Lesson 2

MATHEMATICS IN

EGYPT and MESOPOTAMIA

Mr. Reymen B. De Jesus

Lesson 2: Mathematics in Egypt and Mesopotamia

EGYPT
 Egypt

Number Recording of the Egypt

The History of Herodotus

About 450 BCE, Herodotus, the inveterate Greek traveler and

narrative historian, visited Egypt. He viewed ancient monuments,
interviewed priests, and observed the majesty of the Nile and the
achievements of those working along its banks.
 Egypt

Number Recording of the Egypt

The History of Herodotus

The writing of history, as we understand it, is a Greek invention; and

foremost among the early Greek historians was Herodotus.
 Egypt

Herodotus (circa 485–430 B.C.)

- was born at Halicarnassus, a largely Greek

settlement on the southwest coast of Asia Minor

- was involved in political troubles in his home city

and forced to be in exile to the island of Samos, and
thence to Athens.
 Egypt

Herodotus (circa 485–430 B.C.)

- Herodotus became a citizen of Thorium in

southern Italy

- In Thorium, he seems to have passed the last

years of his life involved almost entirely in finishing
the History of Herodotus, a book larger than any
Greek prose work before it.
 Egypt

Herodotus (circa 485–430 B.C.)

- He is best known for recounting, very objectively,

the Greco-Persian wars of the late 5th century.
 Egypt

Ancient Egyptian Number System

The spectacular emergence of the Egyptian government and

administration under the pharaohs of the first two dynasties could
not have taken place without a method of writing, and we find such a
method both in the elaborate “sacred signs,” or hieroglyphics.
 Egypt

Hieroglyphic

The hieroglyphic system of writing is a picture script, in which each

character represents a concrete object, the significance of which
may still be recognizable in many cases.
Great Pyramid of Giza
Hieroglyphics
Hieroglyphics
 Egypt

In other words, the Egyptian system was a decimal one (from the
Latin decem, “ten”), which used counting by powers of 10.
 Egypt

Special pictographs were used for each new power of 10 up to

1,000,000:

100 by a curved rope,

1000 by a lotus flower,
10,000 by an upright bent finger,
100,000 by a tadpole,
1,000,000 by a person holding up two hands as if in great
astonishment
Ancient Egyptian Hieroglyphic Numeral System
Ancient Egyptian Hieroglyphic Numeral System
Ancient Egyptian Hieroglyphic Numeral System
Ancient Egyptian Hieroglyphic Numeral System
Ancient Egyptian Hieroglyphic Numeral System
EXAMPLES:

Solution:

3 (1,000)= 3,000
1 (100)= 100
4 (10)= 40
5 (1)= 5
Answer: 3,145
EXAMPLES:

Solution:

3 (100)= 300
8 (10)= 80
8 (1)= 5
Answer: 385
EXAMPLES:

Solution:

2 (100,000)= 200,000
4 (1,000)= 4,000
2 (100)= 200
4 (10)= 40
6 (1)= 6
Answer: 204, 246
EXAMPLES:

Solution:
1 (1,000,000)= 1,000,000
1 (100,000)= 100,000
1 (10,000)= 10,000
2 (1,000)= 2,000
1 (100)= 100
3 (10)= 30
3 (1)= 3

Answer: 1,112,133
 Egypt

Egyptian Hieratic Numeration

Hieratic Script

The Hieratic script was invented and developed more or less at

the same time as the hieroglyphic script and was used in
parallel with it for everyday purposes such as keeping records
and accounts and writing letters.
 Egypt

Egyptian Hieratic Numeration

Rhind Papyrus
Rhind Papyrus, was written in Hieratic script, and was kept in the
British Museum in London, from which we know a lot about
Egyptian mathematics.
It is named after the Scottish archeologist, Alexander Henry
Rhind, who found it, and was written in ink on papyrus by an
Egyptian scribe called Ahmes.
Rhind Papyrus
 Egypt

Egyptian Addition and Subtraction

 Egypt

Egyptian Hieratic Numeration

Rhind Papyrus
Rhind Papyrus, was written in Hieratic script, and was kept in the
British Museum in London, from which we know a lot about
Egyptian mathematics.
It is named after the Scottish archeologist, Alexander Henry
Rhind, who found it, and was written in ink on papyrus by an
Egyptian scribe called Ahmes.
Lesson 2: Mathematics in Egypt and Mesopotamia

MESOPOTAMIA
 Mesopotamia

The fourth millennium before our era was a period of

remarkable cultural development, bringing with it the use of
writing, the wheel, and metals.
 Mesopotamia

There the Sumerians had built homes and temples decorated

with artistic pottery and mosaics in geometric patterns.
Powerful rulers united the local principalities into an empire
that completed vast public works, such as a system of canals to
irrigate the land and control flooding between the Tigris and
Euphrates rivers, where the overflow of the rivers were not
predictable, as was the inundation of the Nile Valley.
 Mesopotamia

The Mesopotamian civilizations of antiquity are often referred

to as Babylonian, although such a designation is not strictly
correct. The city of Babylon was not at first, nor was it always
at later periods, the center of the culture associated with the
two rivers, but convention has sanctioned the informal use of
the name “Babylonian” for the region during the interval from
about 2000 to roughly 600 BCE.
 Mesopotamia

Sumerian/Babylonian Mathematics

Sumer (a region of Mesopotamia, modern-day Iraq) was the

birthplace of writing, the wheel, agriculture, the arch, the plow,
irrigation and many other innovations, and is often referred to
as the Cradle of Civilization.
 Mesopotamia

A cradle of civilization is a location and a culture where

civilization was created by mankind independent of other
civilizations in other locations.
 Egypt and Mesopotamia

What is the primary characteristic of a society that can be

characterized as "civilized“?
 Mesopotamia

Cuneiform Script

The Sumerians developed the earliest known writing system – a

pictographic writing system known as cuneiform script, using
wedge-shaped characters inscribed on baked clay tablets – and
this has meant that we actually have more knowledge of ancient
Sumerian and Babylonian mathematics than of early Egyptian
mathematics.
 Mesopotamia

Sumerian Clay Cones

Starting as early as the 4th millennium BCE, they began using a

small clay cone to represent one, a clay ball for ten, and a large
cone for sixty.
 Mesopotamia

Sumerian Clay Cones

Over the course of the third millennium, these objects were

replaced by cuneiform equivalents so that numbers could be
written with the same stylus that was being used for the words
in the text.
Sumerian Clay Cones
 Mesopotamia

Babylonian Numerals
(The Babylonian Positional Number System)
Sumerian and Babylonian mathematics was based on a
sexagesimal, or base 60, numeric system.

Unlike those of the Egyptians, Greeks and Romans, Babylonian

numbers used a true place-value system, where digits written in
the left column represented larger values, much as in the
modern decimal system, although of course using base 60 not
base 10.
 Mesopotamia

Babylonian Numerals
(The Babylonian Positional Number System)

To represent the numbers 1 – 59 within each place value, two

distinct symbols were used, a unit symbol ( ) and a ten symbol
( ) which were combined in a similar way to the familiar system
of Roman numerals (e.g. 23 would be shown as ( )
 Mesopotamia

Babylonian Numerals
(The Babylonian Positional Number System)

Babylonian advances in mathematics were probably facilitated

by the fact that 60 has many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15,
20, 30 and 60
Babylonian Numerals (The Babylonian Positional Number System)
 Mesopotamia

Example:
For example, the Babylonian 3 25 4 might stand as:

(𝟑 ⋅ 𝟐
𝟔𝟎 ) + (𝟐𝟓 ⋅ 𝟏
𝟔𝟎 ) + (𝟒 𝟎
⋅ 𝟔𝟎 )
= (𝟑 ⋅ 𝟑𝟔𝟎𝟎) + (𝟏𝟓𝟎𝟎) + (𝟒 ⋅ 𝟏)
= (𝟏𝟎, 𝟖𝟎𝟎) + (𝟏𝟓𝟎𝟎) + (𝟒)
= 𝟏𝟐, 𝟑𝟎𝟒
 Mesopotamia

Example:

For example, the Babylonian 2 24 might stand as:

In cuneiform, it looks
(𝟐 ⋅ 𝟏
𝟔𝟎 ) + (𝟐𝟒 ⋅ 𝟏) like this

= 𝟏𝟐𝟎 + 𝟐𝟒
= 𝟏𝟒𝟒
 Mesopotamia

Example:
For example, the Babylonian 4 10 3 might stand as:

(𝟒 ⋅ 𝟐
𝟔𝟎 ) + (𝟏𝟎 ⋅ 𝟏
𝟔𝟎 ) + (𝟑 𝟎
⋅ 𝟔𝟎 )
= (𝟒 ⋅ 𝟑𝟔𝟎𝟎) + (𝟏𝟎 ⋅ 𝟔𝟎) + (𝟑 ⋅ 𝟏)
= (𝟏𝟒, 𝟒𝟎𝟎) + (𝟔𝟎𝟎) + (𝟑)
= 𝟏𝟓, 𝟎𝟎𝟑
 Mesopotamia

The Babylonians also developed another revolutionary

mathematical concept, something else that the Egyptians,
Greeks and Romans did not have, a character for zero, although
its symbol was really still more of a placeholder than a number
in its own right.
Mesopotamia
 Mesopotamia

Babylonian Clay tablets

We have evidence of the development of a complex system of

metrology in Sumer from about 3000 BCE, and multiplication
and reciprocal (division) tables, tables of squares, square roots
and cube roots, geometrical exercises and division problems
from around 2600 BCE onwards.
 Mesopotamia

Babylonian Clay tablets

Later Babylonian tablets dating from about 1800 to 1600 BCE

cover topics as varied as fractions, algebra, methods for solving
linear, quadratic and even some cubic equations, and the
calculation of regular reciprocal pairs (pairs of number which
multiply together to give 60).
 Mesopotamia

Babylonian Clay tablets

One Babylonian tablet gives an approximation to √2 accurate to

an astonishing five decimal places. Others list the squares of
numbers up to 59, the cubes of numbers up to 32 as well as
tables of compound interest. Yet another gives an estimate for
π of 3 1⁄8 (3.125, a reasonable approximation of the real value
of 3.1416).
Babylonian Clay tablets
Babylonian Clay tablets
 Mesopotamia

The Babylonians used geometric shapes in their buildings and

design and in dice for the leisure games which were so popular
in their society, such as the ancient game of backgammon.
 Mesopotamia

Their geometry extended to the calculation of the areas of

rectangles, triangles and trapezoids, as well as the volumes of
simple shapes such as bricks and cylinders (although not
pyramids)