by “red auxiliaries.”

COMPLETION PROBLEMS (#21)

Complete to 1

In other words, subtract, or find the difference.

Of course, we would do this with common denominators.

Ahmes did it with “red auxiliaries.”


Ahmes thought like this (we think):

. Use 15 as a

reference number and multiply all the fractions by this: Then has to be 4, and the question is, what fraction of 15 is 4?

We would say , but Ahmes figured it this

way:


So the final answer is 1/5 + 1/15. Red auxiliaries are the Egyptian version of

common denominators. They are called this because they were written in red ink.

1 15

1/10 1 + ½

1/5 3 X

1/15 1 X

1/5 + 1/15 4

AHA (HEAP) PROBLEMS (#26)

“A quantity, its ¼ added to it so that 15 results. What is the quantity?” (A quantity and its quarter add to 15; what is the quantity?)

If the answer were 4, then the sum of 4 and its ¼ would be 5. To get 15, I would have to multiply by 3. So 4 x 3 =12 is the answer.

Called the “method of false position.”

DIVISION OF FRACTIONS (#34) A quantity, its ½ and its ¼ becomes 10. What

is the quantity? 1 112

14

2 317

4 7


is the quantity? 1 112

14

2 312

4 7

17

14

Since 4 x (1+ ½ + ¼) is 7,then (1/7) x (1+ ½ + ¼) is ¼.


is the quantity? X 1 112

14

2 312

X 4 7

X 17

14

14

128

12

x 12

114

1

512

17

114 9

12

14

14

512

17

114

10

AREA (#50)

A round field has diameter 9 khet. What is its area? Take away 1/9 of the diameter, namely 1; the

remainder is 8. Multiply 8 times 8; it makes 64. Therefore it contains 64 setat of land.

Thus the Egyptian “formula” for the area of a circle with diameter d is:

2 22 21 8 64 256

9 9 81 81A d d d d r

AREA (#50)

Note that this implicitly uses the value

Using this produces an error of about 0.6%

256 3.1604981

PESU PROBLEMS

Pesu is a measure of the strength – or as we would think of it, the weakness – of beer or bread in terms of the grain it is made from.

Pesu =

PESU PROBLEMS (#76)

Then one hekat will make 24, or 12 loaves of each kind. The quantity of wedyet flour in the 1000 loaves of pesu 10 is 100 hekats. Multiply 100 by 12; the result is 1200, which is the number of loaves of each kind for the exchange.

EGYPTIANS – SUMMING UP

For the most part, they focused on practical problems. No general rules were stated. No “proofs” were given beyond demonstrations that the number they found satisfied the conditions.

We only have a small snapshot of their mathematics, so we could be way off in our assessments.

EGYPTIANS – SUMMING UP

Solved linear equations, often by false position. A couple of simultaneous equations, one of

second degree, are solved in the Berlin papyrus: and .

In general, not as sophisticated as Babylonian algebraic methods.

Did some squares and square roots.

A LITTLE BIT OF HISTORICAL BIAS:

Morris Kline, Mathematics, A Cultural Approach, 1962. 700 pages, 3 of which focus on Egypt and Babylonia. “[Egyptian] contributions to mathematics were almost

insignificant.” Compared with the Greeks, “the mathematics of the

Egyptians and Babylonians, is the scrawling of children just learning to write, as opposed to great literature.”

“…they barely recognized mathematics as a distinct subject” and “over a period of 4,000 years hardly any progress was made in the subject.”

A LITTLE BIT OF HISTORICAL BIAS:

Dirk Struik, A Concise History of Mathematics, 1987. “….Oriental mathematics, in this respect, never

seems to have been emancipated from the millennial influence of the problems in technology and administration, for the use of which it had been invented.”

“stagnant cultural atmosphere” leading to “the relative ossification of mathematics.”

NOT TO GET ALL PC ON YOU, BUT:

You do have to be careful. Everything is written with a point of view and a cultural context.

It is sort of natural for every age of human beings to believe they are the pinnacle of progress, knowledge, understanding. But they are pretty much proven wrong by each successive generation.

It behooves us to be humble.

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EGYPTIAN MATHEMATICS - Brigham Young Universitywilliams/Classes/300W2012/PDFs/PPTs...EGYPTIAN MATHEMATICS ... Notice the intermediate numerators: 7>4>1. Fibonacci proved these always