Maths and Music

Rational and Irrational Numbers

Why rational numbers aren't enough
Irrational numbers
Proof that root 2 is irrational

Types of numbers

Q1 Sort the following numbers into groups in some way:

• Find two more examples for each of your groups.
• How many different ways of representing numbers do you know?
• There are two different ways to represent fractions. What are they? What do you think the advantages and disadvantages of each are?
• Why do we need to have negative numbers?
• Why do we need irrational numbers?

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Early people counted on their fingers. That gave us numbers like 1, 2, 3, 4 and 5 which we can count on an abacus. Mathematicians call these numbers natural numbers (and give them the symbol N) and they served early people well for a long time, because that is all you needed to count your crops, count your goods, count your animals and count how many wives you had.

However with the invention of money there came a problem! Suppose you have ten cows. The tax man then comes and says your taxes are 11 cows. How many cows do you have now? For a mathematician this means solving the equation x + 11 = 10 and to do this we have to invent new numbers. These numbers were called negative numbers and if you combine the negative and positive whole numbers you get what we call integers, which are given the symbol Z (from the German for numbers, zahlen).

Now let’s suppose you have 4 children and you have 3 fields. You are very fair minded and you decide you want to divide your fields up equally amongst your children, so how many fields does each child get? Unfortunately you run out of numbers again. There is no number you can count on your fingers or even have in your bank balance (when the tax man has taken money away) that allows you to divide three fields amongst four children. We all know now that the answer we need is three quarters of a field per child. We have to invent new numbers to allow us to divide fields up amongst children.

If we have q children and p fields, then the number of fields, r, that each child gets is p divided by q, or r = p/q . These numbers, which mathematicians call rational numbers, are given the name Q (for quotient). They are known as rational because they can be expressed as a ratio: p/q is equivalent to p:q.

Rational numbers can be expressed very naturally as fractions, such as 1/2, 7/8 and -4/5. They can also be represented by decimals (which is a short way of saying decimal fractions, and just means the fractions which have 10 as their denominator). 1/2 = 0.5, 7/8 = 0.875, -4/5 = -0.8, and so on. You can tell that a decimal represents a fraction if it either terminates (like these examples do), or if it has a repeating pattern, for instance, 1/3 = 0.3333..., 1/11 = 0.090909..., 5/7 = 0.714285714285.... However, not all decimals which show pattern represent a fraction: 0.10110111011110111110... has a clear pattern, but it does not have a repeating pattern. This number is therefore not rational - mathematicians say it is irrational.

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Why aren't rational numbers enough?

Rational numbers served everyone very well for a long time and the Greeks, who were fantastic mathematicians, thought that these were the only numbers which were needed. In fact they had entire moral and religious systems based on this presumption. However, they came to a serious problem when they asked: Can we solve the equation x² = r where r is a rational number?

In case this looks a bit scary, an example of a practical problem where this would arise would be as follows: I want to double the amount of crops I can grow. This means doubling the area of my field (the area on the right of the image here is exactly twice that on the left - although you may not think it looks as if it is). But how long should the side of my field be, if remains a square? This is a very natural question to ask if you are interested in replying to difficult questions from the tax man. (The answer isn't 2! If you double the length of the side of the field, you will multiply the area by 4: 2L x 2L = 4L² = 4A.)

So here is the question the Greeks were faced with: can we find a fraction which has the property that when you square it you get another fraction? Let’s think of the particular case where this number is two. Can we find a number which is a fraction, which, when you square it, gives the answer two? The Greeks asked this question and were horrified to discover that the answer is NO! This was a great shock to the Greeks, destroying much of what they believed in terms of music (you can find out more about this in The maths of musical scales) and art and social philosophy and literature, so much so, that it is said that the person who discovered it was required to commit suicide!

We now see that there are very naturally occurring problems, the solution to which is not a fraction. The problem we're interested in, what number when squared gives the answer 2, has the solution: 1.4142135623730950488 approximately. In fact, if you wrote down the decimal expansion for the square root of 2, it would go on forever and never repeat. This was a shock to the Greeks and is still quite a shock to anyone who meets it for the first time. The point about this observation was that the Greeks and everyone since has had to use new types of numbers which are not fractions - they are not rational numbers.

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Irrational Numbers

If we take the set of all the numbers with all possible decimal expansions we get what mathematicians call the real numbers. A real number is a number which can have any possible expansion in decimals. Amongst the real numbers we have the natural numbers (1, 2, 3, 4, ...), the integers (..., -3, -2, -1, 0, 1, 2, 3, ...) and the rational numbers (for instance, 1/2, 1/3, 7/8, 1.5, -6.7, -9.8117, 4.434343....). But, as we saw above, there are many more numbers which cannot be represented either as a whole number or as a fraction - the irrational numbers. The square root of 2 or of 3, or the number , are all examples of numbers which are not rational. In fact, it can be shown that most numbers cannot be expressed as fractions - there are far more irrational numbers than there are rational numbers. So the Greeks who originally thought that all numbers could be expressed as fractions couldn't have been further from the truth.

So what? One thing that it's quite easy to show is that it doesn't matter which number you choose, you can always approximate it as closely as you like by a rational number. Going back to the square root of 2, we start its decimal expansion as follows 1.4142135623730950488. If I stop at 1.4, my approximation is closer to root 2 than one part in ten – not very good. If I stop at 1.41, my approximation is closer to root 2 than one part in a hundred – a bit better. If I stop at 1.414, my approximation is closer to root 2 than one part in a thousand. If I stop with 1.4142135623730950488, I have a number that's closer to root 2 than one part in a million, million, million. Surely this should be close enough for anyone.

GH Hardy, who was arguably one of the greatest mathematicians that the UK produced in the last
century, made the following quote:

"It is obvious that irrationals are uninteresting to the engineer since he is only
concerned with approximations and all approximations are rational"
from A Mathematicians Apology, 1940.

(Hardy, in the same book in 1940, said that relativity theory had no warlike purpose and could have no practical use, five years before the atomic bomb exploded. Still a great book and well worth reading.)

Q2 Convince yourself that all approximations are rational numbers.

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Now I have a lot of respect for GH Hardy. He was a fantastic mathematician who proved fantastic results, which I could never have proved myself. But I firmly, and to the bottom of my soul, believe he was wrong in saying this - many engineers are very interested in rationals and irrationals. Because whilst engineers may only be interested in approximations when they come to building things, engineers are most certainly concerned with phenomena. And phenomena in the physical universe seem to depend very much on whether numbers can be expressed in fractions, or whether they can’t. This seems a rather paradoxical statement and this conference tries to explain some of the reasons why this might be the case. Indeed, by looking at music (The maths of musical scales), we’ll try to see why Hardy might have been a bit mistaken.

Q3 2 squared and root 2 squared

• Using your calculator, find . Write this number down. Now enter the number you have written into your calculator and square it (don't just press the x² key again). What do you notice? Can you explain what you have observed?
• Write a list of the integers between 1 and 20 in one column, and next to them write a list of their squares. In the list of squares, mark off those which are multiples of 3. Now do the same for your original list. What do you notice? What happens if you mark all the multiples of 2 in each list? What about multiples of 4? Can you explain why things are different for 4?

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Q4 See if you can work through the argument that the Greeks used to show that the square root of 2 cannot be expressed as a fraction.

• Step 1: Start by supposing you can express root 2 as a fraction, and put it equal to p/q.
• Step 2: If p and q had a common factor, you would divide through by it (eg. 2/4 = 1/2). So convince yourself that p and q must be integers which cannot have any common factors [this step is crucial to the argument].
• Step 3: Multiply both sides of your equation by q.
• Step 4: Now square both sides.
• Step 5: What does this tell you about p², and what does it tell you about p? [Think about numbers which have 2 as a factor].
• Step 6: If p is an even number, you can express it as, say, 2r. Substitute this for p. Think carefully about what you should substitute for p².
• Step 7: Simplify the resulting equation. What does this tell you about q² and about q?
• Step 8: Look back at the beginning of this argument, in Step 2. Do you see a problem here? What do you think this means?

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