Post on 23-Jan-2018
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:
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
2.9 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