Maths and Music

Rational and Irrational Numbers

Question 1
Question 2
Question 3
Question 4

Q1 Sort the following numbers into groups in some way:

There are many ways you may have chosen to sort these numbers. However, using the classification of natural numbers, integers, rational numbers and irrational numbers, I have sorted the numbers like this:

natural numbers: 12, 139, root 25
integers: 12, 139, -26, -4, root 25
rational numbers: 12, 139, -8.9, 1/3, 0.24187, -26, -4/7, root 25, 12.66, -4,
irrational numbers:

The rational numbers include the integers, which include the natural numbers. However, the irrationals and the rationals do not ever include each other. (Make sure you see why the square root of 25 is a natural number, not an irrational number like root 2, and why 0.24187 is rational).

There are two different ways to represent fractions. What are they? What do you think the advantages and disadvantages of each are?

To illustrate the two ways to represent fractions, both these mean the same number, a quarter: 0.25 and ¼ (1/4). Either is fine for this number. But what about this number: 0.3333333..... and 1/3. Here the representation as a fraction is much more concise than the decimal representation.

Why do we need to have negative numbers?

There are two important uses of negative numbers. One is to express debt: if I have -£10 in my bank account, I know I am in debt to the bank. Another use is to indicate motion to the left or down. A translation of +10 means move 10 units to the right or up, whereas a translation of -10 means move 10 units to the left or down.

Why do we need irrational numbers?

As you will see in Musical scales, we need irrational numbers if we are to tune instruments so that they can play in all keys. We also need irrational numbers to represent the diagonal of a square, unless its area is a square number. There are many other uses for irrational numbers also.

Go to top

Q2 Convince yourself that all approximations are rational numbers.

If I use my calculator to find the square root of 2, it gives me the number 1.414213562. This means 1 + 4/10 + 1/100 + 4/1,000 + 2/10,000 + 1/100,000 + 3/1,000,000 + 5/10,000,000 + 6/100,000,000 + 2/1,000,000,000. These are all rational numbers, and so their sum is also rational. Although root is not rational, as soon as you attempt to write it down as a decimal, you inevitably write down a rational approximation to the actual irrational value. The same is true for all irrational numbers. This is why mathematicians prefer to use precise notation such as for irrational numbers.

Go to top

Q3 Square numbers

Using your calculator, find the square root of 2. 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? Depending on your calculator, you might get back to 2, but you might get 1.999999... Why do you think this might happen?

2² = 4 and $(\sqrt{2})^2 = 2$ $\sqrt{2} = 1.414 213 562$ but

You may not have quite the same number of decimal places as this, but you should see that when you square the value of $\sqrt{2}$ given you by the calculator, you don't get back to 2, as you might expect. This is because you are squaring a number which is not quite equal to the square of 2, so you don't quite get back to 2.

Now 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?

Number	Multiple of 3?	Multiple of 2?	Multiple of 4?	Square	Multiple of 3?	Multiple of 2?	Multiple of 4?
1	No	No	No	1	No	No	No
2	No	Yes	No	4	No	Yes	Yes
3	Yes	No	No	9	Yes	No	No
4	No	Yes	Yes	16	No	Yes	Yes
5	No	No	No	25	No	No	No
6	Yes	Yes	No	36	Yes	Yes	Yes
7	No	No	No	49	No	No	No
8	No	Yes	Yes	64	No	Yes	Yes
9	Yes	No	No	81	Yes	No	No
10	No	Yes	No	100	No	Yes	Yes
11	No	No	No	121	No	No	No
12	Yes	Yes	Yes	144	Yes	Yes	Yes
13	No	No	No	169	No	No	No
14	No	Yes	No	196	No	Yes	Yes
15	Yes	No	No	225	Yes	No	No
16	No	Yes	Yes	256	No	Yes	Yes
17	No	No	No	289	No	No	No
18	Yes	Yes	No	324	Yes	Yes	Yes
19	No	No	No	361	No	No	No
20	No	Yes	Yes	400	No	Yes	Yes

You should be able to see that both the squares are multiples of 3 or 2 only if the original number was a multiple of 3 or 2. However if the original number is not a multiple of 4, but is a multiple of 2, then its square is a multiple of 4. This difference occurs because all integers which are multiples of 2 can be written in the form 2k (ie. 2 multiplied by another integer). When you square 2k, you get 4k², and so the square is a multiple of 4, even though the original number may not have been a multiple of 4.

Go to top

Q4 Proof that the square root of 2 is irrational

You have just worked through the actual argument that the Greeks used to show that the number that when squared gives 2 actually couldn't be expressed as a fraction. The argument goes as follows.

They said, let's suppose that the square root of 2 is a fraction expressed as a ratio of the numbers p and q so that $\sqrt{2} = p/q$ . Here p and q are integer numbers which have no factor in common. You know that 2/4 = 1/2, and that you wouldn't normally write 2/4. We are simply insisting on this condition here, that all fractions are reduced to their lowest terms.

Let

$\begin{displaymath}\sqrt{2} = p/q \end{displaymath}$

Then

$\begin{displaymath}2 = p^2/q^2 \end{displaymath}$

$\begin{displaymath}2q^2 = p^2 \end{displaymath}$

To explain this in words, $\sqrt{2}$ is a number which when squared makes 2. So if you square both sides you get the result that 2 is p² divided by q². If you multiply by q² you get 2q² equals p². This tells us that the number p² is even. Now if p was a number like 3, 3 squared is 9 and 9 is an odd number. If it was 5, 5 squared is 25, that's an odd number as well. Any odd number when squared is always another odd number. So the only way you can have p ² being even is if p itself is even.

To continue:

$\begin{displaymath}p = 2r \end{displaymath}$

$\begin{displaymath}p^2 = 4r^2 \end{displaymath}$

$\begin{displaymath}2q^2 = 4r^2 \end{displaymath}$

$\begin{displaymath}q^2 = 2r^2 \end{displaymath}$

So if it's even, it's two times some other number, so let's call it 2r. If p is 2r then p² is 4r². But then 4r² equals 2q² so we can divide through by 2 to say that q² is now equal to 2r². The same argument as before tells you that q² is an even number and therefore q is an even number.

Why is that a problem? Well, we've shown that p is an even number and q is an even number but we made the initial assumption that p and q could have no factor in common using a completely logical argument. So we've reached a contradiction. What we've found is that if you make this assumption that $\sqrt{2} = p/q$ and follow this by a logical argument, you end up with a contradiction. There is nothing wrong with the logical argument, therefore the initial assumption must be wrong, and therefore the square root of 2 cannot be expressed in the form p/q. This means that it is not a rational number, since all rational numbers can be expressed in the form p/q.

Exactly the same argument shows that the square root of 3 or 5 or any square root of any number which is not itself a square cannot be expressed in terms of a fraction. So the Greeks were right, we know there are numbers out there which can't be expressed as fractions.

top of page

Back to : Maths and Music Main Page