Babylon numbers
Can you make a hypothesis to explain these ancient numbers?
Age
11 to 18
Challenge level
Problem
This image shows an ancient stone tablet containing rows of numbers, dated from 1800BC. It is written in Babylonian cuneiform script. The modern day versions of the last three columns of numbers are written out to the right.
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[image taken from http://commons.wikimedia.org/wiki/Image:Plimpton_322.jpg] |
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What do these numbers mean? Are there any patterns? Why are they in this order? Is there some underlying structure? The writer of the numbers went to a great deal of trouble to make the heavy clay tablet, so we can be fairly sure that they are important in some way.
Investigate these numbers. Create a hypothesis for their meaning or to explain any patterns. Test your hypotheses.Why is it not possible to prove that your hypothesis is true?
Note that although there is a very likely answer to the meaning of the numbers, four rows are thought to contain errors in one of the numbers. Can you work out which these are, and can you correct them?
If you find the rule which explains all but four of the rows, you might like to consider whether the author had other plans for those numbers or whether, in actual fact, he made mistakes in his calculations or writing.
NOTES AND BACKGROUND
Historians believe that the Babylonians were the first mathematically sophisticated culture. They were active around 3000 - 1000 BC. Whilst their knowledge was basic compared to today's standards, they had discovered many of the properties of numbers and geometry which are still studied in schools today.
Attaching meaning to an archeological discovery can be a difficult process. Whilst we will probably never be certain as to the meanings of the numbers in the tablet, mathematicians have found plenty of numerical evidence that they were generated according to a specific rule. This supports the underlying hypothesis to a sufficient degree that many mathematicians are confident that the mis-matched numbers in the table are, in fact, errors which can be corrected. It is these corrected numbers which are often displayed in articles concerning Plimpton 322.
If you enjoyed this challenge, you might like to investigate the Ishango Bone or Stone Age Counting
Getting Started
The Babylonians were interested in building. The maths in these
tables helped them with this task. What maths might be relevant to
this?
Student Solutions
We recieved two very different interpretations of these numbers. Do theyconvince you?
James and Ryan said that
We think that the numbers are two of a pair of pythagorean triples $(a, b, c)$, so that
$$
a^2+b^2=c^2
$$
In each case the third number is
$$120, 3456, 4800, 13500, 72, 360, 2700, 960, 600, 6480, 60, 2400, 240, 2700, 90$$
We wondered why these numbers had been picked in this order, but then found out that they are arranged in order of decreasing angle of the triangle. So, the first row is a (119, 120, 169) triangle which has an angle of $45.2^\circ$ and the last but one has an angle of $33.3^\circ$.
Image
Our hypothesis shows that the errors are in rows 2, 9, 13, 15. These should read
4825, 11521
541, 481
25921, 161
53, 106
The Babylonians might have used these numbes to help to build things.
Michael writes suggesting that the numbers are related to astronomical measurements
The numbers are the measurements of time.
- The number 3541 which is the equivalent of 10 lunar years with just 2 leap days.
- That leads to 4961 which compares to 14 lunar years.
- The number 8161 is the equivalent of 23 lunar years with 11 leap days.
- Is the 10 in the third row intended as leap days?
- 1771 equates to 5 lunar years.
- The number 2929 is practically identical to 8 solar years at 2922 days.
- That 8 year period is also 13 orbits of Venus.
- It is also 5 orbits of the Synodic cycle at 2920 days.
- 4961 is also 22 orbits of Venus making it a common target with 14 lunar years.
- 18,541 equates to 27 orbits of mars.
- The total of the second column at 57613 equates to 99 synodic cycles while the totals of the three columns at 116,798 equates to 200 synodic cycles and the totals of column 1 and 3 equates to 101 synodic cycles.
- There are more synodic cycles evident with the addition of numbers e.g. 2291 + 3541 = 5832 or 10 synodic cycles in 16 years. (+ the 7 as leap days)
- But what about 1679? It equates exactly to one orbit of the Asteroids which were only discovered in 1801 with the aid of a telescope?
- The number 25,921 is just one above the cycle of the Zodiac at 25,920 years, so the numbers are neutral to represent days or years i.e. = 71 solar years.
Teachers' Resources
Why do this problem ?
This short task gives an interesting historical context and a
way to understand the difference between hypothesis and conjecture;
convincing argument and proof. It will encourage mathematical
creativity and also put students in a situation in which there is
no obvious answer or way forward.
Whilst the number skills required are basic, the investigative
aspect of the problem would make an interesting challenge for any
students who want to develop problem solving skills.
Possible approach
Discuss the problem. Does anyone have any suggestions? Can
others test the suggestions? You will need to use calculators.
Students will want to write out the tables and then calculate sums,
differences, products and other combinations of the values until
they spot a possible pattern.
Key questions
- Describe what you see in the numbers
- What ways of linking numbers do we know?
- Are there any obvious rows in the table to look at first?
- If there are mistakes in four rows, how many rows do we need to test before rejecting a hypothesis?
Possible extension
If we find a rule which explains all but the 4 erroneous rows
of numbers, does that prove that our rule is the one that the
person making the table was thinking of at the time? For what sorts
of reasons might a Babylonian have been writing down mathematics?
What sort of mathematics might they have needed to know
about?
Possible support
Some students might initially find it difficult to suggest a
line of enquiry and then to test it.
Start the group off with this statement 'My hypothesis is that
the third column is the sum of the previous two columns' and then
ask the class to test out this hypothesis.
For example, is one row the sum of the other two? (No, but it
starts a chain of enquiry)