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Babylonian Mathematics

Introduction

'Babylonian' is a general word to describe the people living in Mesopotamia, a fertile plain between the Tigris and Euphrates rivers (present day Turkey and Syria). The Babylonian civilisation (dating from around 2000-600BC) replaced that of the Sumerians and Akkadians, and so inherited their sexagesimal (i.e. base 60) number system. The Sumerians had created a form of writing based on cuneiform (wedge-shaped) symbols, which the Babylonians also adopted. This is how most of their texts have come down to us: as symbols written on wet clay tablets which were then baked in the hot sun so the clay set and the symbols were permanent. Thousands of these tablets have survived to the present day.

Babylonian mathematics was, in many ways, more advanced than Egyptian maths. They could extract square and cube roots, work with Pythagorean triples 1200 years before Pythagoras, had a knowledge of pi and possibly e (the exponential function), could solve some quadratics and even polynomials of degree 8, solved linear equations and could also deal with circular measurement. Babylonian mathematics was based much more on algebra and less on geometry, in contrast to the Greeks.

Babylonian Numerals

Cuneiform numbers could be written using a combination of two symbols: a vertical wedge for '1' and a corner wedge for '10'. The Babylonians had a sexagesimal system and used the concept of place value to write numbers larger than 60. So they had 59 symbols for the numbers 1-59, and then the symbols were repeated in different columns for larger numbers. For example, a '2' in the second column from the right meant (2 x 60)=120, and a '2' in the column third from the right meant (2 x 602)=7200.
The numbers 1-59 are written below:

Babylonian cuneiform numbers

To use the sexagesimal notation in modern language we separate the 'columns' by commas, so that the number
7267 = 2(602) + 1(60) + 7
would be written as 2,1,7.

There are some problems with this system. The first is that in practice there is no way of separating the 'columns' except by a gap in the numbers, so '2' looks very similar to 61 (=1,1). A more serious problem is that there was no symbol for zero to put into an empty column, so '1' is indistinguishable from '60 (=1,0). Generally we can work out what the numbers were from the context of the probltaem, but this isn't exactly a very satisfactory way of doing things. Later Babylonian civilisations did eventually invent a symbol for zero, so obviously they were aware of this deficiency in their system too.

The base 60 number system of the Babylonians was successful enough to have worked its way through time to appear in our present day modern world. We still have 60 minutes in an hour, 60 seconds in a minute, 360 degrees in a circle and 60 minutes in a degree. Even our 24 hour clock is a legacy from the ancient Babylonians.

Babylonian Number Tables

One aspect of Babylonian mathematics shared with the Egyptians is that of making tables to ease the effort of calculations. They made tables of many things which allowed them to develop their maths further than previous civilisations, and to calculate things like square roots with as much accuracy as mathematicians in the times of the Renaissance.

Reciprocal Tables

The Babylonians had no special algorithm for long division, and instead used the fact that
a/b
= a x (1/b)
They created tables of reciprocals converted to sexagesimal notation. In the notation introduced earlier, we can use a semi-colon to indicate a decimal point. Then the number 1/2 would be written as (0;30)= 0(1)+30(60-1). Thus division was a lot easier than the rather messy duplation method of the Egyptians and made arithmetical calculations much easier to carry out.

60 is a useful base here because many numbers have finite base 60 fractions, e.g. 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/12, 1/15 and 1/20. However, some numbers (such as 1/7, 1/13) were infinite fractions, and only their approximations were given. It is a shame the Babylonians did not consider these numbers further, since they would have yielded periodically repeating sexagesimal fractions that could have provoked investigation into infinite series.

Tables of Squares

The Babylonian method of multiplication is quite ingenious and only relies on knowing the squares of numbers. They used the formulas
ab = [(a+b)2 - a2 - b2]/2

ab = [(a+b)2 - (a-b)2]/4
for easy multiplication of two numbers. They didn't always use this method though; sometimes it was just as simple to multiply and add, e.g. to multiply by 39 you multiply by 30 and 9 and add the results together.

Square and Cube Roots

It is quite amazing to find, on an ancient stone tablet, a highly accurate approximation to the square root of 2. In sexagesimal notation this is (1;24,52,10) which in decimal is 1.41421296, and differs by about 0.0000006 from the true value. Accuracy in these kinds of computations was quite easy with the fractional notation they had, and approximations to other square roots were also given.

There are two possible methods of approximating square roots. [Here I shall call the square root of x sqrt(x) because I can't find a way of getting the square root symbol to work on these web pages!] The first of these uses the approximation
sqrt(a2 + b) approx= a + b/2a
which is derived from the first few terms of the expansion of the binomial series.

The second method uses an algorithm which was later ascribed to the Greeks.
Let a = a1 be an initial approximation. If a1 < sqrt(2) then 2/a1 > sqrt(2). So as a better approximation take a2 = (a1 + 2/a1)/2. Repeat the process until you have an answer as accurate as you want.

Quadratic Equations and the n3 + n2 table

One important table for Babylonian algebra was that of the values of n3 + n2 for integer values of n from 1 to 30. These tables could be used to solve cubic equations of the form
ax3 + bx2 = c
although note that the Babylonians would not have had this algebraic notation.
Multiplying through by a2/b3 gives:
(ax/b)3 + (ax/b)2 = ca2/b3
Putting y = ax/b gives us the equation
y3 + y2 = ca2/b3
which can be solved by looking up in the table to find the value of y and then substituting back.
It is amazing that without the use of modern notation for these equations the Babylonians could recognise equations of a certain type and the methods for solving them.

It is hardly surprising then to find that the Babylonians were also proficient at solving quadratic equations. If linear problems are found in their texts then the answers are simply given without any working; these problems were obviously thought too elementary for much attention. To solve quadratic equations the Babylonians used a method equivalent to using our quadratic formula. Many quadratics are arrived at from considering simultaneous equations such as x+y=p, xy=q, which yields the quadratic x2 + q = px. The Babylonians could even reduce equations of the form ax2 + bx = c to the normal form y2 + by = ac using the substitution y = ax, which is quite astounding given that they had no formal algebraic system.

Exponentials and Logarithms

By now, the mathematical achievements of the Babylonians should have impressed you enough so that you won't be surprised by yet more remarkable things. Ancient tablets have been found listing successive powers of numbers. The question was then asked in a problem text, to what power must a certain number be raised to yield a given number, i.e. the logarithm to a certain base. However, 'logarithm tables' were not used for general calculation but were only used to solve specific problems.

Pythagorean Triples

Plimpton Tablet

The Plimpton Tablet pictured to the left dates from about 1700BC, and although a large chip has been broken off it the numbers contained in the table are still recognisable as Pythagorean Triples. A Pythagorean triple consists of three integers which satisfy the equation a2 + b2 = c2.

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