The
Mathematics of
Ancient
Mesopotamia
Background
• Mesopotamia: Greek ,
“between the rivers,” specifically the
Tigris and Euphrates. This area
occupies most of what is present-day
Iraq, and parts of Syria, Turkey,
Lebanon, and Iran.
Background
• Thought to be the (or at least a)
“cradle of civilization.”
• Delta region extremely fertile – The
“Fertile Crescent”
• Semi-arid climate required extensive
irrigation projects
Four Empires
• Four civilizations flourished here, from
3100 BCE to 539 BCE. These included the
early Sumerian (3100 – 2400 BCE) and
Akkadian (2400-2100 BCE) empires, and
the later Old Babylonian (1800-1200 BCE)
and Assyrian (1200 -612 BCE;
Ashurbanipal) empires. There followed a
brief Neo-Babylonian period from 612 –
539 BCE. Then Persia. Then Alexander
the Great. Then….
Timeline
Archaic Old Kingdom Int M iddle Kingdom Int Ne w Kingdom EGYPT
3000 BCE 2500 BCE 2000 BCE 1500 BCE 1000 BCE
Sumaria Akkadia Int Old Babylon Assyria M ESOPOTAM IA
Some names you might
recognize
• Hammurabi, founder
of the Old Babylonian
Empire
• Code of Hammurabi -
232 laws, lex talionus,
an eye for an eye
– If anyone strikes
the body of a man
higher in rank than
he, he shall receive
sixty blows with an
ox-whip in public.
Some names you might
recognize
• The Epic of Gilgamesh
– Poem relates story of Gilgamesh, ruler of
Uruk, who seeks out survivor of great
flood in quest of immortality.
• Ur of the Chaldees, Birthplace of
Abraham.
• King Nebuchadnezzar (Neo-
Babylonian Empire)
Sources
• Most of what we know about
Mesopotamian mathematics comes
from several hundred clay tablets
belonging to the Old Babylonian
kingdom, around roughly 1800-1600
BCE.
• Tablets are of two kinds:
– Table texts
– Problem texts
But Before We Go There…
• We need to understand a little about
the number system used in that Old
Babylonian era. The theories about
how it evolved the way it did are
interesting in themselves.
Babylonian Number System
• A base-60 positional system with
individual numbers formed by two
different wedge-shaped marks: a
horizontal wedge worth 10 and a
vertical wedge worth 1.
• Numbers less than 60 were written using
these two symbols in a purely additive
fashion.
Babylonian Number System
Notational Aside:
• Notice that the marks from the
previous table don’t look exactly like
the and the that I used a while
ago. They look even less like the
marks I’ll end up using from here on
because they are easier: ‹ and ˅.
There is considerable variation in both
the original texts and the modern
interpretations.
Babylonian Number System
• These 59 symbols would be written in
a place value system based on
powers of 60. Powers of 60
increased from right to left, just as
powers of 10 increase from right to
left in our system.
Babylonian Number System
• Thus, writing ‹˅ ‹‹‹˅˅˅˅ ‹‹˅˅˅ would
most likely represent
ଶ
or 41,663.
The First of Two Problems
• There was no “0” or placeholder so
we really can’t be sure which power of
60 is being used. Thus, ‹˅˅ ˅˅‹ could
represent either:
, or
ଶ
, or
ଶ
, or many other
possibilities.
The Second of Two
Problems
• Even though the Babylonians used this
system to write fractions as sexagecimals,
there was no “sexagecimal point” or other
way of marking where the fractional part
began. So, again, ‹˅˅ ˅˅‹ could mean any
of:
, or
ିଵ
, or
ିଵ ିଶ
Resolution of Problems
• These two problems were usually
quite easily resolved by the context of
the arithmetic being done, so it
bothers us much more than it did the
Babylonians. Also, there were very
frequently units attached. For
example, any ambiguity in writing
1 1 is resolved if we say $1 1₵.
Resolution of Problems
• In about 300 BC there was a
placeholder symbol invented and
used, but only between symbols,
never at the end.
• In our notation, it
used for 604 but
never 640 or
6400.
Our Babylonian Notation
• We will use a comma to separate
place values, use a 0 when we need
it, and use a semicolon as a
“sexagecimal point.” Thus,
ଵ
But Why 60? Why? Why?
Some suggested reasons:
• Lots of non-repeating sexagecimals,
since 60 as more divisors than 10
(btw, how do you tell if one of our
fractions will terminate or repeat when
converted to a decimal?).
• Sacred or Mystical numbers
• Combination of two number cultures.
Why 60?
• Well, actually, we aren’t sure.
• But we’ll talk about one suggested
solution.
• According to Peter Rudman in his
book How Mathematics Happened:
The First 50,000 Years, it’s probably
more like 6’s and 10’s than 60.
Example:
60x60 60 1 1/60
(carrying row) ٧ ‹ ٧ ‹
(1st number) ‹ ٧٧٧٧٧٧٧ ‹‹‹ ٧٧٧٧٧٧٧٧ ‹‹‹‹ ٧٧٧
(2nd number) + ‹ ٧٧٧٧ ‹‹‹‹ ٧٧٧٧٧ ‹‹‹ ٧٧٧٧٧٧٧٧
(Sum) ‹‹ ٧٧ ‹‹ ٧٧٧٧ ‹‹ ٧
60x60 60 1 1/60
(carrying row) ٧ ‹ ٧ ‹
(1st number) ‹ ٧٧٧٧٧٧٧ ‹‹‹ ٧٧٧٧٧٧٧٧ ‹‹‹‹ ٧٧٧
(2nd number) + ‹ ٧٧٧٧ ‹‹‹‹ ٧٧٧٧٧ ‹‹‹ ٧٧٧٧٧٧٧٧
(Sum) ٧
60x60 60 1 1/60
(carrying row) ٧ ‹ ٧ ‹
(1st number) ‹ ٧٧٧٧٧٧٧ ‹‹‹ ٧٧٧٧٧٧٧٧ ‹‹‹‹ ٧٧٧
(2nd number) + ‹ ٧٧٧٧ ‹‹‹‹ ٧٧٧٧٧ ‹‹‹ ٧٧٧٧٧٧٧٧
(Sum) ‹‹ ٧
60x60 60 1 1/60
(carrying row) ٧ ‹ ٧ ‹
(1st number) ‹ ٧٧٧٧٧٧٧ ‹‹‹ ٧٧٧٧٧٧٧٧ ‹‹‹‹ ٧٧٧
(2nd number) + ‹ ٧٧٧٧ ‹‹‹‹ ٧٧٧٧٧ ‹‹‹ ٧٧٧٧٧٧٧٧
(Sum) ٧٧٧٧ ‹‹ ٧
60x60 60 1 1/60
(carrying row) ٧ ‹ ٧ ‹
(1st number) ‹ ٧٧٧٧٧٧٧ ‹‹‹ ٧٧٧٧٧٧٧٧ ‹‹‹‹ ٧٧٧
(2nd number) + ‹ ٧٧٧٧ ‹‹‹‹ ٧٧٧٧٧ ‹‹‹ ٧٧٧٧٧٧٧٧
(Sum) ‹‹ ٧٧٧٧ ‹‹ ٧
60x60 60 1 1/60
(carrying row) ‹ ٧ ‹ ٧ ‹
(1st number) ‹ ٧٧٧٧٧٧٧ ‹‹‹ ٧٧٧٧٧٧٧٧ ‹‹‹‹ ٧٧٧
(2nd number) + ‹ ٧٧٧٧ ‹‹‹‹ ٧٧٧٧٧ ‹‹‹ ٧٧٧٧٧٧٧٧
(Sum) ٧٧ ‹‹ ٧٧٧٧ ‹‹ ٧
60x60 60 1 1/60
(carrying row) ‹ ٧ ‹ ٧ ‹
(1st number) ‹ ٧٧٧٧٧٧٧ ‹‹‹ ٧٧٧٧٧٧٧٧ ‹‹‹‹ ٧٧٧
(2nd number) + ‹ ٧٧٧٧ ‹‹‹‹ ٧٧٧٧٧ ‹‹‹ ٧٧٧٧٧٧٧٧
(Sum) ‹‹‹ ٧٧ ‹‹ ٧٧٧٧ ‹‹ ٧
Alternating 10-for-1
and 6-for -1 Exchanges
Ok, so…..
• We can understand using groups of
10. But we have to ask:
“Why the freak are there groups of 6?”
• Well, let’s look at Ancient Sumer:
• First, realize that these folks used
different measures for different things,
and that these measures had different
“exchanges” from larger to smaller
units.
• We did this too:
Weight
16 ounces = 1 pound
14 pounds = 1 stone
8 stone = 1 hundredweight
20 hundredweight = 1 ton
(except for us 100 pounds = 1
hundredweight, and 20 hundredweight = 1
ton = 2000 pounds)
Capacity
8 (fluid) ounces = 1 cup
2 cups = 1 pint
2 pints = 1 quart
4 quarts = 1 gallon
Length:
12 inches = 1 foot
3 feet = 1 yard
22 yards = 1 chain
10 chains = 1 furlong
8 furlongs = 1 mile
3 miles = 1 league
And then you have rods and links and
thous. . . .
Land Measures
• Originally, communal plots of land
were laid out in rectangular plots of 1
furlong by 1 chain (660 by 66 feet), or
10 chains by 1 chain (= 1 acre). The
furrows ran in the long direction, so
the plots were a “furrow long.” So
actually furlongs were an agricultural
measure that were independent of
feet, which was a body-part measure.
• By the way, a cricket pitch is still 66
feet long, or 1 chain, or a tenth of a
furlong.
Moving on….
• Eventually small measures based on
body parts had to be reconciled with
large agricultural measures like
furlongs, so things were shifted and
fudged in the measures so that
everything was an integral multiple of
everything else.
The Same Thing Happened
in Babylon:
• A body-part measure called a kush,
about 1 2/3 feet, was the basis for a
nindan, which needed to be
reconciled with two agricultural
measures, the eshe and the USH.
The eshe and the USH came pre-
loaded with a 6-to-1 exchange, and
the nindan and the eshe became an
easy 10-to-1 exchange.
10-for-1 and 6-for-1
• Because units of both length and area
were exchanged for larger units in
both groups of 10 and groups of 6,
using counters that reflected those
exchanges greatly facilitated
calculations with lengths and areas.
And the number system went along
for the ride.
10-for-1 and 6-for-1
• So Rudman claims that units of length
and area that came pre-loaded with
exchanges gave rise to a system of
arithmetic with alternating 10-to-1 and
6-to-1 exchanges, and then to a base
60 system. In any event, there was
real genius in moving to a place-value
system.
• Now, back to Babylon:
Babylonian Tablet Texts:
• Table Texts
• Problem Texts
Table Texts
• Multiplication tables, of which about
160 are known.
• Single tables have the form:
p a-rá 1 p
a-rá 2 2p
a-rá 3 3p
And so on, up to 20p, then:
a-rá 30 30p
a-rá 40 40p
a-rá 50 50p
Table Texts
• Combined tables (of which there are
about 80) have several single tables
included on one tablet.
• One of them (A 7897) is a large
cylinder containing an almost
complete set of tables written in 13
columns. There is a hole through the
center of the cylinder so that it could
be turned on some kind of peg.
Table Texts
• Reciprocal Tables had reciprocals of
numbers from 2 to 81 (provided their
sexagecimal representations did not
repeat.
• These were used to divide, which they
did by multiplying by reciprocals.
Table Texts
• There are a few tables of squares,
square roots, cube roots, powers,
sums of squares and cubes, … .
• Also some conversions and a few
special tables used for particular
business transactions (finding market
rates).
Table Texts
• It is likely that many of the table texts
we have are “exercises” from students
learning to be scribes, or perhaps
tables copied and made by students
for use in computations.
Problem Texts
• Also probably intended for
educational purposes.
• Story problems aimed at computing a
number.
• Often contrived or “tricky” problems:
– If camel A leaves Phoenicia travelling at
nindan per day. . . .
• Kinda like our modern story problems
or recreational math problems.
Problem Texts
• Largely algebraic problems, focusing
on what we would call on linear and
quadratic equations (though that’s not
necessarily how the Babylonians
thought about this).
• Mainly focused on algorithms, but not
on general procedures. Instead, they
gave several worked examples.
A Little Arithmetic
• The book mentions that we don’t
really know how the Babylonians did
arithmetic like adding and subtracting
and we don’t know their algorithms
for multiplication and division, except
that they divided by multiplying by
reciprocals.
Multiplication: 43 x 1,15
Babylonian Style
• This would be broken into 43 x 1 and
43 x 15. The 43 x 1 would be easy; it
would just be 43 but moved one place
value over, in the 60’s place. The 43
x 15 would become 40 x 15 + 3 x 15,
both of which would be available from
tables; 40 x 15 = 10, 0; 3 x 15 = 45;
so 43 x 1,15 = 43, 0 +10, 0 + 0, 45
= 53, 45
• This is why the Babylonian
multiplication tables had, for the
principle number p, multiples of p up
to 20p, then 30p, 40p, and 50p. The
distributive property was used to split
up larger multiples so these were all
that was necessary. (Why 20p
instead of 10p? Who knows?)
Division
• We’ll do 1029 divided by 64:
• In our language, we multiply 1029 by
1/64, or
• 17, 9 by 0; 0, 56,15
0; 0, 56, 15
x 17, 9
2, 15 9 x 15 from table
7, 30 9 x 50 from table; shift
54 9 x 6 from table; shift
4, 15, 17 x 15 from table; shift
14, 10, 17 x 50 from table; shift2
1, 42, 17 x 6 from table; shift2
16; 4, 41, 15
Or in our system, 16.078125
What About Division by 7?
ଵ
• As a Babylonian scribe would say, “7 does
not divide.”
• So, they used an approximation:
ଵ
ଵ ଵ ଵ
ଵଷ ଽଵ ଽଵ ଽ
ଵ
( )
ଵଷ
Story Problems, 1
• I have added the area and two-thirds
of [ the side of ] my square and it is
0;35. What is the side of my square?
ଶ
• The solution follows the standard
procedure for completing the square:
Babylon, 2000 BC Provo, 2011 AD
• You take 1. Two-thirds of
1 is 0;40.
• Half of this, 0; 20, you
multiply by 0;20 and it is
0;6,40,
• you add to 0;35
• and the result 0;41;40
has 0;50 as its square
root.
Babylon, 2000 BC Provo, 2011 AD
• The 0;20 which you have
multiplied by itself, you
subtract from 0;50, and
0;30 is the side of the
square.
Is This Algebra?
• Another textbook author, Victor Katz,
suggests the method was mainly
geometric.
• I believe most scholars assume the
methods were geometric, since
algebraic symbolism was not common
to any ancient culture – with the
exception of Diophantus, perhaps.
Is This Algebra?
• You take 1. Two-
thirds of 1 is 0;40.
x ;40
• Thus, the area on
the right represents ;40x
x x2
the situation; 0;40
is two-thirds. And
the area of the
figure is 0;35.
Is This Algebra?
x ;40
• Half of this, 0; 20,
• The idea is to take x x2 ;40x
half of the rectangle
and rearrange it to
form a gnomon – a x
square with a smaller
square missing from x x2 ;20x
the corner. It still has
area 0;35. ;20x
Is This Algebra?
• you multiply by 0;20
and it is 0;6,40,
• Here, you find the x
small square that is
missing, and… x x2 ;20x
• you add to 0;35
• Because the gnomon ;20x
is still 0.35, you now
missing 0;6,40
have the area of the
big square.
Is This Algebra?
• and the result
0;41;40 has 0;50 as
its square root. x
• The areas of the big
square is 0;41,50, x x2 ;20x
and its side is 0;50.
• We are now finding ;20x
0;20
the side of the big
square, which is
(x + 0;20).
Is This Algebra?
• The 0;20 which you
have multiplied by
itself, you subtract x
from 0;50, and 0;30
is the side of the ;20x
x x2
square.
• Since we know that
, 0;20 ;20x
we now subtract
the 0;20 to find x.
Story Problems, 2
(2/3)(2/3)x+100=x
• First multiply two-thirds by two thirds:
result 0;26,40
• Subtract 0;26,40 from 1: result
0;33,20
• Take the reciprocal of 0;33,20: result
1;48
• Multiply 1;48 by 1,40: result 3,00.
• 3,00 (qa) is the original quantity.
Story Problems, 3
• I found a stone but did not weigh it.
After I weighed out 8 times its weight,
added 3 gin. [Then] one-third of one-
thirteenth I multiplied by 21, added it
and then I weighed it. Result 1 mana.
What was the original weight of the
stone?
• The weight was 4;30 gin. (1 mana =
60 gin).
Babylonian “Algebra”
• Whether or not their “algebra” was
geometric or not, they were skilled in
solving quadratic equations. We
should note, however, that
– All quantities were positive
– Problems were often given in terms of
areas and perimeters of rectangles
Some Geometry-YBC 7289
Error: 0.000000423847 ish
30
1; 24, 51, 10 = 1.414212963...
42; 25, 35 = 42.42638889...
Babylonian Astronomy
• Ancient peoples of Mesopotamia
could easily track the movement of
the celestial sphere as it revolved
around the earth every year. They
could also track the movement of the
sun in a wiggly path (the ecliptic)
against the celestial sphere.
Babylonian Astronomy
Babylonian Astronomy
Babylonian Astronomy,
600 BC
• Early version of the Zodiac, 12 areas
of 30 ush each; the sun travels 1 ush
per day.
• So, there were 360 ush in a full circuit
of the sun.
• The beginning of there being 360
degrees in a circle.
Babylonian Astronomy,
600 BC
• Two different descriptions of how the
sun (and moon) changed speeds
along their path. One was a step
function (two speeds); the other had a
linear change over time and was quite
accurate.
Babylonian Astronomy,
600 BC
• The Babylonians divided the day into
twelve intervals called "kaspu". The
solar kaspu was the span of thirty
degrees which the Sun travels in two
hours of daily motion across Earth's
sky.
Babylonian Astronomy,
600 BC
• The Babylonians also predicted
certain celestial phenomena, such as
eclipses and lunar periods. They
began their studies with the eclipse of
March 19, 721 BC. Calculations were
difficult because the astronomers had
no instruments of high accuracy.
Babylonian Astronomy,
600 BC
• Both the Chaldeans and Babylonian
eclipse records are used in studying
long-term variations in the lunar orbit
in modern theories.
• Records of new moons, eclipses, and
the rising of Venus were kept from
very early times.
Some Astrology
• 2. If in Nisannu the sunrise (looks)
sprinkled with blood and the light is
cool: rebellion will not stop in the
country, there will be devouring by
Adad.
• 3. If in Nisannu the normal sunrise
(looks) sprinkled with blood: battles
Some Astrology
• 4. If in Nisannu the normal sunrise
(looks) sprinkled with blood: there will
be battles in the country.
• 5. If on the first day of Nisannu the
sunrise (looks) sprinkled with blood:
grain will vanish in the country, there
will be hardship and human flesh will
be eaten.
Some Astrology
• 6. If on the first day of Nisannu the
sunrise (looks) sprinkled with blood
and the light is cool: the king will die
and there will be mourning in the
country.
• 7. If it becomes visible on the second
day and the light is cool: the king's ...
high official will die and mourning will
not stop in the country.
Oh Happiness!
• 8. If a normal disk is present and one
disk stands to the right (and) one to
the left: if the king treats the city and
his people kindly for reconciliation and
they become reconciled,
Oh Happiness!
• 8. If a normal disk is present and one
disk stands to the right (and) one to
the left: if the king treats the city and
his people kindly for reconciliation and
they become reconciled, the cities will
start vying with each other, city walls
will be destroyed, the people will be
dispersed.
Constellations
• On the 1st of Nisannu the Hired Man
becomes visible. On the 20th of Nisannu
the Crook becomes visible.
• On the 20th of Ayyaru the Jaw of the Bull
becomes visible.
• On the 10th of Simanu the True Shepherd
of Anu and the Great Twins become visible.
• On the 5th of Du'uzu the Little Twins and
the Crab become visible.
Secret Knowledge
• "Secret tablet of Heaven, exclusive knowledge of
the great gods, not for distribution! He may teach it
to the son he loves. To teach it to a scribe from
Babylon or a scribe from Borsippa or any other
scholar is an abomination to Nabu and Nisaba.
• ...a Babylonian or a Borsippan or any other
scholar.......whoever speaks...
• [Nabu and] Nisaba will not confirm him as a
teacher. In poverty and deficiency may they put
an end to his ......; may they kill [him] with dropsy."
Babylonian Calendar
• The problem with calendars is
coordinating the different cycles:
days, months, years, and seasons.
They don’t come in nice integral
multiples.
• By the way, there were two seasons
in Babylon, Summer (barley harvest)
and Winter (roughly our fall/winter).
Babylonian Calendar
• Months in Babylon started when a
new moon (actually, a visible
crescent) first appeared. So the
priest-astronomers would watch and
announce the beginning of the month.
• This was common in other cultures,
too.
“Calendar”
• In Rome, a Pontifex (priest) observed
the sky and announced a new moon
and therefore the new month to the
king. For centuries afterward Romans
referred to the first day of each new
month as Kalends from their word
calare (to announce solemnly, to call
out). The word calendar derived from
this custom.
Babylonian Calendar
• Calendar based on cycles of the
moon, and needed to be reconciled
with the solar year.
• Alternated 29- and 30-day months,
and added an extra month three times
in every 8 years.
• This still necessitated the King adding
an extra month every now and then
when the seasons shifted too far.
Babylonian Calendar
• In the reign of king Nabû-Nasir, the
astronomers of Babylon recognized
that 235 lunar months are almost
identical to 19 solar years. (The
difference is only two hours.) They
concluded that seven out of nineteen
years ought to be leap years with an
extra month.
Babylonian Calendar
• In the reign of king Nabû-Nasir, the
astronomers of Babylon recognized
that 235 lunar months are almost
identical to 19 solar years. (The
difference is only two hours.) They
concluded that seven out of nineteen
years ought to be leap years with an
extra month.
Babylonian Calendar
• By about 500 BC, there were six
years when a second month Addaru
is added, and one year with an extra
Ululu. The result is that the first day of
the month Nisanu (New year's day)
was never far (< 27 days) from the
vernal equinox, so that the civil
calendar and the seasons were never
far out of step.
Wrapping Up
• Babylonian mathematics was often
practically-oriented, aimed at solving
problems of commerce, calendaring,
and so forth. However, there is also
evidence that Scribes developed a
culture of doing difficult problems to
show off their skill, or just to have
“good clean fun.” (Think about the “I
found a stone” problems.)
Wrapping Up
• We don’t know exactly how the Scribes
came up with their solutions, since they
only wrote down the numerical steps of a
solution.
• They could solve linear and quadratic
equations, some cubic equations,
understood right triangle relations, had
some efficient and accurate arithmetic
capabilities. They were also pretty good
astronomers.
Vestiges of Babylon in Our
Culture
• Zodiac
• 360 degrees, 60 minutes, 60 seconds
• 12 hour clocks
• Decimal numbers
• Others?
