This unit uses a Babylonian clay tablet and the mathematics found on it as a catalyst to investigate a variety of mathematical ideas. This same catalyst is also used for the unit: Babylonian Mathematics 2. Areas under enlargement are discussed in the present unit, and lying behind the various activities in both these units is the idea of incommensurability, which means, roughly, ‘things which cannot be measured no matter how accurate the ruler’.
- understand, use, and calculate simple surd numbers
- understand, use, produce, and work with prime numbers and prime factorisation
- produce simple proof
- work with numbers in bases other than 10
- use Heron’s method to calculate square roots
- understand the construction and point of Babylonian clay tablets and produce one of their own
This unit covers a great deal of material as the specific learning outcomes above demonstrate. Students will need to know about simple surd expressions such as √2, √3 and so on; know about prime factors and how to find them using Eratosthenes’ sieve; understand unique prime factorisation; work with numbers in bases other than base 10; and use Heron’s method to find successively better approximations to a square root. We therefore suggest that this unit be used as a revision or consolidation unit after the class has met the separate ideas and has some familiarity with them. One unit that should be done before this unit is All Shapes and Sizes, Algebra, Level 6.
This unit will help students to see how a number of seemingly different mathematical ideas can be drawn together to solve a single problem. The problem in this case is an archaeological one – the mystery of Babylonian clay tablets.
Copymaster 2: Meaning of Babylonian symbols
Copymaster 1: Clay tablet and diagram
Before this session starts we expect that students will have explored a period of history and that they will report this back to the rest of the class. We then consider prime numbers, surds, prime factorisation, and the squares of odd and even numbers.
In order for students to appreciate the significance of the clay tablet we will shortly discuss, they need to have some sense of what 2000BC means in concrete terms. We are talking about the mathematics done by people 4000 years ago! What was life like back then?
- Activity: In groups, undertake some reserach to create a thumbnail sketch of life and times at each of the following periods of history: 100 years ago; 500 years ago; 1000 years ago; 2000 years ago; 3000 years ago; and 4000 years ago:
Which famous people were alive around then?
What sort of transportation was typically used?
What major inventions occurred around this time?
When was paper first invented?
When was the printing press invented?
As groups report their thumbnail sketches create a timeline documenting the key achievements and historical benchmarks.
- Show the students the photograph of a Babylonian clay tablet containing some mathematics (see Copymaster 1 and below). It is taken from Plate 6 in O. Neugebauer (1957) The Exact Sciences in Antiquity, published by Brown University Press,
. See also the diagrammatic version (Copymaster 1 and below). This is taken from O. Neugebauer and A. Sachs (1945) Mathematical Cuneiform Texts, published jointly by the American Oriental Society and the American Schools of Oriental Research, New Haven, Connecticut, page 42. Providence, Rhode Island
- This mathematics was done 4000 years ago. And it was done on clay. This was before paper and pens were around. The Babylonians stamped special shapes onto the wet clay before it set hard. Many similar tablets have been found. They are equivalent to what we would call mathematics exercise books today. They were used by students of mathematics 4000 years ago. What do the symbols mean? Consult the diagram: Meaning of Babylonian Symbols (Copymaster 2 and below) and see if you can decode the numbers (taken from the internet). The answer is on the diagram: Interpretation of the Symbols (see Copymaster 3 and below). This is taken from O. Neugebauer and A. Sachs (1945) Mathematical Cuneiform Texts, published jointly by the American Oriental Society and the American Schools of Oriental Research, New Haven, Connecticut, page 43.
- Before we go on to discuss the reason why these symbols are displayed like this it is necessary to revise the meaning of ‘square root’ and ‘prime number’.
What does square root mean?
Which numbers between 1 and 1000 have square roots that are whole numbers? What is a prime number?
Why are they similar to the primary colours in painting?
Use the ‘Sieve of Eratosthenes’ to find all the primes between 1 and 200. That is, give each group a 10 by 20 grid with the numbers 1 to 200 listed in order in the grid squares. Students cross out 1, all the even numbers except the first, all the multiples of 3 except the first, all the multiples of 5 except the first (why don’t they need to cross out the multiples of 4?), all the multiples of 7 except the first, all the multiples of 11 except the first, and all the multiples of 13 except the first (why need they not go further?). What is left is the primes less than 200.
- All numbers can be uniquely factorised into primes. For example, 40 = 2 x 2 x 2 x 5. And 36 = 2 x 2 x 3 x 3. The order can change but these will always be the prime factors. Choose some numbers of your own (even large ones) and write them as the product of primes. Get the whole class to find the prime factorisation for 716625. Now verify that all these are really the same factorisation; in other words check out for a single case that the prime factorisation is unique. Perfect squares have symmetry in this respect (eg, 36 has a pair of 2s and a pair of 3s, ie, 36 = (2 x 3) x (2 x 3). The square root of 36 is just one of the pairs, thus: √36 = 2 x 3. 40 is not a perfect square because it does not have this symmetry. The best that can be done with 40 is to isolate the pair of = (2 x 2) x (2 x 5). The square root of 40, in a similar way, could be written as: √40 = 2 x √(2 x 5) = 2√10. There is a name for this way of writing a square root; it is called the surd form of the square root. Find the square roots of the following in surd form: 48, 120, 100, 28, 27, 64, 150, 1000.
- Prove that the square of an even number is even (hint: an even number is 2 x m = 2m, where m is a natural number). Prove that the square of an odd number is odd (hint: an odd number is 2 x m + 1 = 2m + 1, where m is a natural number).
- Construct alternative proofs using the fact that every number can be uniquely written as a product of primes. Hint: If n is even, when it is written as a product of primes listed from smallest to largest (n = p1.p2.p3. . . . . . . .pk), then p1 must be 2.
- Assume that n is even and m is a natural number. Prove that, if m is the square root of n, then m must be even. Assume that n is odd and m is a natural number. Prove that, if m is the square root of n, then m must be odd. Hint: write m as a product of primes.
By folding and photocopying, we see why enlarging an area by a factor of 2, enlarges a length by √2.
- It has been appropriate to discuss square roots and primes because the Babylonian Tablet displays the square root of 2 in both a numerical and a geometric form. Activity for groups of 3. Hand out to each group 4 congruent cardboard isosceles right-angled triangles. Ask them to prove, without using Pythagoras, that, if the length of the shorter side is 1, then the length of the hypotenuse must be the √2.
- Let the students work on this until some success and rigour of logic has been achieved, then discuss the results. The proof is based on the idea that two of the triangles form a square with area 1 (how?). But it can also be shown that 4 of the triangles can also be placed together to form a single larger square (how?). This larger square must have an area of 2 (why?). And it is evident that the length of the side of the larger square is the hypotenuse of one of the original triangles. The proof follows easily.
- This is a ‘nice’ proof and it ought to be dwelt upon to let its significance sink in. It could be displayed as a poster. Also the students should be encouraged to remember that the hypotenuse of an isosceles right-angled triangle with shorter side 1 is √2.
- Activity for groups of 3. An A3 piece of paper is made by placing two A4 pieces of paper side by side in such a way that the A3 has the same shape as the original A4. (Similar work is done in the unit All Shapes and Sizes, Algebra, Level 6.) Samantha draws a line exactly 1cm long onto an A4 piece of paper. She then enlarges the paper on a photocopier by pressing the ‘enlarge A4 to A3’ button. How long will the line be when it is enlarged onto paper with 2 times the area (ie, from A4 to A3)? Students should be encouraged to form a conjecture (many will conjecture than the new line will be 2cm long). Students could check this on the photocopier and get an approximate answer. They should also be encouraged to generate a proof based on what happens to length when a figure is enlarged with scale factor 2 (a square 1cm by 1cm will become a square with area 2cm2, and what will be the length of the side of this square?).
- Prove that, for A3 and A4 paper, the longer side must be √2 times the shorter side. Hint: if the sides of an A4 piece of paper are a and b, then the sides of the corresponding A3 piece of paper must be in the same ratio.
- When the shorter side of an A4 piece of paper is folded onto the longer side (the line AE is folded onto the line AB) prove that the length of the fold (AD) is the same as the length of the longer side.
Here we calculate simple surds using approximate methods. We also work with numbers in bases other than 10. These two things are related to the Babylonian clay tablets.
- You will see, if you check your calculator, that the square root of 2 is 1.414214 (accurate to 6DP). Guess what value the ancient Babylonians calculated 4000 years ago using wet clay and stamps. Let the students discuss their guesses and then record them on the board. In fact they got: 1.414213. Hopefully this will surprise the students. It can readily be seen that this result is remarkably accurate; especially given the fact that they worked only with clay and stamps.
- Activity. If you had to work out √2 to a 5DP level of accuracy and you did not have a calculator, how would you go about it in a systematic way? (Eg, one systematic way involves trying squaring, using pencil and paper, the numbers 1, 1.1, 1.2, 1.3, 1.4, 1.5 (which is too big); then 1.41, 1.42 (too big); then 1.411, 1.412. 1.413. and so on.) Use your method to find √2 and √3.
- Activity. Draw an accurate graph of y = x2 for x greater than or equal to 0 and less than or equal to 4. Use this graph to work out approximate square roots for numbers less than 16.
- What level of accuracy can you get by this method; that is, how many decimal places of accuracy?
- The ancient Babylonians used a base 60 number system (the sexagesimal number system). Our number system is base 10 (why do you think this is?). Thus 3.659 [base 10] means 3 + 6/10 + 5/102 + 9/103. If 3.659 was in base 12, that is, 3.659 [base 12], it would mean 3 + 6/12 + 5/122 + 9/123. Convert this number to base 10. Convert 4.56 [base 8] to base 10. Convert 5.68 [base 10] to base 6.
- Return to the clay tablet. The numbers 1, 24, 51, 10 appear above the horizontal diagonal. These numbers are in base 60. Show that they give the square root of 2. Let the students work this out for themselves by trial and error. The answer is: 1 + 24/60 + 51/602 + 10/603. Note that this can be calculated more easily in nested form, working from right to left on the left and side of the equal sign: 1 + 1/60(24 + 1/60(51 + 1/60 x 10)) = 1.414213.
Here we use Heron’s Method to calculate square roots. This is again linked to the clay tablets.
- On the clay tablet some other numbers appear: 30 and 42, 25, 35. Find out what these number represent. Students need to use their imaginations and investigative capacities, just as if they were archaeologists. The answer is that the square has side 30 and diagonal 42, 25, 35 in base 60. They can readily calculate this to be 42.426389 (which happens to be 30 times √2).
- Archaeologists have been able to work out what method the ancient Babylonians used to make their calculations of √2. They were able to do this by examining, not the tablets that gave the correct answers, but those that contained calculation errors. The maths students in Babylonian times, it appeared, made errors in calculation. And when mathematical archaeologists asked themselves what method would have led to these students of mathematics making these errors in calculation, they concluded that the method must have been Heron’s method. Here is how this method works to find the square root of 10. (i) Make a sensible guess, say 3. (ii) Is 3 the correct answer? Find out by dividing 10 by 3 and seeing if the answer is 3. In fact it is 3.333. (iii) So the correct answer must be somewhere between 3 and 3.333. Choose the number exactly half way between 3 and 3.333 as the next estimate. This number can by found by adding and halving the result: half of 3 + 3.333 = 3.1665. (iv) Is this the correct answer? Find out by dividing it into 10. The answer is 3.1581. (v) The required answer must be between 3.1665 and 3.1581. Choose the number half way between by adding and halving: 3.1623. (vi) Take this as the next estimate and continue as before.
- Use Heron’s method to find the square root of 20 accurate to 5 decimal places. Use it to find the square root of 5 to 5 DP.
- The square on the Babylonian tablet has a side of 30. The number 1, 24, 51, 10 appears above the diagonal; and 42, 25, 35 below the diagonal. If the side had been 20 and not 30 what would these numbers have been instead? Show these numbers using the Babylonian symbols and draw your own clay tablet. It is not intended that students use Heron’s formula to calculate the square root. They can use the electronic calculator. But they must show the answer in base 60 using Babylonian symbols.
- Repeat this for a square of side 40.
Students now construct clay tablets of their own.
- This session rounds off the first unit based on the Babylonian clay tablet, and it is devoted to designing and constructing a clay tablet of the Babylonian sort. Students work individually (or in pairs) and design, on paper, a tablet; complete with Babylonian numerals, with a side length of their choosing (a whole number between 10 and 59 say).
- Students swap with others and check that the tablets are correctly designed.
- Finally they devise some means of stamping the Babylonian shapes onto a ‘tablet’ and then make a tablet. Here the class could use any one of a number of methods of making a tablet. Here are some suggestions: concrete, a biscuit with icing, a cake with icing, plaster of paris, clay, plasticine, paint on cardboard, paint on papier mache, computer graphics package, and screen printing.