Babylonian
Mathematics
Babylonian mathematics
Also know as Assyro-Babylonia Mathematics.
Developed or practice by the people of Mesopotamia, from
days of the early Sumerians to fall of Babylon in 539 BC.
The Babylonians used base number system is 60. the base
number system used today for measuring time, 60 seconds in
a minute, 60 mins in an hour and for measuring 360 degrees
in a full turn.
It is range of numeric and more advanced mathematical
practice in the ancient near east, written in cuneiform
script.
Babylonian mathematics is derived from some 400 clay
tablets unearthed since the 1850s.
Otto E. Neugebauer
AustralianAmerican
By studying clay tablets, he
discovered that the ancient
Babylonians knew much more
about mathematics and astronomy.
Old babylonians
most clay tablets that describe Babylonian
mathematics belong to them.
Their excellent sexagesimal notation enabled to
calculate with fractions as readily as with integers and
led to a highly developed algebra.
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. Numerous
tables give the squares of numbers 1 to 50 and also the
cubes, square roots, and cube roots of these numbers.
Sexagesimal notation
Technique of expressing
numbers in base.
Sexagesimal means base 60.
Babylonian numerals 1-59
Babylonian Numerals
We will use for 10 and for 1, so
the number 59 is
For numbers larger than 59, a “digit”
is moved to the left whose place value
increases by a factor of 60.
So 60 would also be .
Babylonian Numerals
Consider the following number
We will use the notation (3, 25, 4)60.
This is equivalent to
3 60 25 60 4 12,304
2
Babylonian Numerals
Drawbacks:
The lack of a sexagesimal point
Ambiguous use of symbols
The absence of zero, until about 300 BC
when a separate symbol was used to act
as a placeholder.
These lead to difficulties in determining the
value of a number unless the context gives an
indication of what it should be.
Babylonian Numerals
Tosee this imagine that we want to determine the
value of
This could be any of the following:
2 60 24 144
2 60 24 60 8640
2
24 2
2 2
60 5
Babylonian Numerals
TheBabylonians never achieved an absolute
positional system.
We will use 0 as a placeholder, commas to
separate the “digits” and a semicolon to indicate
the fractional part.
For example, (25, 0, 3; 30)60 will represent
30 1
25 60 0 60 3 90,003
2
60 2
More Examples
(25, 0; 3, 30)60 represents
3 30 7
25 60 0 2 1500
60 60 120
(10, 20; 30, 45)60 represents
30 45 41
10 60 20 2 620
60 60 80
More Examples
(5; 5, 50, 45)60 represents
5 50 45 1403
5 2 3 5
60 60 60 14400
Note: Neither the comma (,) nor the
semicolon (;) had any counterpart in the
original Babylonian cuneiform.
Babylonian Arithmetic
Babylonian tablets contain evidence
of their highly developed
mathematics
Some tablets contain squares of the
numbers from 1 to 59, cubes up to 32,
square roots, cube roots, sums of
squares and cubes, and reciprocals.
Babylonian Arithmetic
For the Babylonians, addition and subtraction
are very much as it is for us today except that
carrying and borrowing center around 60 not
10.
Let’s add (10, 30; 50) + (30; 40, 25)
60 60
10, 30; 50, 0
30; 40, 25
11, 1; 30, 25
Babylonian Multiplication
Using
tablets containing squares, the
Babylonians could use the formula
ab [ a b a b ] 2
2 2 2
Or, an even better one is
ab [ a b a b ] 4
2 2
Babylonian Multiplication
10 1,40 19 6,1 Using the table at the right,
11 2,1 20 6,40 find 1112.
12 2,24 21 7,21 Followingthe formula, we
13 2,49 22 8,4 have 1112 =
14 3,16 23 8,49 (232 – 12) 4 =
15 3,45 24 9,36 (8, 48)60 4 =
16 4,16 25 10,25
(2, 12)60.
17 4,49 26 11,16
18 5,24 27 12,9
Babylonian Multiplication
Multiplication can also
be done like it is in our
10; 50
number system.
Remember that carrying 30; 20
centers around 60 not 3, 36, 40
10.
For 5, 25, 0
example,
5, 28; 36, 40
5 1
10 30
6 3
Another large group of
tables deals with the
reciprocals of numbers.
The standard format of
such a table usually
involves two columns of
gures, such as where the
product of each pair of
numbers is always 60
Babylonian Division
Correctly seen as multiplication by the
reciprocal of the divisor.
For example,
2 3 = 2 (1/3) = 2 (0;20)60 =
(0;40)60
Babylonians approximated reciprocals
which led to repeating sexagesimals.
Babylonian Division
44 12 = 44 (1/12) = 44 (0;5)60 = (3;40)60.
Note: 5 44 = 220 and 220 in base-60 is
3,40.
12 8 = 12 (1/8) = 12 (0;7,30)60 =
(1;30,0)60.
25 9 = 25 (1/9) = 25 (0;6,40)60 =
(2;46,40)60.
Babylonian Division
When fractions generated repeating
sexagesimals, they would use an
approximation.
Since 1/7 = (0;8,34,17,8,34,17,...)60.
They would have terminated it to
approximate the solution and state that it
was so, “since 7 does not divide”.
They would use 1/7 (0;8,34,17,8)60.
Babylonian Algebra
Babylonian could solve linear
equations, system of equations,
quadratic equations, and some
cubic as well.
The Babylonians had some sort of
theoretical approach to
mathematics, unlike the
Egyptians.
• The Babylonians were
aware of the link
between algebra and
geometry.
• They used terms like
length and area in their
solutions of problems.
• They had no objection to
Old Babylonian cuneiform text
combining lengths and
containing 16 problems with areas, thus mixing
solutions. (Copyright British
Museum.) dimensions.
Polybius
Greek historian
Known for The Histories,events of the
Roman Republic, 220-146 BC
tells us that in his time unscrupulous
members of communal societies cheated
their fellow members by giving them land
of greater perimeter (but less area) than
what they chose for themselves.
quadratic equation form x2 + ax = b where a
and b were not necessarily integers but b
was always positive. The solution for this
equation is
Although the Babylonian mathematician
had no “quadratic formula” that would
solve all quadratic equations.
Quadratic Equations
I have added the area and two-
thirds of the side of my square and
it is 0;35. What is the side of the
square?
They solved their quadratic
equations by the method of
“completing the square.”
The equation is 2 35
x x
2
3 60
Completing the Square
You take 1 the coefficient [of x].
Two-thirds of 1 is 0;40. Half of this,
0;20, you multiply by 0;20 and it [the
result] 0;6,40 you add 0;35 and [the
result] 0;41,40 has 0;50 as its square
root. The 0;20, which you multiplied
by itself, you subtract from 0;50, and
0;30 is [the side of] the square.
Amazing! But what is it really saying?
Babylonian Geometry
Many problems dealt with lengths, widths, and
area.
Given the semi-perimeter x + y = a and the area xy
= b of the rectangle. Find the length and width.
Giventhe the area and the difference between
the length and width. Find the length and width.
Babylonian Geometry
The length exceeds the width by 10.
The area is 600. What are the length
and width?
We would solve this by introducing
symbols.
Let x = the length and y = the width,
then the problem is to solve:
x – y = 10 and xy = 600
Babylonian Geometry
Answer: length = 30
width = 20
The End
Reported by:
Joanna Marie M. Echane
