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.
In mathematics, the Babylonians were somewhat more advanced than the Egyptians.
For enumeration the Babylonians used two symbols.
All numbers were forms from these symbols.
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?
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.
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:
The product of 30 by 1;24,51,10 is precisely 42;25,35.
Method of the mean.
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.
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.
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.
Problem. Solve x(x+p)=q.
Solution. Set y=x+p Then we have the system
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!
The Babylonians must have had extraordinary manipulative skills and as well a maturity and flexibility of algebraic skills.
Solving linear systems.
Solution. Select such that
So, . Now make the model
So, d=300 and thus
Can you generalize this algorithm to arbitrary systems??!!
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.)
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