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Babylonia
Stories from the 400 Mathematical Tablets
The Tigris and Euphrates Rivers
• 3000 BC: Ending of Stone
Age
• Savannas were shrinking
• Hunting and Gathering became
inefficient
•
Why?
• Overcrowding around oases
•
Danger of Starvation
•
Followed fleeing animals
• Found Cradle between Rivers
•
Agriculture was born
Eves, Howard. History of Mathematics
Agriculture
• Written Language necessary
• Coordinate engineering tasks
•
Dams
•
Irrigation systems
• Record-Keeping Systems
•
Weather Almanacs
•
Flood Seasons
• New Technology
• Leisure Time
• For scribes, merchants, priests and
royalty
• Science and Math was born
Ploughing -http://www.crystalinks.com/sumeragriculture.html
Babylonian Numerals
One
Ten
Babylonian Numbers 1 - 59
• Grouping System
• Used only two symbols to make all of
these numbers
• Try to fill in the missing blanks on your
white boards.
Babylonian Numbers >60
• Positional System
• Positional System
• Without placeholder “zero”
• With placeholder “zero”
Multiplication
Tables by 9
For what number is
this a multiplication
table?
Babylonian representation of
Rational Numbers
• Division: multiplication by the reciprocal
• Used sexagesimal system for describing
reciprocals
• Reciprocal Tables
• Found incredible approximation for
2
Babylonian representation of
Rational Numbers
• Didn’t understand repeating decimals
• Only made tables for factors of 60
• Top line reads: 1, its 2/3, 40 it is
• Second line reads: its half, 30 it is
• The opposite of:
• 3 is :20
• 4 is :15
• 5 is: 12
• 6 is :10
On your white boards
Represent the
numbers to the right
in sexagesimal
1



2
8
60
60
603
1



2
9
60
60
603
1



2
10
60
60
603
1



2
15
60
60
603
1



2
16
60
60
603
How did Babylonians make
their multiplication table?
Number
Square of Number
1
1
2
4
• They had a square table
• Used
( a  b) 2  ( a  b) 2
ab 
4
3
9
4
16
5
25
6
36
7
49
8
1,4
9
1,21
10
1,40
11
2,1
12
2,24
Number
Square of Number
18
324
19
361
20
400
21
441
22
484
23
529
24
576
25
625
26
676
27
729
28
784
29
841
30
900
On your whiteboard
Use babylonian method of multiplication
1) 4(13)
2)12(11)
3)(24)(35)
Yale Collection #7289
• Very high approximation of 2
• Convert 1:24, 51,10 to decimal
• Yale tablet:
Why would Babylonians introduce the
square root of 2 with this problem?
Remember, they were practical farmers!
Plimpton
322
Pythagorean Triples
Literal Translation
• Filled in missing piece
• Convert to decimal
• Take third column squared
minus second column squared
• Divide the result by the third
column squared
http://public.csusm.edu/Aitken_html/m330/Meso/Plimpton322.trans.gif
Generating Pythagorean
Triples
Simplify a  b and c
2
2
a  2uv
2
b  u v
2
c u v
2
2
2
Could this be how the Babylonians generated
their Pythagorean triples?
Algebraic Problem Solving
• Could solve algebraic
equations
• Didn’t use variables
• To the right is a
translation of a
Babylonian tablet
1 2
1 2
( x  )  870  ( )
2
2
1. Take half of 1, which is 0;30,
2. Multiply 0;30 by 0;30, which
is 0;15
3. Add this to 14,30 to get
14,30;15.
4. This is the square of 29;30
5. No add 0;30 to 29;30 and the
result if 30 – the side of
square
Babylonian solution to
2
x  px  q
2
p
 p
x    q 
2
2
Babylonian solution to
2
x  px  q
2
p
 p
x    q 
2
2
Sources
Felluga, Dino. Guide to Literary and Critical Theory.
Purdue U, 28
Nov. 2003. Web. 10 May 2006.
Eves, Howard. Introduction to the History of Mathematics. Pacific Grove,
Thomson Brooks/Cole: 1990. Print
Katz, Victor. A History of Mathematics, An Introduction. Boston,
Addison-Wesley: 2009. Print
Boyer, Carl. A History of Mathematics. Canada,
Wiley: 1989. Print
O’Connor, John. MacTutor History of Mathematics Archive, University of St Andrews, Scotland
JOC/EFR July 2015. Web. 12 Sept 2015
Allen, G. Donald. The History of Mathematicss, University of Texas A&M
2002-2014. Web. 12 Sept 2015