This document provides an overview of ancient mathematics in Babylon and Egypt. It describes how early mathematics developed out of practical needs in early civilizations along rivers like the Nile, Tigris, Euphrates, Indus, and Huangho. Archaeologists have uncovered hundreds of thousands of clay tablets in Mesopotamia containing early mathematical concepts. These include arithmetic, algebra, geometry, and early use of tables and formulas. Egyptian mathematics is also discussed and sources of early mathematical knowledge from Egypt are described, including papyri, monuments, and other inscriptions.

1. BABYLONIAN AND
EGYPTIAN MATHEMATICS
CHAPTER 2

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

3. 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

4. 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

5. 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

6. 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.

7. 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

8. 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

9. 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

10. 2.4 GEOMETRY
 They have been familiar with the:
General rules for the area of rectangle
Area of right and isosceles triangles

11. 2.4 GEOMETRY
They have been familiar with the:
Area of trapezoids
Volume of parallelepiped

12. 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.

13. 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.

14. 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.

15. 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.

16. 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.

17. 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

18. 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

19. 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.

20. 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.

21. 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.

22. 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.

23. 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.

24. Some tangible items bearing on the
mathematics of Egypt
3.1850 B.C Moscow papyrus
 Mathematical text contained 25 problems.

25. Some tangible items bearing on the
mathematics of Egypt
4.1850 The Oldest Extant Astronomical
Instrument.
 A combination of plumb line and sight rod.

26. 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.

27. 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

28. Some tangible items bearing on the
mathematics of Egypt
7. 1500 B.C Egyptian Sundial
 Oldest sundial extant
 Preserved in Berlin museum.

29. Some tangible items bearing on the
mathematics of Egypt
8. 1350B.C The Rollin Papyrus
 Contains some bread accounts
 Preserved in louvre

30. Some tangible items bearing on the
mathematics of Egypt
9. 1167 B.C Harris Papyrus
 A document prepared for Rameses IV.

31. 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.

32. 2.8 ARITHMETIC AND ALGEBRA
 Although the Egyptians had symbols
for numbers, they had no generally
uniform notation for arithmetical
operations. In the case of the famous
Rhind Papyrus (dating about 1650
B.C.),the scribe did represent addition
and subtraction by the hieroglyphs
and , which resemble the legs of
a person coming and going.

33. 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

34. 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

35. 2.8 ARITHMETIC AND ALGEBRA
 Fractions
There were special symbols for the fractions
1/2 , 2/3 , 3/4.

36. 2.8 ARITHMETIC AND ALGEBRA
 Hieratic numerals

37. 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

38. THANK YOU…
PREPARED BY
 MARVEN LAUDE