Babylon GROUP 2

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BABYLONION CILIVIZATION





Babylonian MathematicsHistorical Facts about the

Babylonians Babylonian refers to theancient peoples who livedin the area between theTigris and Euphratesrivers.

Civilization developedabout the same time as inancient Egypt -- between3500 and 3000 B.C.

This area was referred toas Mesopotamia by the

Greeks and currentlyoccupied primarily by Iraqand Syria.

Knowledge of thecivilization there is stillgrowing from thedeciphering and



Babylonians Math

Babylonians used clay tabletsto do math in them. The claytablets included fractions,algebra, quadratic and cubicequations.

Also the Babylonian tabletgives an aproximation to thesquare root of 2 to 6 decimalplaces.

The Babylonian mathematicalsystem was sexagesimal.Babylonian mathematicians alsodeveloped algebraic methods ofsolving equations.

The Babylonians also used

http://asaoh.blogspot.com/2007/07/babylonians-math.html

http://asaoh.blogspot.com/2007/07/babylonians-math.html



Here is one of their tablets

use of a stylus on a clay mediumthat led to the use of cuneiformsymbols since curved lines couldnot be drawn.

The later Babylonians adopted thesame style of cuneiform writingon clay tablets.

Many of the tablets concern topics

which, although not containingdeep mathematics, or example wementioned above the irrigationsystems of the early civilisationsin Mesopotamia.





Babylonian Mathematics

-Fraction-



Babylonian using a sexagesimal numeral

system(base 60).

Generally the only fractions permittedwere such as

, , , …



Irregular fractions such , , …etc werenot normally not used.

There are some tablets that remark, “7

does not divide” or “11 does not divide”etc.



2 30 16 3,45

3 20 18 3,20

4 15 20 3

5 12 24 2,30

6 10 25 2,25

8 7,30 27 2,13,20

9 6,40 30 2

10 6 32 1,52,30

12 5 36 1,40

15 4 40 1,30

Table of a products equal to

sixty



EXAMPLE

= 30 =

= 7,30 = +



SQUARE

ROOT



Babylonion Square Root

Also called Heron’s method.

Involve dividing and averagingmethod.

Simple method and give accurateanswer.



Involve 4 steps:

Find Square Root of Q1. Make a guess (X₀)

2. Divide original number(Q) with your guess(X₀).

Q/X₀ = S

3. Find average of these number.

(X₀ + S)/2

4. Use this average as your new guess, X₁.

Repeat from step 2 THREE times to get accurate

value.



Example:

Find square root of 10.

Let Q=10

First Process

1. X₀=32. Q/X₀ = 10/3 = 3.333333333 S₀

3. (X₀ + S₀)/2 = (3+3.333333333)/2

= 3.1666666667

4. new guess, X₁



Second Process

1. X₁ = 3.166666667

2. Q/X₁ = 10/3.166666667

= 3.157894737 S₁

3. (X₁ + S₁)/2 = (3.166666667 + 3.157894737)/2

= 3.162280702

4. New guess, X₂



Third Process

1. X₂ = 3.162280702

2. Q/X₂ = 10/ 3.162280702

= 3.162274619 S₂

3. (X₂ + S₂)/2= (3.162280702 + 3.162274619)/2

= 3.16227766

Lets check using calculator,

Square root of 10 is equal to 3.16227766



Linear

System



Linear System

What is linear system?◦ The same set of variables



Linear System

Set A

Set B

1800

5002

1

3

2

E A

E A

1

132 2

Z Y X

Y X

1

2

3

4



Linear System

SET AFrom

900ˆ

1800ˆ2ˆˆ

ˆˆ

A

A E A

E ALet

Equation 2



Linear System

Now, make a new model

d E E

d A A

ˆ ii)

ˆ i)

Then, substitute model i) and ii) into equation1

500ˆ

2

1ˆ

3

2

d E d A



Linear System

500ˆ

61

67

500ˆ

2

1ˆ

3

2

6

7

500ˆ

2

1ˆ

3

2

6

7

5002

1ˆ

2

1

3

2ˆ

3

2

Ad

A Ad

E Ad

d E d A

A E ˆˆ

replace



Linear System

900ˆ A

300

3506

7

d

d

600

1200

:Therefore

B

A



Linear System

Change the answer into sexagesimalform

0,10

600

0,20

1200

Y

X





It all started around

3000 BC with theBabylonians.

They were one of the

world's firstcivilizations, andcame up with somegreat ideas likeagriculture,irrigation and writing.



Babylonians for the rather less pleasantinvention of the (dreaded) taxman.

And this was one of the reasons thatthe Babylonians needed to solvequadratic equations.



Let's suppose that you are aBabylonian farmer.



Somewhere on your farm you have asquare field on which you grow somecrop.



Double the length of each side of thefield and you find that you can grow four times as much of the crop as before.

What

amount of your cropcan you growon the field?



In mathematical terms,

c = mx² whereby;

x : the length of the side of the fieldm : the amount of crop you can grow on

a square field of side length 1c : the amount of crop that you can

grow



c

x

m



This is first quadratic equation.

Quadratic equations and areas arelinked together like brothers andsisters in the same family.

However, at the moment it don't haveto solve anything – until the tax manarrives, that is! Cheerily he says tothe farmer



The farmer nowhas a dilemma:how big a field

does he need togrow that amount

of crop?

I want you togive me crops topay for the taxeson your farm !



We can answer this question easily, infact........

m

c x



Now, not all fields are square.

Let's now suppose

that the farmerhas a more oddlyshaped field with

two triangularsections as shownon the right.



Then you can

imply how to findthe solution of theproblem ……

But then perhaps

our ol’ Babylonianpals have to waitfor Al Khwarizmi in

about a thousandyears or so to getthe following

formula….









The word geometry ( geo=earth and metria=measurement)means “earth measuring” .

Babylonian civilizations had the origin in simpleobservations from human ability to recognize physical formand to compare shapes and size.

Babylonian people were forced to take on geometric topics,although it may not have been recognized as such.

For example: Man had to learn with situations involving

distance, bounding their land, and constructing walls andhomes. These types of situations were directly related to thegeometric concepts of vertical, parallel, and perpendicular.



The Babylonians were aware of the connections betweenalgebraic calculations and geometry.

Geometrical terms such as length and area in theiralgebraic solutions. It is apparent, however, that these termsserved only to give names to unknown quantities as they hadno objection to mixing dimensions ( adding lengths to areas).



From the examples given by the Babylonians we know thatthey must have been familiar with:

The general rules for the area of a rectangle, right angledtriangle, isosceles triangle, trapezoid (with one sideperpendicular to the base) and parallelograms

The Pythagorean Theorem

The fact that in an isosceles triangle, the line joining thevertexto the midpoint of the base is perpendicular to the base

The proportionality of the corresponding sides of similartriangles.



For a circle:

Circumference of a circle= 3x DiameterArea =1/12 x square of circumference

which would be correct if π is estimatedas 3

http://en.wikipedia.org/wiki/%CE%A0

http://en.wikipedia.org/wiki/%CE%A0



Thank you..

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Babylon GROUP 2