Download Roughly five thousand years ago in a region the Greeks called

Babylonian Mathematics: Crossing the Chasm from
Concrete Counting to Abstract Concepts
Paper 3
History of Mathematics
9-18-00
Problem 2.3
ABSTRACT NOT REQUIRED DURING THIS SEMESTER.
Roughly five thousand years ago in a region the Greeks called Mesopotamia, the “land
between the rivers,” (Burton 19), a civilization emerged that spawned an empire whose
inhabitants are generally referred to in modern discussions as Babylonians. The Babylonian
empire “stretched (800 miles) from the Persian Gulf in present-day Iraq to the headwaters of the
Euphrates River in Turkey” (Kerr 1998). The map in the Appendix shows cities of ancient
Mesopotamia overlaid on modern Iraq (Oriental Institute). Since the fertile land between the
twin rivers, Tigris and Euphrates, was well suited for civilizations to thrive, over time the
Babylonians established written languages, developed metalworking skills, surveyed the land
and built cities, and became skilled tradesmen and engineers (Eves 37). Indeed, based on
transcriptions of ancient findings done thus far, this prosperity empowered the Babylonians to
pursue some interesting facets of mathematics. Specifically, by developing an efficient
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mathematical language and immersing themselves in computing numerous mathematical tables,
they made appreciable inroads into investigating the abstract nature of mathematics.
Since the Babylonians could not easily draw curved lines, they developed a mathematical
language based on cuneiform, wedge-shaped, symbols (O’Connor). Using a stylus to create the
symbols, their numbers are comprised of two basic signs:
unit and
, a vertical wedge to depict the ones
, a corner wedge to depict the tens unit (Melville). Furthermore, Aaboe explains the
value of each symbol depends on its position such that moving one sign to the left of the unit
position multiplies the symbol’s value by 60 (9). In modern notation, these cuneiform symbols
are transcribed by separating the number signs, or digits, with commas. For example, an
inscribed number may appear as
. Then, its transcription would appear as
1, 25, 30 and, its value would be
1 60 2  23  601  30  60 0
 3600  1380  30  5010 (Aaboe 9).
Likewise, as moving one digit to the left of the unit position multiplies the value of a number by
60, moving one digit to the right divides its value by 60 (Aaboe 15). Modern transcription, in
order to clarify this distinction, uses a semicolon to separate the whole from the fractional part of
a number. For example,
1, 25; 30 = 1  601  25  600 
30
601
1
 85 ,
2
and,
1;25,30  1  600 
25
30
 2
1
60
60
2
 1
17
(Aaboe 16).
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According to Vogeli, the Babylonians effectively used a base-sixty or sexagesimal
number system in combination with a base-ten system: numbers less than 60 were denoted by a
simple base-ten grouping and numbers greater than 60 were denoted by a positional system
based on groups of 60 (36). As Burton points out, however, the Babylonians actually used a
partial positional system, because in its early development they omitted a symbol for zero at the
end of a number or within a number (23). “One could only rely on the context to relieve the
ambiguity “(23). Later on, the Babylonians used either a space between symbols or two
symbols, one placed vertically atop the other (Aaboe 10). Despite some possible confusion due
to the lack of a zero in the early Babylonian texts , “this system of numeration was superior to
other ancient systems, since it was the first to employ the concept of place value” (Vogeli 37).
Not only was the use of place value a noteworthy contribution to mathematics but also, as
Aaboe explains, the use of the sexagesimal number system provided many benefits when
compared to a base-ten system used by other ancient civilizations, such as the Greeks. In
comparison, both systems use a finite number of symbols to express all integers and both assign
importance to a symbol’s position (16). But, this is where the similarities end. The Babylonians
could express more fractions as finite sexagesimal fractions, which means their denominators are
comprised of only multiples of the prime factors of 60: 2, 3, and 5 (19). For example, given “the
denominators 2, 3, 4, . . . , 20 , seven will give finite decimal equivalents (2, 4, 8, 5, 10, 16, 20)
. . . while thirteen have finite sexagesimal expansions (2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16,18, 20)”
(Aaboe 19). Burton recognized that, “The key to the advances the Babylonians made appears to
have been their remarkably facile number system. The excellent sexagesimal notation enabled
them to calculate with fractions as readily as with integers and led to a highly developed algebra”
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(61). The Babylonian number system, therefore, was an efficient tool that provided the latitude
to further explore numbers computationally.
In fact, the Babylonians capitalized on the relative computational ease afforded by the
base-sixty system by inscribing their findings on a vast number of clay tablets. “The
Babylonians, freed by their remarkable system of numeration from the drudgery of calculation,
became indefatigable compilers of arithmetic tables, some of them extraordinary in complexity
and extent” (Burton 61). To date, out of the half-million tablets archaeologists have unearthed to
date from Mesopotamia, 400 are strictly mathematical (Eves 39). Moreover, at least half of the
mathematical tablets contain tables and the remaining contain lists of problems (41). The
mathematical tables include huge databases of numbers depicting multiplication, reciprocals which were used to reduce multiplication to division – squares, square and cube roots and some
powers (Melville). Conversely, “ A problem text often consists of many problems of a similar
kind, pedagogically arranged in order of increasing difficulty” (Aaboe 23).
Rather than using these word problems to teach students how to create a geometric figure or
develop a proof, they were typically meant to teach students how to arrive at a specific numerical
result (Melville). In fact, “there are scores of clay tablets that indicate that the Babylonians of
2000 B.C. were familiar with our formula for the quadratic equation” (Burton 64). Baumgart
adds that the Babylonians seem to use a recipe to solve algebraic problems (236):
1.
2.
3.
4.
5.
State the problem
List the given data
State the answer
Explain the methodology used
Check the answer.
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Interestingly, the answer is given before the problem is worked out, perhaps underscoring the
importance placed on getting the answer. In the course textbook Eves, problem 2.3(a) provides
an example of how the Babylonians would solve a quadratic equation (57):
A Babylonian problem asks for the side of a square if the area of
the square diminished by the side of the square is the (sexagesimal)
number 14,30.
The tablet specifies the following procedural solution of the word problem:
‘Take half of 1, which is 0;30; multiply 0;30 by 0;30, which is
0;15; add the 0;15 to 14,30 to obtain 14,30;15. This last is the
square of 29l30. Now add 0;30 to 29;30; the result is 30, which
is the side of the square.’
It can be shown that this Babylonian solution is equivalent to the modern algebraic equation
x 2 - px  q
(1)
such that
2
x

p
  q
2

p
.
2
(2)
Substituting the value of x defined in equation (2) into (1) yields

x 2 - px  


2
p
  q
2

p 
2

2

p



2
p
  q
2

p 
2

(3)
Expanding the right side of equation (3)
2
  p 2

p p
     q   2     q
 2 

2 2



p2
4
2
p
 q  p   q
2
5

p2
4
2
p
 p   q
2
2

p2
p
 p   q
4
2


p2
2
p2
4
(4)
(5)
Upon collecting like terms in equation (5), all terms cancel except for q. Thus,
 x 2 - px  q
Now it can be shown how the words used in that problem can be parametrically fitted into the
quadratic equation. The stated problem asked for “the side of a square if the area of the square
diminished by the side of the square is the (sexagesimal) number 14,30. Substituting into the
general equation
x 2  px  q , with p = 1 and q = 14, 30,
yields
x 2  x  14 ,30 .
(6)
Careful analysis of the Babylonian solution stated in words to equation (6) yields the
following results:
“Take half of 1, which is 0;30; multiply 0;30 by 0;30, which is 0;15 “ yields
2
 30  30  15
p
 0;15    .
   
 60  60  60
2
Next“. . . Add the 0;15 to 14,30 to obtain 14,30;15 “ yields
2
p
0;15  14,30  14,30;15     q .
2
Then, “ . . . this last is the square of 29,30” yields
2
14,30;15  29;30   p   q  29;302
2
2
or equivalently,
6
14,30;15
2
2
 p
  q
2
 29;30 
 29;30 .
Finally, “This last is the square of 29;30. Now add 0;30 to 29;30; the result is 30.” This yields
2
29;30  0;30  30 
 p
  q
2
 0;30 
30
Although Babylonian transcriptions did not indicate that symbols were manipulated in
formulas to obtain an answer, they do demonstrate the Babylonians understood the concept of
the quadratic equation. The Babylonian mathematician may not have used a general “quadratic
formula”; however, “the instructions in these concrete examples are so systematic that we can be
pretty sure they were intended to illustrate a general procedure”(Burton 64). Baumgart notes
that, although the Babylonians also knew how to solve systems of equations by elimination,
oftentimes they chose to use a parametric method of solution (236). As indicated in Eves,
problem 2.3(b) on page 58, this parametric method can be demonstrated by showing that any
quadratic equation of the form ax2 + bx + c = 0 can be reduced to one of the following forms:
y 2  py  q
y 2  py  q
(1)
(2)
(3)
y2  q  py ,
such that p and q are both nonnegative.
Given ax2 + bx + c = 0, such that a > 0, prove that this general quadratic equation can be
transformed into one of equations (1), (2) or (3) above. Define the following variables:
Let y  x a , so x 
y
;
a
7
p
and,
b
a
, so b  p a ;
q = c.
Substituting these three parameters into the general quadratic equation (1),

 y 
a

 a
y2
2


p a 
y 
  q  0
 a
py

q  0.
Since the general equation, ax2 + bx + c = 0, was written so that a > 0, there exists four possible
cases for the signs of b and c. Specifically,
(i)
(ii)
(iii)
(iv)
b < 0 and c < 0,
b > 0 and c < 0,
b < 0 and c > 0,
b > 0 and c > 0.
If case (i) were true, then p < 0 and q < 0 would yield equation (2):
y2
 py  q .
If case (ii) were true, then p > 0 and q < 0 would yield equation (1): y 2
 py  q
If case (iii) were true, then p < 0 and q > 0 would yield equation (3): y 2
 q  py
If case (iv) were true, the result would yield either a negative or complex answer for y. Since the
Babylonians, as well as other ancient civilizations, neglected the negative root, Babylonian
mathematics disregarded negative solutions to quadratic equations (Burton 68). Furthermore,
Eves points out that not only could Babylonians solve quadratic equations but they investigated
some third degree and fourth degree equations (42).
In summary, modern transcriptions of clay tablets unearthed from ancient Babylonian
sites demonstrate a noteworthy appreciation of some algebraic concepts, which is greatly buoyed
by a highly efficient numerical system as well as a weariless compilation of databases. Though
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there is no evidence the Babylonians used symbols to manipulate algebraic expressions, their
systematic algorithms document an understanding of the principles of the quadratic equation.
Notably, as Ritter states, “Numbers had become free-floating and the detachment of number and
measure was complete. The end of the third millennium B.C. saw the birth of the concept of
number, abstracted from any particular unit” (16).
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References
Aaboe, Asger. Episodes From the Early History of Mathematics. New Mathematical Library,
no.13. New York: Random House and L.W. Singer, 1964.
Baumgart, John K. “The History of Algebra.” Historical Topics for the Mathematics
Classroom. Reston, VA: The National Council of Teachers of Mathematics, 1989: 233-260.
Burton, David M. The History of Mathematics: An Introduction. New York, The McGraw-Hill
Companies, Inc., 1997.
Eves, Howard. An Introduction to the History of Mathematics. New York, Saunders College
Publishing, 1990.
Kerr, Richard A. “Sea-floor Dust Shows Drought Felled Akkadian Empire.” Science, Jan
1998: 325-327.
Melville, Duncan J. “Mesopotamia Mathematics.” http://it.stlawu.edu/~dmelvill/mesomath/html (1999)
O’Connor, J.J. “Babylonian and Egyptian Mathematics.” http://www-groups.dcs.stand.ac.uk…opics/Babylonian-and-Egyptian.html (1996)
Oriental Institute Map Series. “Iraq Site Map” http://wwwoi.uchicago.edu/OI/INFO/MAP/SITE/Iraq_Site300dpi.gif (2000)
Ritter, James. “Sumerian Sums.” UNESCO Courier, Nov 1993: 14-17.
Vogeli, Barry D. “Babylonian Numeration System.” Historical Topics for the Mathematics
Classroom. Reston, VA: The National Council of Teachers of Mathematics, 1989: 36-38.
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