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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 60^{2})=7200.

The numbers 1-59 are written below:

To use the sexagesimal notation in modern language we separate
the 'columns' by commas, so that the number

7267 = 2(60^{2}) + 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} - *a*^{2} - *b*^{2}]/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(a^{2} + 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 n^{3} + n^{2}
table* *

One important table for Babylonian algebra was that of the values of n^{3}
+ n^{2} for integer values of n from 1 to 30. These tables could be
used to solve cubic equations of the form

*ax*^{3} + *bx*^{2} = *c*

although note that the Babylonians would not have had this algebraic notation.

Multiplying through by a

^{2}/b

^{3} gives:

(*ax*/*b*)^{3} + (*ax*/*b*)^{2}
= *ca*^{2}/*b*^{3}

Putting

*y* =

*ax*/

*b* gives us the equation

*y*^{3} + *y*^{2} = *ca*^{2}/*b*^{3}

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
x^{2} + *q* = *px*. The Babylonians could even reduce equations
of the form *ax*^{2} + *bx* = *c* to the normal form
*y*^{2} + *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

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 *a*^{2} + *b*^{2}
= *c*^{2}.

http://www.bath.ac.uk/~ma2jc/babylonian.html

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