2 Babylon/Mesopotamia
Babylon was an ancient city located near modern-day
Baghdad, Iraq. The term also serves as a shorthand
for the many Mesopotamian
3
empires/civilizations
dating back at least to 3000 BC: Sumeria, Akkadia,
Babylonia, etc.
Babylonians used cuneiform (wedge-shaped) script,
typically indentations on clay tablets. Most recovered
tablets date from the time of Hammurabi (c. 1800 BC)
or the Seleucid dynasty (c. 300 BC) which ruled after
the conquests of Alexander the Great.
Mathematical tablets are of two main types: tables
of values (multiplication, reciprocals, measures) and
worked problems.
Sexagesimal (base-60) Positional Enumeration Our modern decimal system is base-10 positional.
This means two things, both of which are easy to see with reference, say, to the number 3835.
• The symbol 3 represents both 3000 and 30: the meaning of a symbol depends on its position.
• The position of a symbol denotes the power of 10 by which it should be multiplied. Thus
3835 = 3 ·10
3
+ 8 · 10
2
+ 3 · 10
1
+ 5 · 10
0
Positional enumeration makes for efficient calculations and easy representation of numbers of vastly
different magnitude. Contrast with the difficulty of performing calculations using (non-positional)
Egyptian hieroglyphic notation.
One of the key Babylonian contributions to mathematical history is the creation of (arguably) the
first positional system of enumeration, dating to at least 2000 BC. Rather than our ten symbols 0–
9, Babylonians used only two: roughly ∨ for 1 and ∢ for 10, likely made by the same stylus. Any
number up to 59 could be written with combinations of these symbols, e.g.,
53 =
∢∢∢
∢∢
∨∨∨
The picture shows a typical cuneiform representation. Larger numbers were represented base-60.
For instance, the sexagesimal decomposition of 3835 is
3835 = 1 · 60
2
+ 3 · 60
1
+ 55 · 60
0
which the Babylonians would have written
∨ ∨∨∨
∢∢∢
∢∢
∨∨∨
∨∨
(∗)
Rather than using cuneiform, we’ll instead write 1, 3, 55;
3
‘Between two rivers,’ namely the Tigris and Euphrates. As indicated on the map, these rivers formed the backbone of
the fertile crescent, a region of early civilization, farming, crop and animal domestication.
7
Just as, for us, ‘3’ might mean 30, 3000 or even
3
1000
, for the Babylonians ∨ could mean 1, 60, 3600,
216000, or fractions such as
1
60
,
1
3600
depending on its position. There was no symbol for zero (as a
placeholder) until very late in Babylonian history, nor any sexagesimal point, so determining position
on ancient tablets can be difficult. For instance, rather than 3835, (∗) might instead have represented
60 + 3 +
55
60
= 63
11
12
or 60
3
+ 3 · 60
2
+ 55 · 60 = 230100
To make things easier to read, we use commas to separate terms and, if necessary, a semicolon to
denote the sexagesimal point. Thus
23, 12, 0; 15 = 23 ·60
2
+ 12 · 60 +
15
60
= 83520
1
4
Why base-60? There are many theories, but we cannot be certain. Here are some ideas.
• The Babylonians might have combined two systems (base-10 and base-12) inherited from older
cultures.
• Since 60 has many proper divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30), more numbers have exact
representation than with decimal arithmetic: for instance
1
3
= ; 20, as a terminating sexagesimal,
is much simpler than the decimal 0.33333.
• As prolific astronomers and astrologers, the Babylonians might have chosen 60 as a divisor of
360, approximately the number of days in a year. Our modern usage of degrees-minutes-seconds
for angle, hours-minutes-seconds for time, and the standard zodiac are all of Babylonian origin.
Indeed Babylonian units of measure often used factors of 60 for magnitude similarly to how
modern science uses 1000 (e.g., joules → kilojoules → megajoules).
This sort of historical question is rarely answerable in a satisfying way. Likely no-one ‘decided’ to
use base-60; like most cultural issues, it likely happened slowly and organically, without fanfare.
Basic Sexagesimal Calculations Addition and subtraction would have been as natural to the Baby-
lonians as decimal calculations are to us. For instance, we might write
21, 49
+ 3
1
, 37
25, 26
(in decimals 1309 + 217 = 1526)
Note how we carry 60 just like we are used to doing with 10 in decimal arithmetic: 49 + 37 = 1, 26.
Multiplication is significantly harder. To mimic our familiar long-multiplication process would re-
quire memorizing up to the 59 times table! For small factors this might have been fine. For larger
factors there is evidence of the Babylonians using two representations of a product in terms of squares
xy =
1
2
(x + y)
2
− x
2
− y
2
=
1
4
(x + y)
2
− (x −y)
2
Tablets consisting of tables of squares greatly aided the computation of large products. For instance,
31 × 22 =
1
4
53
2
−9
2
=
1
4
46, 49 − 1, 21
=
1
4
45, 28
= 11, 7 + 15 = 11, 22 (= 682)
8
Fractions & Division As we’ve already seen, the Babylonians also represented non-integers using
sexagesimals. Tables of reciprocals
1
n
were used to quickly evaluate division using multiplication!
m ÷ n = m ×
1
n
For example
1
18
= 0; 3, 20 =⇒
23
18
= 23(0; 3, 20) = 1; 9 + 0; 7, 40 = 1; 16, 40
This works nicely provided n has no prime divisors other than 2, 3 or 5, since any such
1
n
will be
an exact terminating sexagesimal.
4
Approximations were used for other reciprocals; a scribe would
choose a nearby denominator with an exact sexagesimal and state that the answer was approximate
11
29
≈
11
30
= 11(0; 2) = 0; 22
For more accuracy, one could choose a larger denominator. For instance, if a scribe wanted to divide
by 11, they might observe that 11 · 13 = 143 ≈ 144, from which
5
1
144
= 0; 0, 25 =⇒
1
11
≈
13
144
= 0; 5, 25
Scribes were explicit in acknowledging the approximation by stating, say, “11 does not divide.” Re-
member that a single digit in the second sexagesimal place means only
1
3600
, so even the most de-
manding application doesn’t require many terms (the above is 99.3% accurate!). The denominators
in some of these reciprocal tables were enormous, so far greater accuracy was often possible.
Another table listed all the ways an integer < 10 could be multiplied exactly to get 10.
1 10 5 2
2 5 6 1 40
3 3 20 8 1 15
4 2 30 9 1 6 40
We omit the commas for separation and the sexagesimal point as they did not exist. Moreover 7 is
missing since
1
7
(and thus
10
7
) is not an exact sexagesimal. It should be clear from the table that
10
6
= 1; 40 and
600
9
= 1, 6; 40
In the latter case, note that 600 = 10 ·60 would be written the same as 10, so this amounts to moving
the sexagesimal point in
10
9
= 1; 6, 40.
4
Analogous to the fact that
1
n
has a terminating decimal if and only if n has no prime divisors other than 2 or 5.
5
Being rational,
1
11
= 0.09090909 . . . = 0; 5, 27, 16, 21, 49, 5, 27, 16, 21, 49, . . . has a periodic sexagesimal expansion as can
be found using a pocket-calculator:
60
11
= 5 +
5
11
,
5 · 60
11
= 27 +
3
11
,
3 · 60
11
= 16 +
4
11
,
4 · 60
11
= 21 +
9
11
, . . .
9
Linear Systems of Equations These could be solved by a combination of the method of false posi-
tion (guess and modify as per the Egyptians) and the consideration of homogeneous equations. For
instance, here is a (suitably modernized) Babylonian approach to solving the system
(
3x + 2y = 11
2x + y = 7
1. Choose one equation, say the second, and set
ˆ
x =
ˆ
y. Solve this (say using false position) to
obtain
ˆ
x =
ˆ
y =
7
3
= 2; 20.
2. Since (d, −2d) is the general solution to the homogeneous equation 2x + y = 0, all solutions to
the second equation have the form x =
ˆ
x + d and y =
ˆ
y −2d. Substitute into the first equation:
11 = 3
7
3
+ d
+ 2
7
3
−2d
= 11 +
2
3
− d =⇒ d =
2
3
3. Finally compute x =
ˆ
x + d =
7
3
+
2
3
= 3 and y =
ˆ
y −2d =
7
3
−
4
3
= 1.
Step 2 should should remind you of the ‘nullspace’ method from modern linear algebra: all solutions
to the matrix equation ( 2 1)
(
x
y
)
= 7 have the form
x
y
=
x
0
y
0
+ n
where
x
0
y
0
is some particular solution (here x
0
= y
0
=
7
3
) and n =
d
−2d
lies in the nullspace of the
(row) matrix (2 1).
The Yale Tablet (YBC 7289) and Square-root Approximations One of the most famous tablets con-
cerns an approximation to
√
2. YBC stands for the Yale Babylonian Collection which contains over
45,000 objects. YBC 7289 is shown below alongside an enhanced representation of the numerals.
The tablet depicts a square of side 30 and labels the diagonal in two ways:
• 1; 24, 51, 10 is an approximation to
√
2, an underestimate by roughly 1 part in 2.5 million!
• 42; 25, 35 is an approximation to the diagonal when the side is 30.
10
The Babylonians more often used the simpler approximation 1; 25 = 1.41666 . . . which is still very
close. Given the impractical accuracy of YBC 7289, it is reasonable to ask how it was found. No-one
knows for certain, but two methods are theorized since both were used to solve other problems. It
should be stressed that no Babylonian proofs of these approaches are known.
1: Square root approximation
√
a
2
± b ≈ a ±
b
2a
. This is essentially the linear approximation from
elementary calculus. If one chooses a rational number a whose square is close to 2, then the
error will also be small. For instance:
•
√
2 =
√
1 + 1 ≈ 1 +
1
2
= 1; 30 (a = 1)
•
√
2 =
s
4
3
2
+
2
9
≈
4
3
+
2/9
8/3
=
4
3
+
1
12
=
17
12
= 1; 25 (a =
4
3
= 1.3333 . . .)
•
√
2 =
s
7
5
2
+
1
25
≈
7
5
+
1/25
14/5
=
99
70
= 1; 24, 51, 25, 42, 51, 25, 42, . . . (a =
7
5
= 1.4)
2: Method of the Mean It may be checked (Exercise 11) that any sequence defined by the recurrence
relation a
n+1
=
1
2
a
n
+
2
a
n
converges to
√
2. We apply this when a
1
= 1.
a
1
= 1, a
2
=
3
2
= 1; 30, a
3
=
17
12
= 1; 25
a
4
=
577
408
= 1 +
169
408
= 1; 24, 51, 10, 35, 17, . . .
a
5
=
665857
470832
= 1; 24, 51, 10, 7, 46 . . .
It seems incredible that any ancient culture would have bothered to go as far as this to obtain
the observed accuracy.
The same approach can be used to approximate other roots. For example, we can approximate
√
11 via a
n+1
=
1
2
(a
n
+
11
a
n
) and a
1
= 3:
a
2
=
10
3
= 3; 20, a
3
=
199
60
= 3; 19, a
4
=
79201
23880
= 3; 18, 59, 50, 57, 17, . . .
Quadratic Equations The above methods were applied to solve general quadratic equations. A
question might be phrased as follows:
I added twice the side to the square; the result is 2, 51, 40. What is the side?
In modern language, we want the solution to x
2
+ 2x = 2 · 60
2
+ 51 · 60 + 40 = 10300.
Questions such as these were solved using templates, typically as worked examples. The above
problem requires the template for solving x(x + p) = q where p, q > 0. Since the Babylonians did not
recognize negative numbers, the other types of quadratic equation (x
2
= px + q, etc.) had different
templates.
11
To make things a little easier, we apply their approach to the simpler equation x
2
+ 4x = 2:
Set y = x + p (y = x + 4) and decouple the equation:
(
xy = q
y − x = p
(
xy = 2
y − x = 4
Use this to solve for x + y:
4xy + (y − x)
2
= p
2
+ 4q 4xy + (y − x)
2
= 4
2
+ 4 · 2
(y + x)
2
= p
2
+ 4q (y + x)
2
= 24
x + y =
q
p
2
+ 4q x + y =
√
24 ≈ 4; 54
where the square-root was approximated using one of the earlier algorithms, e.g.
√
24 =
p
5
2
−1 ≈ 5 −
1
10
= 4; 54
Since x + y and x −y are now known, we have a linear system which is easily solved:
x =
p
p
2
+ 4q − p
2
x ≈ 0; 27
The method of completing the square and the quadratic formula are at least 4000 years old!
While we’ve written this abstractly, in practice scribes would be copying from a particular example
of the same type. There were no abstract formulæ and everything was done without the benefit
modern notation. There was moreover typically no written commentary to explain the method; often
all historians have to work with is a single column of numbers!
Note also that the template only found the positive solution; the Babylonians had no notion of nega-
tive numbers. Amazingly, they were able to address certain cubic equations similarly.
Pythagorean Triples The Plimpton 322 tablet (also at Yale) lists a large number of Pythagorean
triples (albeit with some mistakes). Due to the strange manner of encoding, it took scholars a long
time to realize what they had.
As an example, line 15 describes the Pythagorean
triple 53
2
= 45
2
+ 28
2
:
• The first entry 1; 23, 13, 46, 40 is the exact sex-
agesimal for
53
45
2
.
• The second entry is 28.
• The third entry is 53.
• The last two entries indicate line number 15.
The first three (interesting) entries are therefore
(
c
a
)
2
, b, c
where c
2
= a
2
+ b
2
. Since the table is
broken on the left side it is possible that a missing column explicitly mentioned a.
12
It is not known how the table was completed, though the first column exhibits a descending pattern
that provides clues to its construction. One theory is that a scribe found rational solutions to the
equation v
2
= 1 + u
2
(equivalently (v + u)( v − u) = 1) by starting with a choice of v + u and using a
table of reciprocals to calculate v − u.
To revisit our example, if v + u =
9
5
= 1; 48, then
v − u =
1
v + u
=
5
9
= 0; 33, 20
We therefore have a linear system of equations in u, v whose solutions are
v = 1; 10, 40 =
53
45
, u = 0; 37, 20 =
28
45
We investigate this further in Exercise 7. The Plimpton tablet has been the source of enormous schol-
arship; look it up!
Geometry The Babylonians also considered many geometric problems. They used both π ≈ 3
and π ≈ 3
1
8
to approximate areas of circles. They had calculations (both correct and erroneous)
for the volume of a frustrum (truncated pyramid). They also knew that the altitude of an isosceles
triangle bisects its base, and that the angle in semicircle is a right-angle (Thales’ Theorem). None
of these statements were presented as theorems in a modern sense; we merely have computations
and applications that make use of these facts. We simply do not know the depth of Babylonian
understanding of such concepts.
Summary
• Sexagesimal positional enumeration. No zero. Fractions also used sexagesimal representation.
• More advanced than Egyptian mathematics but still practical/non-abstract. Perhaps Babylo-
nian mathematics only appears more advanced because we have so much more evidence: thou-
sands of tablets versus only a handful of papyri. Like Egypt, we have worked examples without
abstraction or statements of general principles.
• Some distinction (‘does not divide’) between approximate and exact results.
• Some geometry, but algorithmic/numerical methods predominate.
Exercises There is no single correct way to do Babylonian calculations. Play with the ideas and use
modern notation to get a feel for things without torturing yourself.
1. Convert the sexagesimal values 0; 22, 30, 0; 8, 6, 0; 4, 10 and 0; 5, 33, 20 into ordinary (modern)
fractions in lowest terms.
2. (a) Multiply 25 by 1, 4 (b) Multiply 18 by 1, 21.
(Either compute directly (long multiplication) or use the difference of squares method on page 8)
3. (a) Use reciprocals to divide 50 by 18. (b) Repeat for 1, 21 divided by 32.
4. Use the Babylonian method of false position to solve the linear system
(
3x + 5y = 19
2x + 3y = 12
13
5. (a) Convert the approximation
√
2 ≈ 1; 24, 51, 10 to a decimal and verify the accuracy of the
approximation on page 10.
(b) Multiply by 30 to check that the length of the diagonal is as claimed.
6. Babylonian notation is not required for this question.
(a) Use the square root approximation (pg. 11) with a =
8
3
to find an approximation to
√
7.
(b) Taking a
1
= 3, apply the method of the mean to find the approximation a
3
to
√
7.
7. Recall that v
2
= 1 + u
2
in the construction of the Plimpton tablet.
(a) If v + u = α, show that u =
1
2
(α − α
−1
) and v =
1
2
(α + α
−1
).
(b) Suppose v + u = 1; 30 =
3
2
. Find u, v and the corresponding Pythagorean triple.
(c) Repeat for v + u = 1; 52, 30 =
15
8
.
(d) Repeat for v + u = 2; 05 =
25
12
. This is line 9 of the tablet.
8. Solve the following problem from tablet YBC 4652. I found a stone, but did not weigh it; af-
ter I subtracted one-seventh, added one-eleventh (of the difference), and then subtracted one-
thirteenth (of the previous total), it weighed 1 mina (= 60 gin). What was the stone’s weight?
(The meaning of the problem isn’t completely clear: make your best guess!)
9. Solve the following problem from tablet YBC 6967. A number exceeds its reciprocal by 7. Find
the number and the reciprocal.
(In this case, two numbers are reciprocals if their product is 60)
10. For this question it is helpful to think about the corresponding facts for decimals.
(a) Explain the observation on page 9 regarding which reciprocals n have a terminating sexa-
gesimal. Can you prove this?
(b) Find the periodic sexagesimal representation of
1
7
and use geometric series prove that you
are correct.
11. For this question, look up the AM–GM inequality and remind yourself of some basic Analysis.
(a) Suppose (a
n
) is a sequence satisfying the recurrence a
n+1
=
1
2
(a
n
+
2
a
n
). Prove that a
n
≥
√
2
whenever n ≥ 2.
(b) Prove that lim a
n
=
√
2.
12. (a) The Babylonians used an approximation of the form A ≈
1
12
c
2
for the area of a circle in
terms of its circumference. To what approximation for π does this correspond?
(b) Use a Babylonian method to show that
√
3 ≈
7
4
.
(c) A ‘bulls-eye’ (pictured) is constructed using congruent circu-
lar arcs built from a circumscribed equilateral triangle. If the
arc-length is a, use Babylonian approximations to prove that
the area of the bulls-eye is ≈
9a
2
32
. What, approximately, are its
dimensions (width/height)?
(Use as much modern trigonometry as you like!)
a
14
