Mathematics in Early Civilizations

Historical Facts About Ancient Egypt

 

Egypt was able to flourish because of the exceptional climate and topography along the Nile River. Agriculture developed in the Nile valley approximately 5000 B.C.E.  Initially separate, independent agricultural communities joined together into larger groups until there were only two kingdoms:  Upper Egypt and Lower Egypt.  In 3100 B.C. military conquest by Menes joined the two.  Menes was the first of the long line of pharaohs  (32 dynasties) until Cleopatra's end in 31 B.C.   These dynasties allowed Egypt to flourish, since Egypt was protected from outside invasion by the deserts that surrounded her. Map from E-Museum at Minnesota State University

Under single leadership,  Egypt developed a powerful, vast administration system with census taking, taxation, and maintenance of an army all requiring work with numbers.  Vast construction projects also required the development of mathematics.

Our major knowledge of Egyptian culture comes  extensive sources of Egyptian hieroglyphics  (sacred signs).  Hieroglyphics remained indecipherable until 1799 when in Alexandria the trilingual Rosetta Stone was discovered.  The discovery was due to Napoleon Bonaparte's attempted seizure of Egypt in 1798.  While  the invasion was a military disaster, it was a great scientific achievement.  Napoleon had taken 167 scholars with him to make a complete investigation of ancient and modern Egypt.  One of their major finds was the black basalt Rosetta Stone.   It contains three panels  -- each with a different form of writing ( Greek, demotic --Egyptian script used post 7th Century, hieroglyphic ) of the same text.    Jean Francois Champollion (1790-1832) was able to decipher the hieroglyphics based on the Greek text.
 

Two Egyptian  papyri containing collections of mathematical problems with their solutions provide us with the basis of much of our knowledge about Egyptian mathematics.

1.  The Rhind Mathematical Papyrus named for its owner  A.H. Rhind (1833-1863).  It was written in hieratics ( a cursive form of hieroglyphics better suited to writing with pen on papyrus) in 1650 BC by a scribe named Ahmes.  It was originally a single scroll of 18 feet long 13 inches high.    It resides in the British Museum.

2.  The Moscow  (also called Golenishchev ) Mathematical Papyrus purchased by V. S. Golenishchev, who died in 1947.  This papyrus dates from 1700 BC and  15 ft long and 3 inches high.   It resides in the Museum of Fine Arts in Moscow.
 

Egyptian  Number System

(From  Egyptian Mathematics by Don Allen)

Other numbers were expressed by using the symbols additively -- the number represented by the set of symbols is the sum of the numbers represented by each symbol.  The normal direction of writing was right to left -- with the larger units therefore written to the right, and decreasing to the left.

Primarily only unit fractions (1/n) existed in the Egyptian number system.   Such fractions were written by placing a "mouth" above the number representing the denominator:

(From The MacTutor History of Mathematics)

Note that when the number was large, the mouth could be placed over the start of the number (in this case at the right).
 

Addition was carried out by collecting like symbols and then reducing by exchanging groups of 10 like symbols for the next higher unit.

Subtraction was carried out by the process in reverse.  Borrowing was used when necessary (exchanging a larger unit for 10 of the next smaller unit).
 

Egyptian Arithmetic

Integer Multiplication
Multiplication was accomplished by successively doubling one of the two multiplicands and then addition the appropriate doubles to form the product -- best seen by example:

To multiply 19 and 62   ( using doubling  of  62 ).
 

Note the numbers to the left represent the number of times 62 is multiplied (by the doubling process).   We stop at 16 because 32 would be larger than 19.  We then note that 1 + 2 + 16 = 19 and so use the addition of 62 + 124 + 992 = 1178 to obtain the product.

Integer Division

Egyptian division is multiplication in reverse -- the divisor is repeatedly doubled to give the dividend.  For example to divide 72 by 6:
 

By determining that 24 + 48 = 72,  we conclude that 4 + 8 = 12 is the answer.

Division with Remainders

Double the divisor until the next duplication would exceed the dividend.  Then start halving the divisor to complete the remainder.
To divide 35 by 8.
 
 

Since 35 = 32 + 2 + 1, we use 4 + 1/4 + 1/8 as the answer.   Other examples are more complicated -- since we might have to compute 1/3 ,etc.

Try this link to interactive  Egyptian Calculator:

 Egyptian Calculator

 Exercises:

E1.  Write each of the following numbers in Egyptian hieroglyphics:
a)  1234
b)  1,240,307
c)   1/123
E2.  a)  Write the following two numbers in hieroglyphics and then add using hieroglyphics.  Express the answer in  hieroglyphics, then translate to our number system.
      678 + 345.
 b)  Write the following two numbers in Egyptian hieroglyphics, subtract the second from the first , then translate to our system.
     2631 - 642
E3.  Use the Egyptian method of multiplication by doubling to compute the following products:
a)  26 * 33
b)  17 * 21
E4.  Use the Egyptian method of division to compute the following divisions:
a)  184 / 8
b)  27 / 8
 

Egyptian Geometry

It is accepted that the origin of geometry is in ancient Egypt, arising out of the need to survey land.  Because the Nile frequently flooded, washing away land, property had to be frequently surveyed for taxation. Egyptian surveyors were highly skilled in geometry.  Sources of information for the geometry of ancient Egypt include inscriptions on temples, the geometrical problems of the Rhind Papyrus as well as the Moscow Papyrus, and the written history of Egypt by the Greek Herodotus.  The precise construction of the pyramids also indicates the use of geometry. More information about the pyramids, including the Great Pyramid at Gizeh, erected in 2600 B.C. by Khufu at  Pyramids and Temples

The mathematics of Egyptian geometry is documented by examples of rules for determining areas and volumes of common plane and solid objects.  They appear to be based on trial and error results and observations as opposed to theoretical proof.
 

Examples of Geometrical Problems from the Rhind and Moscow Papyri

The geometrical problems of the Rhind Papyrus are 41 -- 60.

Statement of Rhind Papyrus Problem 50:
Example of a round field of a diameter 9 khet.  What is its area?  Take away 1/9 of the diameter, namely 1;  the remainder is 8.  Multiply 8 times 8; it makes 64.  Therefore it contains 64 setat of land.

In-class Exercise:  Convert this to a formula.   Generalize the formula (for arbitrary diameter d).  Compare it to the formula we know to be the correct formula for the area of a circle with diameter d.   What value do the Egyptians seem to be using for Pi?  Approximately what is the size of error in their estimate for Pi?   What is the approximate area (to three decimal places) with the correct value the area?

Justification of the Egyptians formula:
(Graphic from  Don Allen's Egyptian Mathematics

The area of the inscribed circle is approximated by the octagon inscribed in the same square.
Derive a formula for the area of this octagon.

(This is the presumed derivation based on the problem 48 -- In this problem there was a crude drawing of the square with the four triangles at the vertices,  In the middle was a symbol for the number 9).

Volumes of Pyramids
The Egyptians had the correct formula for the volume of a square pyramid:
V = (h/3)a².

Even more remarkably,  they  had the formula for the volume of a truncated square pyramid.   The correct formula is
V = (h/3)*(a² + ab + b²)  where h is the height, a is the length of the side of the top, and h is the length of the side of the base.   This is indicated in the following  problem from the Moscow Papyrus (Source:   Burton, David. The History of Mathematics)
Example of calculating a truncated pyramid.  If you are told:  a truncated pyramid of 6 for the vertical height by 4 on the base by 2 of the top:  You are to square this 4; result 16.  You are to double 4; result 8.  You are to square this 2; result 4.  You are to add the 16 and the 8 and the 4;  result 28.  You are to take 1/3 of 6; result 2.  You are to take 28 twice; result 56.  See, it is of 56.  You will find it right.    ( Accompanied by a drawing. )
In-class exercise:  Derive this formula  V = (h/3)*(a² + ab + b²)   from the statement of the problem above.

Drawing of Problem from Don Allen's Egyptian Mathematics  site (Upper left is similar to what appeared on the papyrus).

 
 
 

Babylonian Mathematics

Historical Facts about the Babylonians

Babylonian refers to the ancient peoples who lived in the area between the Tigris and Euphrates rivers.  This area was referred to as Mesopotamia by the Greeks and currently occupied primarily by Iraq and Syria.

Civilization developed about the same time as in ancient Egypt -- between 3500 and 3000 B.C.  Sumer in southern Mesopotamia was the first area of civilization.  It consisted of approximately twelve communities of several thousand each.  The cities had large-scale irrigation systems, a governmental bureaucracy, laws, and even postal service.  People had specialized occupations including architects, blacksmiths, cooks, engineers, dancers, priests and tavern owners.  Their technology included plows, wheeled vehicles, boats, wheel-thrown pottery, weaving, and ziggurats (adobe brick towers).

Knowledge of the civilization there is still growing from the deciphering and documentation of archaeological artifacts. An important collection of artifacts resides at the  Oriental Institute at the University of Chicago.  A graphic of a tablet  can be found at YBC 7289   which is referenced in the lecture  Mesopotamian Mathematics  by Duncan Melville at St. Lawrence University.

Babylonians accomplished writing with pictures much like Egyptian hieroglyphics after 3000 B.C.  Their writing medium was moist clay and was written upon via impressions with a stylus with a triangular end.   Because the clay dried quickly,  most documents were short and written at one time.  They were very enduring and many still exist today.

The major breakthrough in the deciphering of the system of writing on the clay tablets came in 1846, when Henry Rawlinson deciphered the equivalent of the Rosetta stone -- called the Behistun Rock.  This equivalent was a massive carving on the Behistun Cliff in Iran four hundred feet above the ground.  The carving gives an account of a military victory by Darius I in Persia in 516 B.C.  It is written in three languages (Elamitic -- Old Persian, Babylonian, and Persian).  Bawlinson deciphered the cuneiform writing system.  However, translating and analyzing mathematical cuneiform tablets commenced with Otto Neugebauer  in the 1930's and is still not yet complete.

Mesopotamia did not have sufficient protection from outside invasions.  From 3000 B.C. and for nearly 3000 years invaders battled for control in Mesopotamia.  Despite this,  the areas culture was maintained, and continued to spread through trade and also through the invasions themselves.

Mathematics was needed for monitoring of grain holdings,  merchant transactions,  and calculations for irrigation and building projects.  Astronomy also made great strides in Mesopotamia.

Babylonian Number System

The Babylonians used cuneiform writing on clay tablets.  This consisted of using a stylus to make impressions with vertical strokes for straight lines, and the end for a triangular shape.   By combining these, wedge -shapes (cuneus in Latin) were created.
For an example of a Babylonian tablet see:  YBC 7289
The Babylonians used a base 60 positional number system with two basic symbols -- a  vertical wedge for the number 1 and a broad side-ways wedge for the number 10 (with 10's written to the left of 1's).    Groups of symbols were spaced to indicate descending powers of 60.    Sometimes a subtractive symbol  (vertical wedge followed by sideways wedge) was used.
 

-- One

-- Ten

 
To see examples of how they combined these see Cuneiform numbers

Because their early positional system had no placeholder for 0, conflicting interpretations were possible.  The numbers 2*60 + 35   = 155 and 2*60² + 35  = 7235  would be written the same way.   Context of the problem being representing would be used to resolve the meaning.   (Because of the large base, 60, this was generally easy to determine.

Later (after 300 B.C.) a placeholder symbol for empty positions was introduced to avoid the conflict above.  Problems still remained because this symbol was only used for empty positions in the middle of the number -- not at the right end.   Hence one could not be sure whether the right-most number representing a unit, multiple of 60, or 60², etc.

In discussing the arithmetic of Babylonian mathematics, it is common to use the following notation:
Separate each position value by a  comma ',' and  integers from fractions by a semicolon, ";".
In this notation the number 31, 0, 20; 30 , 1 would represent
31*60² + 0 * 60 + 3 + 30/60 + 1/60²
 

Exercises Set 1:
B.1.  Write each of the following numbers in base 60 form and then cuneiform notation.
      1234
      23561

B.2.  Convert the following numbers from base 60 (sexagesimal) to decimal:
     2, 31, 45
     2, 12; 12

B.3.   Find via search through references or the Internet suggestions of why the Babylonians might have used base 60 for their number system.

B.4.  Discuss the advantages of the Babylonian number system over the Egyptian number system.
 

Babylonian Algebra

The Babylonian sexigesimal number system made algebraic operations much less tedious.  There numerous cuneiform tablets that contain tables of numbers.
 
 

What is being represented in this table?  (Hint  -- what does each row have in common).
Why are 7, 11, 13, 14, and 17 not in the left column.

Babylonians only dealt  with finite sexagesimal fractions.  Reciprocals of numbers which were nonterminating sexagesimal  (e.g. 1/7 )  would be approximated.

For example,  a table gives an approximation of 1/7 as
 

;8, 34, 16, 59 < 1/7 < ;8,34, 18

Exercise B.5:  Translate the approximations to base 10 and then compare their accuracy to the value of 1/7.
 

The Babylonians used the formula

     ab = ((a + b)² - a² - b²)/2

to make multiplication easier.   They also had extensive multiplication tables and tables of squares to use.

Division is a harder process. The Babylonians did not have an algorithm for long division. Instead they
based their method on the fact that

     a/b = a*(1/b)

That is,  to compute 7/2:

• Find the reciprocal of 2.
• Then multiply the reciprocal of 2  -- i.e. 0;30  by 7:

 7 (0;30)  = ;210  = 3;30   (i.e. 3 1/2  in our notation).

Exercise B.6:
Express the numbers in the following problem in sexagesimal notation -- then compute the division using the method of multiplying by the reciprocal  (also write down the reciprocal first in sexigesimal notation.   Then convert to a fraction in our notation.
a)  31/15

b)  49 / 12
 
 
 

Babylonian Discoveries: The Quadratic Equation

Problem:  I have added the area and two-thirds of my square and it is 0;35.  What is the side of my square?
Solution:  You take 1.  Two-thirds of 1 is 0;40.  Half of this, 0; 20, you multiply by 0;20 and it, 0;6,40, you add to 0;35 and the result 0;41;40 has 0;50 as its square root.  The 0;20 which you have multiplied by itself, you subtract from 0;50, and 0;30 is the side of the square.

Explanation:

Translate the original problem into the form of an equation in modern notation:
x² + (2/3)*x = 35/60  or x² + (2/3)*x - 35/60 = 0
This is a special case of the problem
x² + a*x = b  or x² + a*x - b = 0  with a = 2/3, b = 35/60.
Solving this by completing the square, or simply using the quadratic equation gives:

x = (+/-)squareroot ( (a/2)² + b) - a/2.

Ignoring the negative square root we have for this problem:

x = squareroot ( ((2/3)/2)² +35/60) - (2/3)/2.

or

x = squareroot( (0;40/2)^2 +;35) - 0;40/2
  =  squareroot( (0;20)^2 +;35) - 0;20
  =  squareroot( (0;6,40) +;35) - 0;20
  =  squareroot( 0:41,40) - 0;20
  =  0;50 - 0:20
  =  0;30

Following the text of the solution you can see the calculations used above.
Thus by a specific example they have given the method of solving the quadratic equation of the form

x² + a*x = b .

Other Quadratic Problems:  Dimensions of Rectangles

In-class Exercise:
Read the method of solution for the two quadratic problems.  Write out the formula represented by each of the methods, using l for length and w for width.  Verify that the formulas are correct.
Apply the algorithms to the following problems:

Problem 1:  Sum of length and width = 30.   Area  = 200.
Problem 2:  Difference of length and width = 30.  Area = 2800.

The Babylonian tablets indicate that they could solve other equations simultaneously.  An example of these type of problems as an exercise from Burton, David:  The History of Mathematics follows:

Exercise B.7:  Following the hints, solve the following Babylonian problems:
a) (We will do this in class)   x = 30, xy -(x-y)² = 500.   Hint:  Subtract the second equation from the square of the first to get a quadratic in x - y.  Solving this quadratic leads to a system of the form x - y = a and xy = b.
b)  x + y = 50, x² + y² + (x-y)² = 1400.  Hint:  Subtract the square of the first equation from twice the second equation, again to get a quadratic in x-y.

Babylonian Geometry:  Plimpton 322 Tablet