Download Babylonia - Math with Mrs. Smith

Babylonia
Stories from the 400 Mathematical Tablets
Change of Rule
• Babylonians invaded Mesopotamia replacing the Sumerians from
around 2000 BCE
• This was right after Sumerians revolted against the Akkadians
(who ruled them previously)
• Babylonian writing actually came from the Sumerians
Hammurabi
• Babylonian King
• Wrote the world’s first written code of law
• On a pillar
• Most of the Babylonian way of writing was
much more permanent than Egyptian.
Lots and Lots of Mud
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
Mesopotamia: “Between the Rivers”
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
Babylonian Numbers >60
• Positional System
• Positional System
• Without placeholder “zero”
• With placeholder “zero”
Multiplication
Tables by 9
Table Texts
Babylonian representation of
Rational Numbers
• Division: multiplication by the
reciprocal
• Used sexagesimal system for
describing reciprocals
• Reciprocal Tables
Base 60
• 1. Divide the base-10 number by 60, and record the remainder.
• Divide the quotient from Step 1 by 60 again, and record the remainder.
• Repeat the process until the quotient cannot be divided by 60 (in this case,
the quotient will be 0 with a remainder of the original number).
• The number, in base 60, will be the remainders in the reverse order.
• Ex. Convert 148 to base 60
• Ex. Convert 3, 20, 5 to base 10
Babylonian
representation of
Rational Numbers
• Didn’t understand repeating
decimals
• Only made tables for factors
of 60
• 2  0;30
• 3  0;20
• 9  0;6;40
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
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
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. Now add 0;30 to 29;30 and
the result if 30 – the side of
square
Algebraic Problem
• A canal 5 GAR long, 1 ½ GAR wide, and ½
GAR deep is to be dug. Each worker is
assigned to dig 10 GIN, and is paid 6 SE.
Find the area, volume, number of workers,
and total cost.
Solution
• Multiply length and width to get 7;30 SAR, the
area. Multiply 7;30 by depth to get 45 SAR, the
volume. Multiply the reciprocal of the
assignment, 6, by 45 to get 4,30, which is the
number of workers. Multiply 4,30 by the wages to
get 9 GIN, the total expenses.
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