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### Ancient Numeration Systems, Place Value

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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
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```
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.

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

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

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

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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