Transcript (Babylonian and Egyptian mathematics).

π
Consider the value of π below we recall that two of the
more accurate fractional approximations of π are:
22
 3.142857142857
7
355
 3.141592920353982300884655722124
113
The 7th, 22nd, 113th, and 355th positions in the decimal
value of π are all “2”. Is this coincidental, or does it
have some mysterious meaning?
1
2 – Babylonian & Egyptian
Mathematics
The student will learn about
Numeral systems
from the Babylonian
and Egyptian
cultures.
2
Cultural Connection
The Agricultural Revolution
The Cradles of Civilization – ca. 3,000 – 525 B.C.
Student led discussion.
3
Cultural Connection
The Agricultural Revolution
The Cradles of Civilization – ca. 3,000 – 525 B.C.
Ends at 525 B.C. when Persia conquered
Babylonia.
Climatic changes caused the savannahs to change
into forest or desserts.
Population density prohibited hunter/gathers
(about 40 people per square mile) so man turned to
agriculture.
continued
4
Cultural Connection
The Agricultural Revolution
The Cradles of Civilization – ca. 3,000 – 525 B.C.
Civilization centered about rivers –
Africa
Nile River
Mid-East
Tigrus and Euphrates Rivers
(Mesopotamia) with city of Ur about 24,000 people.
India
Indus River
China
Yellow River
continued
5
Cultural Connection
The Agricultural Revolution
The Cradles of Civilization – ca. 3,000 – 525 B.C.
Civilization needed and developed –
A written language
Engineering skills
Commercial skills
Astronomical skills
Geodetic skills
continued
6
Cultural Connection
The Agricultural Revolution
The Cradles of Civilization – ca. 3,000 – 525 B.C.
Governments were developed –
Oligarchy – small clique of privileged citizens.
Monarchies – king or queen.
Theocracies – rule by religious leaders.
Republics – broad citizen participation
7
§2-1 The Ancient Orient
Student Discussion.
8
§2-1 The Ancient Orient
Calendars.
Weights and measures to harvest, store and
apportion food.
Surveying for canals and reservoirs and to parcel
land.
Financial and commercial practices – raising and
collecting taxes and trade.
9
§2-2 Babylonian
Sources of Information
Student Discussion.
10
§2-2 Babylonian
Sources of Information
About 500,000 clay tablets found in Mesopotamia.
Many were deciphered by Sir Henry Creswicke
Rawlinson in the mid 1800’s.
Tablets were small. Several inches on a side.
11
§2-3 Babylonian
Commercial and Agrarian Math
Student Discussion.
12
§2-3 Babylonian
Commercial and Agrarian Math
Commercial examples – bills, receipts,
promissory notes, interest, etc.
Agrarian examples – field measurement,
crop calculation, sales of crops, etc.
Many tablets were math tables – reciprocals,
squares, cubes, exponents, etc.
continued
13
§2-3 Babylonian
Commercial and Agrarian Math
Remember they worked in base 60 with only
two symbols
and
for 1 and 10
respectively.
meant 11 or 11 · 60 or 11· 60 2 or ….
765 was 12 · 60 + 45 or
.
A fraction was also in base 60 where ½ = 30/60 =
continued
14
§2-3 Babylonian
Commercial and Agrarian Math
There is a modern notation for base 60 which
is quite helpful.
1, 02, 34; 15 means
1 · 60 2 + 2 · 60 + 34 + 15/60 =
3600 + 120 + 34 + 0.25 = 3754.25 ten
15
§2-4 Babylonian
Geometry
Student Discussion.
16
§2-4 Babylonian
Geometry
Area of rectangles, right triangles, isosceles
triangles, and trapezoids was known.
Volume of rectangular parallelepipeds,
and right prisms was known.
π was assumed to be 3 1/8.
Proportions between similar triangles were known.
The Pythagorean theorem was known.
17
§2-5 Babylonian
Algebra
Student Discussion.
18
§2-5 Babylonian
Algebra
Solved some quadratics by substitution
and completing the square.
Solved some cubic, biquadratic and a few
of higher degree.
19
2 by Babalonian Methods
The ancients knew that if 2 < x then 2/x < 2 .
Show why.
This implied:
2/x < 2 < x
First iteration: Let x = 1.5
For a better approximation average x and 2/x:
x
2/x
Average
3/2
4/3
17/12
17/12
24/17
577/408
continued
20
2 by Babalonian Methods
With basically two iterations we arrive at 577 / 408
In decimal form this is 1.414212963
In base sixty notation this is 1 ; 24, 51, 10, 35, . . .
To three decimal places 1 ; 24, 51, 10 is what the
Babylonians used for 2 !
Accuracy to - 0.0000006 or about the equivalency
of about 1 foot over the distance to Boston!
This calculation was on Tablet No. 7289 from the
Yale Collection.
21
YBC 7289
On the Yale Babylonian Collection
Tablet 7289 there are three numbers:
a = 30
b = 1, 24, 51, 10 and
c = 42, 25, 35
Note that c = a ∙ b = 30 ∙ (1, 24, 51, 10)
Instead of multiplying b by 30 the Babylonians no doubt
divided it by 2. Why?
Do it!
b = 1, 24, 51, 10 OR 84, 50, 70 ÷ 2 = 42, 25, 35
Just like in our base ten system multiplying by 5 and
dividing by 2 yield the same numeric results less decimal
22
point placement.
§2-6 Babylonian
Plimpton 322
Student Discussion.
23
§2-6 Babylonian
Plimpton 322
(c/a) 2
1.9326
1.8696
1.8107
…
1.3611
a
120
3456
480
…
90
b
119
3367
4601
…
56
c
A
B
90
b
a
c
169
4825
6649
…
106
1
2
3
…
15
(c/a) 2 is the secant 2 of 44°, 43 °, 42 °, … , 31 °.
Accuracy is from 0.02 to 0.08. We will see the
significance of secant later in the course.
Column a is
regular
sexagesimal
numbers.
Columns b and c
are generated
parametrically
from regular
sexagesimal
numbers.
24
Egyptian
25
§2-7 Egyptian
Sources of Information
Student Discussion.
26
§2-7 Egyptian
Sources of Information
Egypt was more seclude and naturally
protected. Their society was a theocracy with
slaves doing manual labor. The dry climate
preserved many of their documents. It has
been felt recently that they were not as
sophisticated as the Babylonians.
continued
27
§2-7 Egyptian
Sources of Information
3100 B.C.
2600 B.C.
Numbers to millions
Great Pyramid – 13 acres,
2,000,000 stones from 2.5 to 54 tons granite
blocks from 600 miles away. Square to
1/14,000, and right angles to 1/27,000. 100,000
laborers for 30 years
1850 B.C.
1650 B.C.
Moscow papyrus – 25 problems
Rhine papyrus – 85 problems
continued
28
§2-7 Egyptian
Sources of Information
1500 B.C.
1350 B.C.
Sundial
Papyrus with bread accounts.
1167 B.C.
Harris papyrus – Rameses III
196 B.C.
Rosetta Stone – Egyptian
hieroglyphics, Egyptian Demotic, and Greek.
29
by MIKE PETERS
30
§2-8 Egyptian
Arithmetic and Algebra
Student Discussion.
31
§2-8 Egyptian
Arithmetic and Algebra
Duplation and Mediation for multiplication.
26 · 33.
1 33
Pick the numbers in the left
2 66
column that add to 26. Cross
4
8
16
132
264
528
out the remaining rows. The
sum of the right column is the
answer.
858
continued
32
§2-8 Egyptian
Arithmetic and Algebra
Duplation and Mediation – Why It Works!
26 · 33.
(26) x (33)
1 33
= (2 + 8 + 16) x (33)
2 66
4
8
16
132
264
528
= (2)(33) + (8)(33) + (16)(33)
= (66) + (264) + (528)
= 858
858
continued
33
§2-8 Egyptian
Arithmetic and Algebra
Duplation and Mediation for division.
753  26.
Pick the numbers in the right
1 26
column that add to 753 or less.
2 52
Cross out the remaining rows. The
4
8
16
104
208
416
28
728 + 25 = 753
sum in the left column is the
quotient and the difference
between the right column and 753
is the remainder.
continued
Quotient
remainder
34
§2-8 Egyptian
Arithmetic and Algebra
Duplation and Mediation for division. Why it works!
753 ÷ 26
753  26.
1
2
26
52
753 = 28 x 26 + 25
4
8
16
104
208
416
753 = (104 + 208 + 416) + 25
28
728 + 25 = 753
753 = (4 + 8 + 16) x 26 + 25
753 = (728) + 25
continued
Quotient
remainder
35
36
§2-8 Egyptian
Arithmetic and Algebra
Unit fractions to avoid fractional difficulties.
2 1
1
 
7 4 28
3 1
1
 
5 2 10
5
?
18
continued
37
§2-8 Egyptian
Arithmetic and Algebra
Rule of False Positioning.
x – x/3 = 8
Pick a number to try. A good choice would be
a number divisible by three, Why? Try 6.
6 – 6/3 = 4
Notice 4 is one-half the correct answer hence
the correct answer must be double 6 (6 was
your guess) or 12.
38
§2-9 Egyptian
Geometry
Student Discussion.
39
40
§2-9 Egyptian
Geometry
They knew the area of a circle as (8/9 d)2,
area of a triangle as ½ ab, area of a
quadrilateral as (a + c) (b + d) / 4 which is
incorrect.
Knew the volume of a right circular cylinder
as bh,  = (16/9) 2 which is off by 0.0189.
3 1/8 is more accurate. No Pythagorean
Theorem.
41
§2-10 Egyptian
Rhind Papyrus
Student Discussion.
42
§2-10 Egyptian
Rhind Papyrus
Curious Problem.
Knew regular sexagesimal numbers – that is
a number divisible by factors of 60.
This made work with fractions easier since
they produced reciprocals which were
terminating fractions..
43
Assignment
1. Read Chapter 3.
2. Calculate the cost of
building a pyramid at 100,000
laborers, six days a week at
twelve hours a day for 30
years at $7.15 an hour.
3. By Duplation and
Mediation (346)(53)
4. By Duplation and Mediation (7634)  (24)
5. Handouts.
44