J OHN P ULHAM E SSAY
What’s an Irrational Number?
At an early stage of Ancient Greek thought about Mathematics it seems to have been assumed that
any two quantities A, B of the same kind (lengths, areas, temperatures, densities etc) must have
a common measure. That is to say, a quantity U of that kind such that each of A and B are simple
multiples of U . Here is a simple illustration. U goes 5 times into A and 4 times into B.
In this case we would then say that, in the language of ratio, A is to B as 5 is to 4. Or, even more
commonly: the ratio of A to B is 5/4.
(ideas like this are still prevalent in music and horse betting)
Of course, if there is one common measure for two quantities there will be lots of them. A line of
length half of that of U would serve equally well (hence 5/4 = 10/8).
It is suggested that this belief comes from Pythagorean ideas of the ‘atomic’ nature of matter (?). If
you think of a line as being made up of billiard balls then any two pieces of the line will consist of
certain numbers of billiard balls. So a billiard ball is a common measure for all parts of the line.
At some time between Pythagoras (mid 500s BC) and Plato (428-348 BC) somebody, somewhere,
made a horrible discovery. He managed to show that there are pairs of quantities for which it can
be proved that there is no common measure.
The standard example of this phenomenon is the one quoted by Plato: the side and the diagonal of
a square do not have a common measure. There are reasons to believe that this was not the example
that originally caused the problem, but it is the most familiar example now, as it was then.
Let’s rehearse the argument
Suppose that there is a common measure, as the Pythagoraeans would insist that there were. Call
it U . Then, in the notation of the diagram AB = nU and AC = mU . We could always choose U so
that n and m do not have any common factor (absorb that factor into the size of U : if n and m had
hcf 3 then replace U by 3U ).
Now, by Pythagoras’ theorem (hoist with his own petard!) AB2 + BC2 = AC2 or, since we are
dealing with a square, AC2 = 2AB2 . In other words
m2 = 2n2
From here on in the argument is quite simple. Remember that we have arranged that n and m do
not have any common factors.
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The equation tells us that m2 is even (it is twice something). The square of an odd number is clearly
an odd number, so m must be even. Since m is even we can write m = 2k for some number k. So
m2 = 4k 2 . Put this back into the original equation and get
4k 2 = 2n2
or
n2 = 2k 2
So n2 is even (it is twice something) and therefore n is even.
That is a pity. We started by ensuring that n and m had no factors in common and ended by
showing that they were both divisible by 2. That is what you call a logical contradiction! The only
logical possibilities now are that (i) we have committed a logical error in our deduction (I claim
that we have not) or (ii) our original assumption was incorrect—namely, the assumption that AB
and AC had a common measure (we have not assumed anything else apart from the validity of
Pythagoras’ theorem).
So it cannot be the case that AB and AC have a common measure.
AB and AC are clearly quantities of the same kind—unless you wish to claim that lengths of lines at different
slopes are different kinds of thing!
The cat lands amongst the pigeons
This discovery caused huge problems for philosophers and mathematicians. There’s a large section
of Euclid’s Elements of Geometry (circa 300 BC) that shows all the signs of last minute editorial
panic resulting from this problem.
The difficulty comes with the theory of ratio. This is fundamental to geometry. When I was in
school in the 1960s we learned all about similar triangles and their properties at an early stage of
our geometrical education. If you ignore school geometry texts and read Euclid’s ‘original’ (in as
much as we have such a thing) you will get a strange surprise. Euclid gets through all the basic
work on triangles, quadrilaterals and circles before he even mentions similar triangles. Circles
come in Book 4, similar triangles in Book 6. In between comes the astonishing Book 5.
Book 5 of the Elements develops the theory of ratio. So long as a belief in common measures
could be sustained the theory of ratio was reasonably easy because it was ‘just’ the theory of whole
number ratios. Once that belief was lost the theory of ratio suddenly became very difficult indeed
because you were not just doing simple arithmetic calculations. It is a major miracle that the Greeks
(probably Eudoxus) were able to produce a logical theory of ratio even without this prop. Such a
miracle, in fact, that it was almost entirely lost to Western thought in later times. I doubt whether
many mathematicians before the late 19th century were really capable of understanding what Book
5 was all about.
The same in modern language
Consider all lengths on a line. Some have a common measure with a unit length and so their length
can be expressed as a fraction (of the unit length). For example, 22/7 or 1/3. These are called rational
lengths (‘rational’ as in ‘pertaining to ratio’ rather than as in Jane Austen).
The consequence of the earlier argument is that these do not exhaust all lengths. The lengths that
are not rational are called irrational.
The (positive) Real Number system is the collection of all lengths on a line (don’t quote that definition at a mathematician! he may not like it very much—though he ought to). We rather loosely call
these lengths ‘numbers’ by analogy with whole numbers.
So, some real numbers are of the form n/m, where n, m are positive integers, and some are not. The
former are called rational
numbers and the rest are called irrational numbers. We showed above that
√
the number called 2 is irrational. So is the number called π, but that is much harder to prove.
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Decimals
It is traditional in modern Western civilisation to describe numbers in the decimal system. Thus we
write 123.654 to represent the number
1 × 100 + 2 × 10 + 3 +
5
4
123654
6
+
+
=
.
10 100 1000
1000
You were probably told at school that
1
= 0.33333333333333333333 . . .
3
—the list of 3s does not stop. This shows a complication with the decimal system. We have to allow
for ‘infinite’ decimal expansions as well as finite ones if we want to represent all possible numbers
(even numbers as simple as 1/3).
In fact it is easiest to regard all decimal expansions as being infinitely long. It’s just that the expansions of things like 1/2 end in infinitely many zeros!
1
= 0.500000000000000 . . .
2
I can now tell you the difference between rational and irrational numbers from the point of view of
their decimal representations.
The decimal expansion of a rational number is always eventually repetitive. For example
1
= 0.333333333333333 . . .
3
5
= 0.41666666666666 . . .
12
4
= 0.363636363636363 . . .
11
‘eventually’ because there may be a non repeating bit at the start, as with 5/12.
Conversely, any decimal that is eventually repeating must represent a rational number. (This is
easy to prove if you know about summing infinite geometric series).
That leaves us with the conclusion that irrational numbers are characterised by have decimal expansions that are not eventually repeating.
√
2 = 1.41421356237309504880168872420969807856967187537694807317667973799073 . . .
and we know that it is never going to settle into a repetition.
Irrationals outnumber Rationals
Let me first prove a few results about the way in which rational and irrational numbers are distributed on the line. They make it clear that irrational numbers are by no means exceptional.
(1) There are infinitely many rational numbers and any section of the line, no matter how short,
contains infinitely many rational numbers.
(2) There are infinitely many irrational numbers and any section of the line, no matter how
short, contains infinitely many irrational numbers.
To prove the first claim: there are obviously infinitely many rationals (all integers are rational and
there are infinitely many of them). Consider any segment of the line. Since it has positive length,
that length must be greater than 1/2N for some value of N (just make N bigger and bigger until
that is the case). So, of the rational numbers k/2N +2 , at least two of them must land in our chosen
segment. So there are rational numbers in the segment. But it is clear that there are infinitely
many rational numbers between any two different rational numbers (prove that yourself—don’t
be ‘theoretical’, just write down an infinite number of examples!).
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Now that you have seen that, I will leave the proof of the second
claim to you. I will just make the
√
comment that if r is a rational number (and not zero) then r 2 is an irrational number (prove that
as well).
So both rational and irrational numbers are ‘all over the place’ on the line. There are infinitely
many rationals between any two irrationals and there are infinitely many irrationals between any
two rationals.
However, despite that symmetry, I am going to argue that irrational numbers are actually far more
common than rational numbers. In fact rational numbers are, in some ways, as scarce as integers!
There is a ‘proper’ proof of this claim that is based on the concept of countability. I will offer you a
less logical argument that has the virtue of being very simple.
The argument is this. Go back to what I said about the decimal expansions of numbers. The
expansion of any rational number is eventually repeating.
Now imagine choosing the decimal digits of a number at random (tossing a 10-sided dice for each
digit). What do you think are the chances of ending up with a number a block of whose digits
repeats itself infinitely often without a single change? I think that you will agree that the chances
are infinitely remote. So random choice is infinitely more likely to produce an irrational number
than a rational one.
That’s a very dodgy argument, but the result is sound.
Back to the Greeks: the wearing-down process
Suppose we have two quantities A and B of the same type (both areas, masses, lengths or whatever).
Suppose that B is not bigger than A. Take B from A as often as you can. When you stop there will
be either nothing left of A or there will be a residue that is smaller than B (or we could take B away
another time).
If there are still two quantities left (i.e. you didn’t exhaust one of them) then repeat the process.
Take the smaller from the bigger as often as you can.
In fact, repeat the process as often as you can. The process stops when one of the quantities is
reduced to nothing.
There are precisely two possible outcomes: the process stops or it doesn’t.
This is a very natural process and is the most basic way to think of the difference between rational
and irrational ratios.
It should be obvious that the above subtraction process is not really affected by scaling A and B to
λA, λB (you just scale everything in the calculation by the same factor). So the question of whether
or not the process stops just depends on the ratio A/B.
The process stops if and only if the ratio A/B is a rational number.
Some well-known irrational numbers
On the basis of what I said in the previous section, you would assume any number to be irrational
until convinced otherwise!
√
Having been shown the earlier argument
that proved that 2 was irrational, students are usually
√
asked to go ahead and prove that 3 is irrational in much the√same way. It is then suggested that
they should check that their ‘proof’ does not also prove that 4 is irrational, because it ain’t. The
general result is that the square root of a positive whole number is irrational unless the number is
a perfect square (4, 9, 16, 25, . . .).
You can extend the result to other powers and to fractions rather than integers.
It is true that numbers like π and e are irrational, but this is not at all easy to prove.
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