Download Square Root Tablet

History of Mathematics in and for the Curriculum
Babylonian Mathematics: The ‘Square Root’ Tablet
The problem of what we call ‘square roots’ goes back at least to the Mesopotamian
scribes, who showed that finding an arithmetical value for the side of a square whose
area was 2 square units (in our terms, 2 ) was a never-ending process. They had
algorithms for finding successive approximations to the sides of squares whose
whole-number areas were not square numbers. The drawing below is of a tablet from
the Yale Babylonian Collection YBC 7289 dated about 1500 BCE.
How did they do it?
We don’t know exactly, but from evidence of their ‘cut-and-paste’ methods for
calculating the sides and areas of squares, and in terms of arithmetic in our modern
classroom, we might proceed something like this:
We want two numbers (exactly the same), which when multiplied together make 2.
Measuring the diagonal of a square is one approach, but how accurate is the
measurement, and what is the error? Alternatively, we can encourage some ‘strategic
guessing’
1
3
3 3 9
1
1
Guess 1 and write it as ; so x =
but this is 2 ; and too big by
2
2
2 2 4
4
4
1
4
4 4 16
7
2
Guess again: try 1 or ;
so x =
or 1
but this is too small by
3
3
3 3
9
9
9,
so by taking the mean of these two numbers, we can proceed to the next stage:
© Leo Rogers.
July 2009
Oxford University
[email protected]
History of Mathematics in and for the Curriculum
2
17
17
289
1  3 4
 17 
and
is 1.41667 (to 5 decimal places) and   =
which is
   =
 12 
12
12
144
2 2 3
1
2
better, but still too big. The next stage is usually to divide 2 by this number to
144
288
1
1 
 289

get
or 
which is too small by
.

 289 289 
289
289
1
1
and 1 may
2
3
be a useful exercise in manipulating fractions, it is probably more interesting and less
prone to error for the pupils to ask them to use their calculators for subsequent stages.
Clearly we can go on with this process, but while the first stage with 1
By taking some number in between 1.5 and 1.333… pupils can repeat the process and
get successively better numerical values for the diagonal of the square.
We could suppose that the Babylonians had some idea that this procedure could go on
and on, but we don’t really know what they wanted this number for. Was it for
practical purposes? In that case, for building, or land measurement, you would stop
when it was ‘good enough’ for the job. On the other hand, it is quite possible that this
discovery involved a debate about the nature of numbers, all those years ago.
You might think of this as the first ‘Existence Theorem’ - the diagonal of the square
exists, but we cannot measure it exactly!
Proving that this actually was an infinite process had to wait until the Pythagoreans
formed a logical argument in approximately 500 BC.
___________________________________________________________________
In the Classroom.
With pupils, there is always value in discussing the problem of the ‘length’ of the
diagonal of a square, and encouraging experiments to find a suitable number. This
brings out the inevitable problems of measurement, the meaning of ‘accuracy’ and the
introduction of necessary approximations.
Once you get involved with this kind of process, it begins to look as if it is never
going to end. Obviously, that can be true, and so someone has to make a decision
about “how near do you want it to be”? Pupils often have the idea that if you write


many
numbers
after the decimal point, that makes the measurement more ‘accurate’.
But, apart from the practical sense, this is useless if your measuring instrument cannot
match the calculation! There are many opportunities for pupils to engage and be
fascinated with this and other infinite processes.
What is written on the tablet?
The numbers on the tablet are in cuneiform script in a mixture of base 60 and base 10,
and the equivalent in our notation gives a result for 2 as 1.4142129 to seven
decimal places. Not bad for so long ago!
Referring to the diagram, the sexagesimal numbers are 30 for the side, and on the
diagonal the top row is 1; 24, 51, 10 and the row below is 42, 25, 35.
These numbers are expressed as a series of sexagesimal fractions.
© Leo Rogers.
July 2009
Oxford University
[email protected]
History of Mathematics in and for the Curriculum
Dealing with sexagesimal fractions can be tricky, and depends on careful organisation
of the calculation.
The principle is exactly the same as in base 10,
4
2
3 

 2  3
e.g.  1.423 

10 10
10 
and involves plenty of practice in arithmetic as you can see from the calculations here.
Since there was no zero or a ‘sexagesimal point’ guessing where the units were
separated from the fractions is done by examining the context of the calculation.
Duncan Melville’s Mesopotamian Mathematics Website is very comprehensive:
http://it.stlawu.edu/~dmelvill/mesomath/
Go to the section on Special Topics - Cuneiform Numbers for an introduction.
This site shows four excellent photographs of the original tablet:
http://www.math.ubc.ca/%7Ecass/Euclid/ybc/ybc.html
From the diagram,
The numbers on the diagonal 1: 24, 51, 10 become 1 
1
24
51
10
 2  3
60 60
60
2
51
1
8,947
which comes to 1 
or 1.414212963…


5 3600 21,600
21,600
The other numbers are probably used for checking that the calculation is correct:
If the side of the square is 30 units, then the diagonal will be 30
sexagesimal system will be 30 x (1; 24, 51, 10) or
30  1  
2 which in the
51 30  10
30  24   30  51  30  10 

  
  
 = 30  12 
2
3
120
60 3
60
60
60
30  12   
25
1   30  10 
5
 25 30 

 2  2
= 42  
  
2
 60 60  60
60 120
60  60 
which comes to 42, 25, 35.
On my calculator, 2 = 1.414213562 differing from the value they obtained in the
sixth decimal place.
Another value that appears in ancient cultures - particularly in Greece - is 5 which
appears in the study of the ‘Golden Section’. Here is another non-rational number that
arises from an infinite geometrical process.
© Leo Rogers.
July 2009

Oxford University
[email protected]