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Ancient Numeration Systems, Place Value


Date: 6/4/96 at 19:0:10
From: Anonymous
Subject: Ancient Numeration systems, Place Value

I am trying to get some information on ancient number systems. 
Specifically, I am trying to get information on African, and Roman.

Any help would be greatly appreacieated.

Thanks,
Very stuck!
Hakeem Hart


Date: 6/5/96 at 12:44:2
From: Doctor Jodi
Subject: Re: Ancient Numeration systems, Place Value

Hello there!  

On the Internet, I know of two math history sites which might provide 
sources for further exploration:

The St. Andrew's MacTutor 

  http://www-groups.dcs.st-and.ac.uk/~history/   

has a search page which will return a list of several related web pages.

You will find pictures and the full text of this page at: 

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_and_Egyptian.html   

----------------------------
Babylonian and Egyptian mathematics:

The Babylonians lived in Mesopotamia, a fertile plain between the 
Tigris and Euphrates rivers...

They developed an abstract form of writing based on cuneiform (i.e. 
wedge-shaped) symbols. Their symbols were written on wet clay tablets 
which were baked in the hot sun and many thousands of these tablets 
have survived to this day. It was the use of a stylus on a clay medium
that led to the use of cuneiform symbols since curved lines could not 
be drawn. [picture of one of their tablets]

Perhaps the most amazing aspect of the Babylonian's calculating skills 
was their construction of tables to aid calculation. 

The Babylonians had an advanced number system, in some ways more 
advanced than our present system. It was a positional system with base 
60 rather than the base 10 of our present system. Now 10 has only two 
proper divisors, 2 and 5. However 60 has 10 proper divisors so many 
more numbers have a finite form. 

The Babylonians divided the day into 24 hours, each hour into 60 
minutes, each minute into 60 seconds. This form of counting has 
survived for 4000 years. To write 5h 25' 30", i.e. 5 hours, 25
minutes, 30 seconds is just to write the base 60 fraction, 5 25/60 30/
3600 or as a base 10 fraction 5 4/10 2/100 5/1000 which we write as 
5.425 in decimal notation. 

Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 
BC. They give squares of the numbers up to 59 and cubes of the numbers 
up to 32. The table gives 

       8 = 1 4 which stands for 8 = 1 4 = 1.60 + 4 = 64 

and so on up to 59 = 58 1 (= 58.60 +1 =3481). 

One major disadvantage of the Babylonian system however was their lack 
of a 0. This meant that numbers did not have a unique representation 
but required the context to make clear whether 1 meant 1, 61, 3601, 
etc. 

The Babylonians used the formula 

       ab = ((a + b) - a - b)/2 

to make multiplication easier. Even better is the formula 

       ab = (a + b)/2 - (a - b)/2 

which shows that a table of squares is all that is necessary to 
multiply numbers, simply taking the difference of two numbers that 
were looked up in the table. 

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

       a.b = a.(1/b) 

so what was necessary was a table of reciprocals. We still have their 
reciprocal tables going up to the reciprocals of numbers up to several 
billion. Of course the tables are in their number notation,
but translating into our notation, but leaving the base as 60, the 
beginning of one of their tables would look like 

         2         30
         3         20
         4         15
         5         12
         6         10
         8          7    30
         9          6    40
        10          6
        12          5
        15          4
        16          3    45
        18          3    20
        20          3
        24          2    30
        25          2    24
        27          2    13    20

Now the table had gaps in it since 1/7, 1/11, 1/13, etc. do not have 
terminating base 60 fractions.

This did not mean that the Babylonians could not compute 1/13, say. 
They would write 

1/13 = 7/91 = 7.(1/91) =(approx) 7.(1/90) 

and these values were given in the tables. 

One of the Babylonian tablets which is dated from between 1900 and 
1600 BC contains answers to a problem containing Pythagorean triples, 
i.e. numbers a, b, c with a + b = c. It is said to be the
oldest number theory document in existence. [picture of tablet] 


A translation of another Babylonian tablet which is preserved in the 
British museum goes as follows 

       4 is the length and 5 the diagonal. What is the breadth? Its 
size is not known. 4 times 4 is 16. 5 times 5 is 25. You take 16 from 
25 and there remains 9. What times what shall I take in order to get 
9? 3 times 3 is 9. 3 is the breadth. 

The Egyptians and the Romans had number systems which were not well 
suited for arithmetical calculations. Addition of Roman numerals is 
not too bad but multiplication is essentially impossible. The Egyptian 
system had similar drawbacks. 

The Egyptians were very practical in their approach to mathematics. 

You can see an example of Egyptian mathematics (the Rhind papyrus) 
and of another papyrus (the Moscow papyrus) with a translation into 
hieroglyphics. 

The Rhind papyrus is named after the Scottish Egyptologist A Henry 
Rhind, who purchased it in Luxor in 1858. The papyrus, a scroll about 
6 metres long and 1/3 of a metre wide, was written around 1650 BC by 
the scribe Ahmes who is copying a document which is 200 years older. 
This makes the original papyrus and the Moscow papyrus both date from 
about 1850BC. 

Unlike the Greeks who thought abstractly about mathematical ideas, the 
Egyptians were only concerned with practical arithmetic. In fact the 
Egyptians probably did not think of numbers as abstract quantities but 
always thought of a specific collection of 8 objects when 8 was 
mentioned. To overcome the deficiencies of their system of numerals 
the Egyptians devised cunning ways around the fact that their numbers 
were unsuitable for multiplication as is shown in the Rhind
papyrus which date from about 1700 BC. 

The the Rhind papyrus recommends that multiplication be done in the 
following way. Assume that we want to multiply 41 by 59. Take 59 and 
add it to itself, then add the answer to itself and continue:- 

            41          59
            ______________
             1          59
             2         118
             4         236
             8         472
            16         944
            32        1888
            ______________

Since 64 > 41, there is no need to go beyond the 32 entry. Now go 
through a number of subtractions 

41 - 32 = 9, 9 - 8 = 1, 1 - 1 = 0 

to see that 41 = 32 + 8 + 1. Next check the numbers in the right hand 
column corresponding to 32, 8, 1 and add them. 

                        59
            ______________
             1          59     X
             2         118
             4         236
             8         472     X
            16         944
            32        1888     X
            ______________
                      2419

Notice that the multiplication is achieved with only additions, notice 
also that this is a very early use of binary arithmetic. Reversing the 
factors we have 

             59          41
             ______________
             1           41     X
             2           82     X
             4           16
             8          328     X
            16          656     X
            32         1312     X
            _______________
                       2419

References: 

    1. A Aaboe, Episodes from the Early History of Mathematics (1964). 
    2. R J Gillings, Mathematics in the Time of the Pharaohs 
(Cambridge, MA., 1982). 
    3. G J Toomer, Mathematics and Astronomy, in J R Harris (ed.), The 
Legacy of Egypt (Oxford, 1971), 27-54. 
    4. O Neugebauer and A Sachs, Mathematical Cuneiform Texts (New 
Haven, CT., 1945). 
    5. A B Chace, L S Bull, H P Manning and R C Archibald, The Rhind 
Mathematical Papyrus (Oberlin, Ohio, 1927-29). 
    6. J Hoyrup, Babylonian mathematics, in I Grattan-Guinness (ed.), 
Companion Encyclopedia of the History and Philosophy of the 
Mathematical Sciences (London, 1994), 21-29. 
    7. C S Roero, Egyptian mathematics, in I Grattan-Guinness (ed.), 
Companion Encyclopedia of the History and Philosophy of the 
Mathematical Sciences 
(London, 1994), 30-45. 
    8. J Friberg, Methods and traditions of Babylonian mathematics. 
Plimpton 322, Pythagorean triples, and the Babylonian triangle 
parameter equations, Historia Mathematica 8 (1981), 277-318. 
    9. B L van der Waerden, Science Awakening (Groningen, 1954). 
   10. B L van der Waerden, Geometry and Algebra in Ancient 
Civilizations (New York, 1983).


-------------

Another math history page is located at 

  http://aleph0.clarku.edu/~djoyce/mathhist/mathhist.html   

Try especially the section on regions at 

  http://aleph0.clarku.edu/~djoyce/mathhist/earth.html   

Best luck!

-Doctor Jodi,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
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High School History/Biography

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