An Overview of Babylonian Mathematics

A.1 Introduction to Babylonian Mathematics

The Babylonians lived in Mesopotamia, a fertile crescent between the Tigris and
Euphrates rivers. Here is a map of the region where the civilization flourished.

The region had been the center of the Sumerian civ- ilization which flourished before 3500 BC. This was an advanced civilization building cities and supporting the people with irrigation systems, a legal system, admin- istration, and even a postal service. Writing developed and counting was based on a sexagesimal system, that is base 60. Around 2300 BC the Akkadians invaded the area and for some time the less civilized culture of the Akkadians mixed with the more advanced culture of the Sumerians. The Akkadians invented the abacus as a tool for counting and they developed somewhat

clumsy methods of arithmetic with addition, subtraction, multiplication and division
all playing a part. The Sumerians, however, revolted against Akkadian rule and by
2100 BC they were back in control.

The Babylonian civilization, whose mathematics is the subject of this article,
replaced that of the Sumerians starting around 2000 BC. The Babylonians were a
Semitic people who invaded Mesopotamia, defeated the Sumerians and by about
1900 BC established their capital at Babylon.

The Sumerians had developed an abstract form of writing based on cuneiform
(i.e. wedge-shaped) symbols. Their symbols were written on wet clay tablets which
were baked in the hot sun. Many thousands of these tablets have survived to this
day. It was the use of a stylus on a clay medium that led to the use of cuneiform
symbols since curved lines could not easily be drawn. The later Babylonians adopted
the same style of cuneiform writing on clay tablets. A picture of one of their tablets
Babylonian mathematics is to the right.
 

Many of the tablets concern topics which are fasci- nating, although they do not contain deep mathemat- ics. For example we mentioned above the irrigation sys- tems of the early civilizations in Mesopotamia. Muroi writes: It was an important task for the rulers of Mesopotamia to dig canals and to maintain them, be- cause canals were not only necessary for irrigation but also useful for the transport of goods and armies. The rulers or high government officials must have ordered Babylonian mathematicians to calculate the number of workers and days necessary for the building of a canal, and to calculate the total expenses of wages of the workers.

There are several Old Babylonian mathematical
texts in which various quantities concerning the digging of a canal are asked for.
They are YBC 4666, 7164, and VAT 7528, all of which are written in Sumerian ...,
and YBC 9874 and BM 85196, No. 15, which are written in Akkadian ... . From the
mathematical point of view these problems are comparatively simple ...

The Babylonians had an advanced number system, in some ways more advanced
than our present systems. It was a positional system with a base of 60 rather than
the system with base 10.

The Babylonians divided the day into 24 hours, each hour into 60 minutes, each
minute into 60 seconds. This form of counting has survived for 4000 years. To write 5h
25' 30", i.e. 5 hours, 25 minutes, 30 seconds, is just to write the sexagesimal fraction,
5 25/60 30/3600. We adopt the notation 5;25,30 for this sexagesimal number. As
a base 10 fraction the sexagesimal number 5;25,30 is 5 4/10 2/100 5/1000 which is
written as 5.425 in decimal notation.

Perhaps the most amazing aspect of the Babylonian's calculating skills was their
construction of tables to aid calculation. Two tablets found at Senkerah on the
Euphrates in 1854 date from 2000 BC. They give squares of the numbers up to 59
and cubes of the numbers up to 32. The table gives 8² = 1, 4 which stands for

and so on up to 59² = 58, 1(= 58×60+1 = 3481). The Babylonians used the formula

to make multiplication easier. Even better is their formula

which shows that a table of squares is all that is necessary to multiply numbers,
simply taking the difference of the two squares that were looked up in the table then
taking a quarter of the answer.

Division is a harder process. The Babylonians did not have an algorithm for long
division. Instead they based their method on the fact that

so all that was necessary was a table of reciprocals. We still have their reciprocal
tables going up to the reciprocals of numbers up to several billion. Of course these
tables are written in their numerals, but using the sexagesimal notation we introduced
above, the beginning of one of their tables would look like:

Now the table had gaps in it since 1/7, 1/11, 1/13, etc. are not finite base 60
fractions. This did not mean that the Babylonians could not compute 1/13. They
would write

and these values, for example 1/90, were given in their tables. In fact there are
fascinating glimpses of the Babylonians coming to terms with the fact that division
by 7 would lead to an infinite sexagesimal fraction. A scribe would give a number
close to 1/7 and then write statements such as

... an approximation is given since 7 does not divide.

Babylonian mathematics went far beyond arithmetical calculations. We now ex-
amine some algebra which the Babylonians developed, particularly problems Babylo-
nian mathematics which led to equations and their solution. The Babylonians were
famed as constructors of tables. Now these could be used to solve equations. For
example they constructed tables for n³+n² then, with the aid of these tables, certain
cubic equations could be solved. For example, consider the equation

Note that we are using modern notation, and nothing like this symbolic representation
existed in Babylonian times. The Babylonians could handle numerical examples of
such equations. They did this by using certain rules, which indicates that they did
have the concept of a typical problem of a given type and a typical method to solve it.
For example in the above case they would (in modern notation) multiply the equation
by a² and divide it by b³ to get

Setting y = ax/b gives the equation y³ + y² = ca²=b³ which could now be solved
by looking up the n³ + n² table for the value of n satisfying n³ + n² = ca²/b³. When
a solution was found for y then x was found by x = by/a. We cannot stress too much
that all this was done without algebraic notation.

Again a table would have been looked up to solve the linear equation ax = b. They
would consult the 1/n table to find 1/a and then multiply the sexagesimal number
given in the table by b. An example of a problem of this type is the following.

Suppose, writes a scribe, 2/3 of 2/3 of a certain quantity
of barley is taken, 100 units of barley are added and the
original quantity recovered.

The problem posed by the scribe is to find the quantity of barley. The solution
given by the scribe is to compute 0; 40 times 0; 40 to get 0; 26, 40. Subtract this
from 1; 00 to get 0; 33, 20. Look up the reciprocal of 0; 33, 20 in a table to get 1; 48.
Multiply 1; 48 by 1, 40 to get the answer 3, 0.

It is not that easy to understand these calculations by the scribe unless we translate
them into modern algebraic notation. We have to solve

which is, as the scribe knew, equivalent to solving

This is why the scribe computed subtracted the answer from 1 to get ,
then looked up and so x was found from multiplied by 100 giving
180 (which is 1; 48 times 1, 40 to get 3, 0 in sexagesimal).

To solve a quadratic equation the Babylonians essentially used the standard formula.
They considered two types of quadratic equation, namely x² + bx = c and
x² - bx = c where here b, c were positive but not necessarily integers. The form that
their solutions took was, respectively

Notice that in each case this is the positive root from the two roots of the quadratic
and the one which will make sense in solving "real" problems. For example problems
which led the Babylonians to equations of this type often concerned the area of a
rectangle. For example if the area is given and the amount by which the length
exceeds the width is given, then the width satisfies a quadratic equation and then
they would apply the first version of the formula above.

A problem on a tablet from Babylonian times states that the area of a rectangle
is 1, 0 and its length exceeds its width by 7. The equation x² +7x = 1, 0 is, of course,
not given by the scribe who finds the answer as follows.

Compute half of 7, namely 3;30, square it to get 12; 15.
To this the scribe adds 1, 0 to get 1; 12, 15. Take its
square root (from a table of squares) to get 8; 30. From
this subtract 3;30 to give the answer 5 for the breadth
of the triangle.

Notice that the scribe has effectively solved an equation of the type x² + bx = c
by using x = ((b/2)² + c) - (b/2). Berriman gives 13 typical examples of problems
leading to quadratic equations taken from Old Babylonian tablets.

If problems involving the area of rectangles lead to quadratic equations, then
problems involving the volume of rectangular excavation (a "cellar") lead to cubic
equations. The clay tablet BM 85200+ containing 36 problems of this type, is the
earliest known attempt to set up and solve cubic equations. Of course the Babylo-
nians did not reach a general formula for solving cubics. This would not be found for
well over three thousand years.