BABYLONIAN AND EGYPTIAN MATHEMATICSCHAPTER 2

2.1 THE ANCIENT ORIENT

Early mathematics required a practical basis for its development & such basis arose with the evolution of more advance forms of society.

Cradle of ancient civilization:

Nile river in Africa

2.1 THE ANCIENT ORIENT

Early mathematics required a practical basis for its development & such basis arose with the evolution of more advance forms of society.

Cradle of ancient civilization:

Tigris and Euphrates

2.1 THE ANCIENT ORIENT

Early mathematics required a practical basis for its development & such basis arose with the evolution of more advance forms of society.

Cradle of ancient civilization:

Indus & Genghis in south-asia

2.1 THE ANCIENT ORIENT

Early mathematics required a practical basis for its development & such basis arose with the evolution of more advance forms of society.

Cradle of ancient civilization:

Huangho & Yangtze in eastern Asia

2.2 SOURCES

ARCHEOLOGIST WORKING IN MESOPOTAMIA

Unearthed half-million inscribed tablets.

50,000 tablets where excavated at Nippur

Can be found in museums in Paris, Berlin,& London, Yale Colombia,& University of Pennsylvania.

300 have been identified a mathematical tablets.

RAWLINSON

Unlocked the puzzle of the inscription in 1847.

Tablets contain the early history of Babylonia.

There are Mathematical text dating from period of:

Sumerian 2100 B.C

King Hammurabi’s era 1600 B.C

Empire of Nebuchadnezzar 600 B.C- 300 A.D

Persian & Seleucidan era

2.3 COMMERCIAL AND AGRARIAN MATHEMATICS

THE TABLETS SHOW THAT THESE ANCIENT SUMMERIAN WERE FAMILIAR WITH LEGAL AND DOMISTIC CONTRACTS:

Bills

Receipt

Promissory notes

Accounts

Simple & compound interest

Mortgage

Deeds of sale

Guarantees

2.3 COMMERCIAL AND AGRARIAN MATHEMATICS

Records of business firms, system of weight and measure.

Out of 300, 200 are table tablets:

Multiplication table

Tables of reciprocals

Tables of square and cubes

Tables of exponentials

2.4 GEOMETRY

They have been familiar with the:

General rules for the area of rectangle

Area of right and isosceles triangles

2.4 GEOMETRY

They have been familiar with the:

Area of trapezoids

Volume of parallelepiped

2.4 GEOMETRY

They have been familiar with the:

Circumference of a circle was taken as three times the diameter.

The area as one-twelfth the square of the circumference.

Babylonians know that….

The corresponding sides of two similar right triangles are proportional.

The perpendicular through the vertex of an isosceles triangles bisects the base.

Babylonians know that….

An angle inscribed in a semicircle is a right angle.

Pythagorean theorem.

31/8 is an estimate for pi.

Division of the circumference of a circle into 360 equal parts.

Babylonians know that….

Babylonian miles- use as along distance & time unit.

Equals to 7 miles

1 day = 12 time-miles

I complete day= one revolution of the sky.

Have been subdivided into 30 equal parts.

Thus , 12(30)= 360 in a complete circuit.

ALGEBRA

2000 B.C Babylonian arithmetic had evolved into a well-developed rhetorical or prose algebra.

Quadratics equations are solved by the equivalent of substituting in general form and by completing the square.

Cubic and biquadratic were discussed.

Tabulations of cubes and square from 1-30.

ALGEBRA

Unsolved problems involving simultaneous equations which leads to biquadratic equations for solution. These can be found in Yale's tablets.

xy= 600, 150 ( x – y ) – ( x + y ) 2 = -1000

xy = a, bx2/y + cy 2 / x + d = 0

Leads to an equation of the sixth degree in x but quadratic in x 3

ALGEBRA

Babylonians gave some interesting approximation to the square roots of nonsquare numbers like

17/12 for 𝟐

17/24 for 1/ 𝟐

Using ( a2 + h)1/2 = a + h/2a

A very remarkable approximation for 𝟐 is

1+24/60 + 51/602 + 10/603 = 1.14213

ALGEBRA

Neugebauer has found two interesting series problems on a louvre tablets about 300B.C.

1.

2.

Found by contemporary Greek

1.

Found by Archimedes

2.

2.6 PLIMPTON 322

Most remarkable Babylonian mathematical tablet.

It is the item with catalog number 322 in the G.A Plimpton collection at Colombia University.

Written old Babylonian script.

EGYPT

2.7 SOURCE AND DATES

Mathematics of ancient Egypt never reached the level attained by Babylonian mathematics

Because it is semi isolated place.

Was long the richest field for ancient historical research

Egyptians respect their dead leads to building of long lasting tombs with richly inscribed walls.

Thus many papyri & objects preserve as well.

Some tangible items bearing on the mathematics of Egypt

1. 3100 B.C Royal Egyptian mace

Has several number in millions & hundred of thousands.

Written in Egyptian hieroglyphs.

Some tangible items bearing on the mathematics of Egypt

2. 2900 B.C The Great Pyramid of Giza.

covers 13 acres, contains 2,000,000 stone blocks averaging 2.5 tons each quarried from near the Nile.

Chamber roof: 54 ton granite block

27ft. Long x 4ft thick.

Quarried 600 miles away

100,000 laborer for 30 years to complete.

Some tangible items bearing on the mathematics of Egypt

3.1850 B.C Moscow papyrus

Mathematical text contained 25 problems.

Some tangible items bearing on the mathematics of Egypt

4.1850 The Oldest Extant Astronomical Instrument.

A combination of plumb line and sight rod.

Some tangible items bearing on the mathematics of Egypt

5. 1650B.C Rhind Payrus

A mathematical text partaking of the nature of a practical handbook & containing 85 problems copied in hieratic writing by the scribe Ahmes.

Some tangible items bearing on the mathematics of Egypt

6. 1500B.C The Largest Existing Obelisk

It is 105 ft long with a square base 10ft.

430 tons

Some tangible items bearing on the mathematics of Egypt

7. 1500 B.C Egyptian Sundial

Oldest sundial extant

Preserved in Berlin museum.

Some tangible items bearing on the mathematics of Egypt

8. 1350B.C The Rollin Papyrus

Contains some bread accounts

Preserved in louvre

Some tangible items bearing on the mathematics of Egypt

9. 1167 B.C Harris Papyrus

A document prepared for Rameses IV.

2.8 ARITHMETIC AND ALGEBRA

Hieroglyphic Representation of Numbers

Hieroglyphs are little pictures representing words.

The Egyptians had a bases 10 system of hieroglyphs for numerals. By this we mean that they has separate symbols for one unit, one ten, one hundred, one thousand, one ten thousand, one hundred thousand, and one million.

2.8 ARITHMETIC AND ALGEBRA

Although the Egyptians had symbolsfor numbers, they had no generallyuniform notation for arithmeticaloperations. In the case of the famousRhind Papyrus (dating about 1650B.C.),the scribe did represent additionand subtraction by the hieroglyphsand , which resemble the legs ofa person coming and going.

2.8 ARITHMETIC AND ALGEBRA

Multiplication is basically binary.

Example Multiply: 47 × 24

47 × 24

47 1

94 2

188 4

376 8 *

752 16 *

Selecting 8 and 16 (i.e. 8 + 16 = 24), we have

24 = 16 + 8

47 × 24 = 47 × (16 + 8)

= 752 + 376

= 1128

2.8 ARITHMETIC AND ALGEBRA

Fractions

The symbol for unit fractions was a flattened oval above the denominator. In fact, this oval was the sign used by the Egyptians for the mouth .

For ordinary fractions, we have the following

1

24

1

7

1

3

2.8 ARITHMETIC AND ALGEBRA

Fractions

There were special symbols for the fractions

1/2 , 2/3 , 3/4.

2.8 ARITHMETIC AND ALGEBRA

Hieratic numerals

2.9 GEOMETRY

26 Of the problems in the Moscow & Rhind papyri are geometric.

Computation of land area and granary volumes.

AC= 8/9 D

V right cylinder = base x height

THANK YOU…

PREPARED BY

MARVEN LAUDE