Mathematics and Ancient Mesopotamia
Deepak kumar
April 18, 2023
Deepak kumar Mathematics and Ancient Mesopotamia
Method of Computation
Base-60 place value system
Babylonians used various system of numbers in different period of
times but in the ”Old Babylonian” period base-60 place value
system together with a grouping system based on 10 to represent
numbers up to 59 was used. In their system a vertical stoke of
stylus (as given below) would represent 1.
A horizontal stroke of stylus would represent 10.
Deepak kumar Mathematics and Ancient Mesopotamia
How Babylonians used to write numbers
Number less than 59
Suppose they have to write the number 37 then would simply draw
Figure: 37
Deepak kumar Mathematics and Ancient Mesopotamia
Number Greater than 59
Now if they have to write numbers greater than 59 like
3 × 602 + 42 × 60 + 9 = 13329
They would write it as
Figure: 13329
From now we would write 3 × 602 + 42 × 60 + 9 to be 3,42,09.
Deepak kumar Mathematics and Ancient Mesopotamia
Zero
The Old Babylonians did not use a symbol for 0, but often left an
internal space if a given number was missing a particular power.
There would not be a space at the end of a number, making it
difficult to distinguish 3 × 60 + 42(3, 42) from
3 × 602 + 42 × 60(3, 42, 00). Sometimes, however, they would give
an indication of the absolute size of a number by writing an
appropriate word, typically a metrological one, after the numeral.
Thus, “3 42 sixty” would represent 3,42, while “3 42 thirty-six
hundred” would mean 3,42,00.
Deepak kumar Mathematics and Ancient Mesopotamia
Addition
Addition
Extensive tables for multiplication and reciprocals have been found
so far but no tablet involving addition has yet been
discovered.Perhaps the scribes knew their addition procedure so
well that they didn’t find it worthy to explictly write down tables
for addition. Suppose one has to add 23,37 (= 1417) to 41,32 (=
2492).
1 First add 37 and 32 to get 1,09(=69)
2 Next add 23 + 41 + 1 = 1, 05(65)
3 Result is 1,05,09.
Deepak kumar Mathematics and Ancient Mesopotamia
Sexagecimal System
Babylonian treated all fractions as Sexagcimal fraction analogus to
our decimal fractions.First place after ”sexagecimal point”(which
we’ll denote by ;) represents 60th place,second place represents
3600ths and so on.
Fractions
1 Reciprocal of 48 is 0; 1, 15 and
1 15
0; 1, 15 = + 2
60 60
2 Reciprocal of 1,21 (= 81) is 0;0,44,26,40 and
44 26 40
0 : 0, 44, 26, 40 = 2
+ 3+ 4
60 60 60
Deepak kumar Mathematics and Ancient Mesopotamia
Geometry
Scribe did not have any symbolism for ”operations” and
”unknowns”.Therefore the solution of the problems are presented
with only verbal techniques.
The number 0; 52, 30(= 7/8) as the coefficient for the height of a
triangle means that the altitude of an equilateral triangle is 7/8 of
the base, while the number 0; 26, 15(= 7/16) as the coefficient for
area means that the area of an equilateral triangle is 7/16 times
the square of a side.
Therefore their approximation for
√ 7
3≡
4
Deepak kumar Mathematics and Ancient Mesopotamia
Circle
Tablet YBC 7302
Instead of radius r of the circle Babylonians used circumfence as
the defining component of a circle. The diameter and area of cicle
are as follows:- Diameter is 0; 20( 13 ) of cicumfrence and
area is 0; 05 of the square of the circumfrence.Therefore they used
3 as an approximation of pi
Figure: Tablet YBC 7302
Deepak kumar Mathematics and Ancient Mesopotamia
Barge and Bull’s Eye
In analogy with the circle, the defining component of these figures
was the arc making up one side. The coefficient of the area of the
barge is 0; 13, 20 ( 29 ), while that of the bull’s eye is
9
0; 16, 52, 30 ( 32 ).
Thus, the areas of these two figures are calculated as 92 a2 and
9 2
32 a , respectively, where in each case a is the length of that arc.
Figure: Barge and bull’s eye
Deepak kumar Mathematics and Ancient Mesopotamia
Volume of Solids
Prism and Pyramidal structures
Babylonian’s knew how to calculate the volume of rectangular
block and how to calculate the volume of prisms with given base
area.In tablet BM 96954, there are several problems involving a
grain pile in the shape of a rectangular pyramid with an elongated
apex, like a pitched roof. The method of solution corresponds to
the modern formula
hw
V = (l + t/2)
3
Deepak kumar Mathematics and Ancient Mesopotamia
Square Roots and The Pythagoras Theorem
√
2
In the tablet YBC 7289 a square is drawn with side indicated as
30 and two numbers, 1 : 24, 51, 10 and 42; 25, 35,written on the
diagonal.The product of 30 and 1 : 24, 51, 10 is precisely
42; 25, 35.It is then a reasonable assumption that the last number
represents √the length of the diagonal and that the other number
represents 2.
Figure: SquareMathematics
Deepak kumar root of 2and Ancient Mesopotamia
Speculation
Now the question arises that how scribes calculated the square
roots of positive integers.One possible method is based on the
identity
(x + y )2 = x 2 + 2xy + y 2
Suppose we have to find square root of the integer N(or side of the
square whose area is N).
1 b = N − a2
2 Now we have to choose a value of c such that b = 2ac + c 2
3 If a2 is close enough to N then we can approximate c by
c = 21 1a b
√ √
4 Then N = a2 + b ≈ a + 12 1a b
Deepak kumar Mathematics and Ancient Mesopotamia
Pythagoras Theorem and Pythagorean Triplets
Pythagoras Theorem
In any right triangle the sum of area of squares on the legs is equal
to the area of the square on the diagonal.
Tablet Plimpton 322
This tablet give an insight to the fact that scribes knew how to
compute Pythagorean triples. The extant piece of the tablet
consists of four columns of numbers. Other columns were probably
broken off on the left.
Deepak kumar Mathematics and Ancient Mesopotamia
Puzzle
Tablet Plimpton 322
Deepak kumar Mathematics and Ancient Mesopotamia
Figure: Completing the Puzzle
Deepak kumar Mathematics and Ancient Mesopotamia
Speculative Answer
x2 + y2 = d2
x2 d2
=⇒ + 1 =
y2 y2
d x
=⇒ v 2 − u 2 = 1 and v = ,u =
y y
=⇒ (v − u)(v + u) = 1
1
=⇒ u = [(u + v ) − (u − v )]
2
p
=⇒ v = 1 + u 2
Deepak kumar Mathematics and Ancient Mesopotamia
Gnomon
Deepak kumar Mathematics and Ancient Mesopotamia
