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 1. 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
- show how to find the fractional form of rational numbers expressed as decimals
- understand the difference between rational and irrational numbers
- realise that there are more irrationals than there are rationals
- produce proofs by contradiction
- work with numbers in base 60
- understand more about the construction and point of Babylonian clay tablets
- find a formula for a set of Pythagorean triples
This unit uses two historical points as a basis for exploring irrational numbers. These are (i) the Babylonian clay tablets that have already been considered in Babylonian Mathematics I, and (ii) negative numbers. It also looks at a practical method of suggesting that √2 is not rational and later goes on to prove this in two ways. The method of finding the fractional form of a rational decimal is also introduced as is a demonstration that there are more irrationals than there are irrationals.
Proof occurs once explicitly and at least once implicitly in this unit. Explicitly proof occurs in the form of Proof By Contradiction (or Reductio Ad Absurdum). This form is used twice in proving that √2 is irrational. The second proof given here is in the classical first context that students meet it. It turns out that this form of proof is frequently found in Number Theory and Discrete Mathematics at university level. Students should appreciate that this proof is not just something that is used once in this context alone.
Implicity proof occurs in finding the fractional form of rationals expressed in decimal form. But this technique of multiplying by the right power of 10 can be used to show that any repeating decimal is rational. What’s more, the technique is also used later in establishing the formula for the sum of a Geometric Progression.
Again then, it is worth pointing out that a technique that we use for a particular reason can be used elsewhere in mathematics. This is frequently the case. It is seldom that something we find useful in one situation is only useful in that situation. The more maths one does, the more individual techniques get used over and over again.
Starting from an historical perspective, we show that √2 is not equal to any rational number (fraction).
- This session is designed to establish some interest in things historical. Students may or may not remember, from the unit Babylonian Mathematics 1, that the Babylonians calculated √2 to a high level of accuracy some 4000 years ago. Stimulate some interest in the history of mathematics by asking the students to think about, and discuss, when in history mathematicians first accepted the use of negative numbers. Ask them to suggest a point in history and to suggest a reason for their choice. If students work in groups they could then feedback their answers to the whole class.
- Now show them the following quotations from mathematicians (Copymaster 1). These are taken from M. Kline, Mathematical Thought from Ancient to Modern Times, vol. 1, pp. 252-253, vol. 2, pp. 592-593: Negative numbers are ‘absurd’ (Chuquet, 15th Century); Negative numbers are ‘absurd’ (Stifel, 16th Century); Negative numbers are ‘impossible’, and ‘fictitious’ (Caradan, 16th Century); Negative numbers should be ‘discarded entirely’ (Vieta, 16th/17th Century); Negative numbers are ‘false’ (Descartes, 17th Century); Negative numbers are ‘utter nonsense’ (Pascal, 17th Century); Negative numbers are ‘larger than infinity’ and ‘less than zero’ (Wallis, 17th Century); Negative numbers are ‘greater than infinity’ (Euler, 18th Century); Negative numbers should be ‘rejected’ (Maseres, 18th/19th Century); Their use will lead to ‘erroneous conclusions’ (Carnot, 18th/19th Century); Negative numbers are evidence of ‘inconsistency’ or ‘absurdity’ (De Morgan, 19th Century). So, while the ancient Babylonians found √2 to 5 or 6 decimal places of accuracy 4000 years ago, less than 200 years ago mathematicians still had not accepted the validity of negative numbers.
- Recall that √2 is the length of the side of an isosceles right-angled triangle with shorter side 1. Carefully draw such a triangle (using 1 unit = 3cm). We are now about to discuss a topic of great historical interest and of importance in senior mathematics. The class is grouped in 4s. Each group is issued with a long strip of paper (say 3m by 10cm). The students draw a 3m straight line down the middle of the strip. In effect they will turn each strip into 2 rulers by marking two different scales, one on each side of the line drawn down the middle. On one side of the line the students carefully mark out unit graduations of 1 unit (where each unit is the length of the non-hypotenuse side of the isosceles right-angled triangle above). On the other side of the line the students carefully mark out unit graduations of unit length equal to the length of the hypotenuse of the triangle (that is with unit lengths of √2).
- The critical question is this: apart from the starting graduation marks, when do the graduation marks next coincide? It is the student’s job to find out. When they think that they have an answer, for instance the 7th small unit and the 5th larger unit appear to coincide, but do they in fact? Check on the calculator. Find out if 5 times √2 is equal to 7 times 1. A quick check will show that although the graduation marks are close, they are in fact not coincident. The students are encouraged to keep checking likely situations of coincidence in the same way. Their goal is to find where the two sets of marks match up. Hopefully they will begin to conclude, as time goes on, that they may never match up.
- If the students become confident of this (and that is what the previous activity was designed to achieve) they are in good company. Mathematicians of ancient times suspected the same, and eventually, in about 500BC the Pythagoreans proved that no matter how long you made the strip of paper, and no matter how accurately you draw the graduation marks, they will never again match up after the starting graduation marks. In effect they proved that the two rulers are incommensurable; one set of graduation marks cannot be make to measure the other. Imagine that the graduation marks did coincide; say, for example, that 7071 of the 1 units matched 5000 of the √2 units. What this is saying is that √2 can be written as the fraction 7071/5000. So another way of thinking about what the Pythagoreans proved is to say that they proved that √2 can never be written as a fraction; and of course, that means that √2 can never be written as a terminating or recurring decimal either. In short, √2 can never be written down precisely as any number; it can only be written as the unary operation ‘take the square root of 2’, that is, √2. For this reason they called √2 an irrational number.
Here we see how to find a rational equivalent to a repeating decimal. We then show that there are a lot more irrationals than there are rationals by generating some.
- We turn now to study infinite decimals, and to discover whether recurring decimals can be written as a fraction and if so how. First set this as a discussion exercise in groups of 3: Is the infinite decimal 0.9999999…… less than 1 or equal to 1 or greater than 1? Give reasons. Give students time to find an answer they can support with arguments. Once these have been developed give the class the chance to discuss the solutions they found, and their reasons. Now show them this: Let S = 0.9999999…… Multiplying both sides by 10 will give 10S = 9.9999999…… Subtracting 10S – S = 9. Thus 9S = 9, and S = 1.
- Exercise: use the same method (multiply by 10 and subtract) to write 0.8888… as a fraction. Do the same for 0.444444….. and 0.11111…..
- Activity: How should this method be adapted to write 0.5656565656….. as a fraction? What about 0.123123123123123123……
- Activity: Find a method for writing 0.435232323232323….. as a fraction.
5. Activity: What infinite decimal do you need to add to 0.7777777….. in order to give an answer of 1? Explain.
- Next we investigate the relative sizes of the two infinite sets: the set of rational numbers, and the set of irrational numbers. Clearly both are infinite, but the intention is to give the students a feeling for the fact that the set of irrational numbers is vastly larger than the set of rationals. This, of course, goes right against what most students instinctively believe. This is because they have only met a few irrationals but have met a very large number of rationals. Here is a way of enriching the intuitions of the students on this topic. Obtain or construct a roulette wheel which will randomly produce digits from 0 to 9. Alternatively obtain a 20 sided dice and number the sides: 0, 0, 1, 1, 2, 2, 3, 3, . . . 9, 9. (It is easy to make one of these out of cardboard by making an icosahedron. In fact, it is useful to make a large icosahedron because this adds to the dramatic quality of this activity. Also a 20 sided dice is useful for a number of other mathematical activities.) Now, with great fanfare, generate an infinite decimal. You do this by randomly generating a digit for each decimal place. For instance, if successive rolls of the dice give 6, 5, 5, 4, 5, 7, 2, … you will be generating the decimal: 6.554572… Now it is important to prepare the students well for what is about to happen. Emphasise that, if the decimal is to be rational it will need to generate an ordered set of numbers such as 4.323232323232…. or 3.0000000000… Alternatively, if the decimal created is not like this it will be irrational. Now generate such a decimal using the random digit generator. It should become clear to the students that the chance of generating a rational number is minute (eg, getting a 3 followed by 0, 0, 0, 0 …. forever) compared to the chance of generating an irrational number. Thus they should have a feeling for the fact that although both sets are infinite, one is enormously larger than the other.
In this session students construct a spiral and find that [√(r+1) – √r] [√(r+1) + √r] = 1.
- What is the name of the shape of this curve? From what you know about the shape of this curve predict how the spiral would look after 100 stages.
- Activity: Use the spiral to show that m2 = 1 + n2 where n is the square root of a natural number and m is the square root of the next larger natural number. Hence show that (m – n)(m + n) = 1. Thus, for instance, (√3 - √2)(√3 + √2) =1. Check this out for a few other cases.
In this session we show that √2 is irrational using two proofs by contradiction.
- Now let’s establish assumption A. We will assume that √2 is rational. If we assume that √2 is rational we can construct the original triangle (the larger one before the fold was made) in such a way that each of the sides is a whole number of centimetres long. Why? If the hypotenuse is m/n just enlarge the triangle by scale factor n so that the shorter becomes n. A consequence of this is that the smaller triangle must have sides that are also a whole number of centimetres long. Why is this? AB is the difference between EB (which is a whole number of centimetres long) and EA (which is a whole number of centimetres long). Thus AB must be a whole number of centimetres long. Because the small triangle is isosceles, AC must also be a whole number of centimetres long. But DC is the same as CA (why?). BC is the difference between DB (which is a whole number of centimetres long) and DC (which is a whole number of centimetres long). Thus CB must be a whole number of centimetres long. So the smaller triangle also has three sides all whole number lengths. But we don’t have a contradiction . . . yet.
- How do we establish a contradiction? If we now start with the smaller triangle which has whole number sides, we can then repeat the whole process and form another even smaller similar triangle which (for all the reasons outlined above) must also have sides that are whole number lengths. And then the process can be repeated and an even smaller triangle can be formed. And so on. Eventually the smaller triangle will be so small that it is smaller than a whole number of centimetres in length. But this is a contradiction because our logic tells us that the smaller triangle will always have whole number lengths. So our assumption must be false, and its negative must be true. Hence √2 cannot be rational; it must be irrational.
- We turn now to the second proof that √2 is irrational. This is the proof found by the ancient Pythagoreans. Assume that √2 is rational (assumption A). Thus √2 = m/n. Squaring we get 2 = m2/n2 and 2n2 = m2. Thus m2 is even. Hence m is even (this was proved during the unit Babylonian Mathematics 1 with a proof based on writing m as a unique product of primes). Since m is even, let m = 2p. Thus 2n2 = (2p)2 = 4p2. Thus n2 = 2p2. Thus n2 is even and so must n be even. Since both m and n are even, the fraction m/n can be simplified to the fraction r/q.
- But we could use the same argument to show that both r and q are even and thus derive an even simpler fraction s/t which will also have both top and bottom numbers even. As this process continues the top and bottom numbers of the fraction become smaller and smaller. Eventually they will become smaller than 2 and thus will no longer be even. This is where the contradiction occurs. Accordingly assumption A must be false; that is, √2 must not be rational. [Note to the teacher: This proof could have been shortened by assuming that m/n were in the simplest form. However, students find this version harder to understand.]
- Extension. Show that √3 is irrational. (Hint: Assume √3 is m/n. Therefore 3 = m2/n2 and 3n2 = m2. Thus m2 is a multiple of 3. Write m as a unique product of primes and show that if m2 is a multiple of 3 then m must also be a multiple of 3.)
- Extension. Show that if p is prime then √p is irrational.
This session links back to the Babylonian clay tablets that contain an accurate value for √2. It then works on a general set of Pythagorean triples.
- This session finishes off this unit by linking it with the earlier unit Babylonian Mathematics 1.
- Basic historical information. The numerical techniques used by the Babylonians were related to their economic needs and subsequently to developments in mathematical astronomy. There are at least 500,000 Babylonian tablets in museums, and this represents only a tiny proportion of those estimated to remain buried in the ruins of Mesopotamian cities. Many of these tablets are clearly ‘school texts’; that is, they were written by people under instruction (students or apprentices). Some tablets were used for reference purposes. Some, for example, contain elaborate and detailed multiplication tables in sexagesimal form (base 60) of course. Others contain tables of squares, square roots, cubes, cube roots, and reciprocals. We will see shortly that they also explored the ‘Pythagorean triples’ 2000 years before Pythagoras.
- Return to the tablet that was used to begin the unit Babylonian Mathematics 1. We noted that 1, 24, 51, 10 is approximately √2 and that when it is multiplied by 30 it gives 42, 25, 35. In fact the Babylonians probably did this calculation, not by multiplying by 30, but by dividing by 2. Explain why this short cut works and explain how the calculation would work out in detail. (30 = 60/2. So they multiplied by 60 implicitly and then divided by 2. In base 10 we can multiply by 5 this way.)
- The Babylonian value for √2 was such a good one it was still used by Ptolemy for computations 2000 years later. Why is Ptolemy famous? Another Babylonian approximation for √2 was 1 + 1/3 + 1/(3 x 4) – 1/(3 x 4 x 34). Write this as a decimal and compare it to the value on your calculator. Show that this value is 1, 24, 51, 10, 37 in sexagesimal. Show it is also 1, 25 – 0, 0, 8, 49, 22.
- ‘Pythagoras’ Theorem’ was known by the Babylonians more than 2000 years before Pythagoras (although probably not proved). They did not prove that √2 is irrational either (the Pythagoreans did this around 500BC). The Chinese knew about ‘Pythagoras’ Theorem’ around 200BC.
- The Babylonians developed tables of ‘Pythagorean triples’ 2000 years before Pythagoras. That is tables of whole number values, a, b and c, such that a2 + b2 = c2.
- Here is the start of a table of triples:
n a b c
1 3 4 5
2 5 12 13
3 7 24 25
4 9 40 41
Form a conjecture that suggests what the relations are between n and a, n and b, and n and c. [Hint: to find the relation between n and b try finding the factor d such that n x d = b. Prove that the relations you have found are actually those that make up (a subset of) the Pythagorean triples.]