Babylonian Mathematics

Our first knowledge of mankind's use of mathematics comes from the Egyptians and Babylonians. Both civilizations developed mathematics that was similar in some ways but also very different in others.

Some basic facts about ancient Babylon.

• The dates of the Mesopotamian civilizations date from 2000-600 BC. Occasionally, the region, at least between the two rivers (Tigris and Euphrates) is described as Chaldea.
• The Sumerians of the Mesopotamian valley built home and temple decorated with artistic pottery and mosaics in geometric patterns.
• Powerful rulers united the local Principates into an empire which completed vast public works, such as irrigation canals. One of the most powerful was Sargon (ca. 2276-2221 BC).

Babylonian Mathematics

• The cuneiform (wedge) pattern of writing that the Sumerians had developed during the fourth millennium may have been the earliest form of written communication. It probably antedates the Egyptian hieroglyphic.
• The Mesopotamian civilizations are often called Babylonian, though this is not correct. Actually, Babylon was not the first great city, though the whole civilization is called Babylonian. Babylon, during its existence, was not always the center of Mesopotamian culture.

• Babylon fell to Cyrus of Persia in 538 BC, but the city was spared. The great empire was finished.
• The use of cuneiform script formed a strong bond. Laws, tax accounts, stories, school lessons, personal letters were impressed on soft clay tablets and then were baked in the hot sun or in ovens.
• From one region, the site of ancient Nippur, there have been recovered some 50,000 tablets.
• Many university libraries have large collections.
• Deciphering cuneiform succeeded the Egyptian hieroglyphic. Indeed, just as for hieroglyphics, the key to deciphering was a trilingual inscription. Found at Behistun by a British office, Henry Rawlinson (1810-1890), stationed as an advisor to the Shah, this inscription depicts the greatness and glory of Darius the Great. In 516 B.C. Darius commissioned his lasting monument to be engraved on a foot surface on a rock cliff, the ``Mountain of the Gods" at Behistun. Like the Rosetta stone, it was inscribed in three languages -- Old Persian, Elamite, and Akkadian (Babylonian). However, all three were unknown. Only because Old Persian has only 43 signs and had been the subject of serious investigation since the beginning of the 19th century was the deciphering accomplished. Nonetheless, progress was very slow. Rawlinson was correctly able to assign correct values to 246 characters and discovered that the same sign could stand for different consonantal sounds, depending on the vowel that followed. (polyphony) It has only been in this century that substantial publications have appeared.

In mathematics, the Babylonians were somewhat more advanced than the Egyptians.

• Their mathematical notation was positional but sexigesimal.
• They used no zero.
• More general fractions, though not all fractions, were admitted.
• They could extract square roots.
• They could solve linear systems.
• They worked with Pythagorean triples.
• They solved cubic equations with the help of tables.
• They studied circular measurement.
• Their geometry was sometimes incorrect.

For enumeration the Babylonians used two symbols.

All numbers were forms from these symbols.

Example:

Note the notation was positional and sexigesimal:

There is no clear reason why the Babylonians selected the sexigesimal system. It was possibly selected in the interest of metrology, this according to Theon of Alexandria, a commentator of the fourth century A.D.: i.e. the values 2,3,5,10,12,15,20, and 30 all divide 60.

Remnants still exist today with time and angular measurement. In fact the Babylonians used a 24 hour clock, with 60 minute hours, and 60 second minutes.

Because of the large base, multiplication was carried out with the aide of a table.

A positional fault??? Which is it?

1. There is no ``gap" designator.
2. There is a true floating point -- its location is undetermined except from context.

The ``gap" problem was overcome by the time of Alexander the Great, rather late in the game for the Chaldeans.

We use the notation:

The values are all integers.

Example

This number was found on the Old Babylonian Tablet (Yale Collection #7289) and is a very high precision estimate of .

The exact value of , to 8 decimal places is = 1.41421356.

Fractions. Generally the only fractions permitted were such as

because the sexigesimal expression was known.

Irregular fractions such as , etc are generally not used.

A table of all products equal to sixty has been found.

The table is also used for reciprocals. For example,

Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 B.C. They give squares of the numbers up to 59 and cubes up to 32. The Babylonians used the formula

to assist in multiplication.

Division relied on multiplication, i.e.

There apparently was no long division.

The Babylonians knew some approximations of irregular fractions.

However, they do not appear to have noticed infinite periodic expansions.

They seemed to have an elementary knowledge of logarithms.

Square Roots Recall the approximation of . How did they get it? There are two possibilities:

• Applying the approximation

  

• Applying the method of the mean.

The product of 30 by 1;24,51,10 is precisely 42;25,35.

Method of the mean.

1. Take as an initial approximation.
2. Idea: If then
3. So take

   

4. Repeat the process.

Example. Take . Then we have

Now carry out this process in sexigesimal, begining with . Remember to us Babylonian arithmetic. Using full decimal arithmetic will not give the value 1;25,51,10. Use Babylonian arithmetic.

Note: was commonly used as a brief, rough and ready, approximation.

Pythagorean Triples

The Plimpton 322 tablet dates from about 1700 BC.

If appears to have the left section broken away. What was found has numbers tabulated as follows.

What it means.

How did they determine these. Assuming they knew the Pythagorean relation , divide by b to get

Choose u+v and find u-v in the table of reciprocals.

Example. Take u+v=2;15. Then u-v=0;26,60 Solve for u and v to get

Multiply by an appropriate integer to clear the fraction. We get a=65, c=97. So b=72. This is line 5 of the table.

It is tempting to think that there must have been known general principles, nothing short of a theory, but all that has been discovered are tablets of specific numbers and worked problems.

References:
O. Neugebauer and A. Sachs. Mathematical Cuneiform Texts. Amer. Oriental Series 29. American Oriental Society, New Haven, 1945

E. M. Bruins. On Plimpton 322, Pythagorean numbers in Babylonian mathematics. Afdeling Naturkunde, Proc. 52 (149), 629-632.

Solving Quadratics

Problem. Solve x(x+p)=q.

Solution. Set y=x+p Then we have the system

This gives

All three forms

are solved similarly. The third is solve by equating it to the nonlinear system, . Moreover, all three date back 4,000 years!

Solving Cubics

• Solving was accomplished using tables and interpolation.
• Mixed cubics were also solved using tables and interpolations
• The general cubic can be reduced to the normal form . To do this one needs to solve a quadratic, which the Babylonians could do.

The Babylonians must have had extraordinary manipulative skills and as well a maturity and flexibility of algebraic skills.

Solving linear systems.

Solve

Solution. Select such that

So, . Now make the model

We get

So, d=300 and thus

Can you generalize this algorithm to arbitrary systems??!!

Circular Measurement

Generally, the Babylonians used for practical computation. But, in 1936 at Susa (captured by Alexander the Great in 331B.C.), a number of tablets with significant geometric results were unearthed.

One tablet compares the areas and the squares of the sides of the regular polygons of three to seven sides. For example

This gives an effective (Not bad.)

Geometry

There are two forms for the volume of a frustum

The second is correct, the first is not.

There are many geometric problems in the cunieform texts. For example, the Babylonians were aware that

• The altitude of an isosceles triangle bisects the base.
• An angle inscribed in a semicircle is a right angle. (Thales)

Babylonian Mathematics
Summary

• Problems contain only specific cases. There seem to be no general formulations.
• There is an absence of clear cut distinctions between exact and approximate results.
• Questions about solvability or unsolvability are absent.
• The concept of ``proof" is unclear and uncertain.
• Overall, there is no sense of abstraction.
• Their mathematics, like the Egyptians, is utilitarian -- but apparently far more advanced.

• About this document ...