Mathematics of Egypt

Transcription

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 Rhind Papyri.
2nd Intermediate Period (1640 – 1550 BCE)
New Kingdom (1550 – 1070 BCE) – temple building,
empire building. Ramses, Tutankamon, Moses.
TIMELINE
Archaic
3000 BCE
Sumaria
Old Kingdom
2500 BCE
Akkadia
Int
M iddle Kingdom Int
2000 BCE
Int
Ne w Kingdom
1500 BCE
Old Babylon
EGYPT
1000 BCE
Assyria
M ESOPOTAM IA
TIMELINE
Archaic
3000 BCE
Sumaria
Old Kingdom
2500 BCE
Akkadia
Int
M iddle Kingdom Int
2000 BCE
Int
Ne w Kingdom
1500 BCE
Old Babylon
EGYPT
1000 BCE
Assyria
M ESOPOTAM IA
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
the key to our
understanding
hieroglyphics.
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 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.
 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
. So tables were provided
fractions:
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? 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
Half of 8
4
Half ...
2
… until you get to 1
1
2
1
2
1
4
1
8
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
Half of 8
4
Half ...
2
… until you get to 1
1
2
1
2
1
4
1
8
1
4
19
2
x
x
x
1
8
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
Half of 12
6
Half of 6 – now
what?
3
2
1
2
1
4

12
1
The usual doubling
24
Half of 12
6
Half of 6 – now
what?
3
2
1
2
1
4
Dag, yo
.

12
1
The usual doubling
24
Two-thirds of 12
8
Half of 8
4
Half
2
Done
1
2
2
3
1
3
1
6
1
12

12
The usual doubling
24
Two-thirds of 12
8
Half of 8
4
Half
2
Done
1
31
1
2
2
2
3
1
3
1
6
1
12
1 1
3 6
x
x
x
x
1
12
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
2/99 = 1/66 + 1/198
2/101 = 1/101 + 1/202 + 1/303 + 1/606
2/97 = 1/56 + 1/679 + 1/776
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
 Do
. Subtract:
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 (quantity) 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
3 divided by 10 yields 1/5 +1/10
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
2
4
8
10
1/2 + 1/10
x 1 + 1/5
2 + 1/3 + 1/15
x 4 + 2/3 + 1/10 + 1/30
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
1
2
1
4
1
1
2
1
4
1
2
1
4
1
8
1
14
1
28
1
56
1
2
1
1
4
.
1
8
1
14
1
28
1
56
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,

So: addition facts for unit fractions were probably
partly memorized, partly taken from tables, partly
determined by “red auxiliaries.”
wouldn’t be too hard.
COMPLETION PROBLEMS (#21)

Complete
 In
to 1
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
Then
fractions by this:
has to be 4, and the question is, what
fraction of 15 is 4?
We would say
way:
, but Ahmes figured it this
1
1/10
1/5
1/15
1/5 + 1/15
15
1+½
3
1
4
X
X
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.

AHA 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
1 1
1
is the quantity?
1
4
2
2
4
1
7
3
7

1 1
1
is the quantity?
1
4
2
2
Since 4 x (1+ ½ + ¼) is 7,
then (1/7) x (1+ ½ + ¼) is
¼.
4
1
7
1
2
3
7
1
4

1 1
1
is the quantity? X
1
4
2
2
X
4
1
7
X
x
5
5
1
4
1
2
1
2
1
2
1
28
1
14
1
7
1
7
1
2
3
7
1
4
1
2
1
1
14
1
14
9
1
2
1
4
10
1
4
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
2
1   8   64  2  256  2

A  d  d    d    d  
r
9   9   81 

 81 
AREA (#50)

Note that this implicitly uses the value
256

 3.16049
81

Using this produces an error of about 0.6%
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
Like Babylonians, focused on practical
problems. No general rules stated. No
“proofs” beyond demonstrations that the
number they found satisfies 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
and
.
papyrus:
 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.


Mathematics of Egypt

Transcription

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