3-7. Irrational Numbers

3. Irrational numbers.

If the reader will mark off on the line all the points corresponding to the rational numbers whose denominators are $1$, $2$, $3,\dots$ in succession, he will readily convince himself that he can cover the line with rational points as closely as he likes. We can state this more precisely as follows: if we take any segment $BC$ on $\mathrm{\Lambda }$, we can find as many rational points as we please on $BC$.

Suppose, for example, that $BC$ falls within the segment $A_1{A}_2$. It is evident that if we choose a positive integer $k$ so that $$\begin{array}{}\text{(1)}& k\cdot BC>1,\end{array}$$ and divide $A_1{A}_2$ into $k$ equal parts, then at least one of the points of division (say $P$) must fall inside $BC$, without coinciding with either $B$ or $C$. For if this were not so, $BC$ would be entirely included in one of the $k$ parts into which $A_1{A}_2$ has been divided, which contradicts the supposition $\text{(1)}$ . But $P$ obviously corresponds to a rational number whose denominator is $k$. Thus at least one rational point $P$ lies between $B$ and $C$. But then we can find another such point $Q$ between $B$ and $P$, another between $B$ and $Q$, and so on indefinitely; , as we asserted above, we can find as many as we please. We may express this by saying that $BC$ includes infinitely many rational points.

The meaning of such phrases as ‘infinitely many’ or ‘an infinity of’, in such sentences as ‘$BC$ includes infinitely many rational points’ or ‘there are an infinity of rational points on $BC$’ or ‘there are an infinity of positive integers’, will be considered more closely in Ch. IV. The assertion ‘there are an infinity of positive integers’ means ‘given any positive integer $n$, however large, we can find more than $n$ positive integers’. This is plainly true whatever $n$ may be, for $n=100,000$ or $100,000,000$. The assertion means exactly the same as ‘we can find as many positive integers as we please’.

The reader will easily convince himself of the truth of the following assertion, which is substantially equivalent to what was proved in the second paragraph of this section: given any rational number $r$, and any positive integer $n$, we can find another rational number lying on either side of $r$ and differing from $r$ by less than $1/n$. It is merely to express this differently to say that we can find a rational number lying on either side of $r$ and differing from $r$ by as little as we please. Again, given any two rational numbers $r$ and $s$, we can interpolate between them a chain of rational numbers in which any two consecutive terms differ by as little as we please, that is to say by less than $1/n$, where $n$ is any positive integer assigned beforehand.

From these considerations the reader might be tempted to infer that an adequate view of the nature of the line could be obtained by imagining it to be formed simply by the rational points which lie on it. And it is certainly the case that if we imagine the line to be made up solely of the rational points, and all other points (if there are any such) to be eliminated, the figure which remained would possess most of the properties which common sense attributes to the straight line, and would, to put the matter roughly, look and behave very much like a line.

A little further consideration, however, shows that this view would involve us in serious difficulties.

Let us look at the matter for a moment with the eye of common sense, and consider some of the properties which we may reasonably expect a straight line to possess if it is to satisfy the idea which we have formed of it in elementary geometry.

The straight line must be composed of points, and any segment of it by all the points which lie between its end points. With any such segment must be associated a certain entity called its length, which must be a quantity capable of numerical measurement in terms of any standard or unit length, and these lengths must be capable of combination with one another, according to the ordinary rules of algebra, by means of addition or multiplication. Again, it must be possible to construct a line whose length is the sum or product of any two given lengths. If the length $PQ$, along a given line, is $a$, and the length $QR$, along the same straight line, is $b$, the length $PR$ must be $a+b$. Moreover, if the lengths $OP$, $OQ$, along one straight line, are $1$ and $a$, and the length $OR$ along another straight line is $b$, and if we determine the length $OS$ by Euclid’s construction (Euc. vi. 12) for a fourth proportional to the lines $OP$, $OQ$, $OR$, this length must be $ab$, the algebraical fourth proportional to $1$, $a$, $b$. And it is hardly necessary to remark that the sums and products thus defined must obey the ordinary ‘laws of algebra’; viz. $$\begin{array}{c}a+b=b+a,a+(b+c)=(a+b)+c,\\ ab=ba,a(bc)=(ab)c,a(b+c)=ab+ac.\end{array}$$ The lengths of our lines must also obey a number of obvious laws concerning inequalities as well as equalities: thus if $A$, $B$, $C$ are three points lying along $\mathrm{\Lambda }$ from left to right, we must have $AB<AC$, and so on. Moreover it must be possible, on our fundamental line $\mathrm{\Lambda }$, to find a point $P$ such that $A_{0}P$ is equal to any segment whatever taken along $\mathrm{\Lambda }$ or along any other straight line. All these properties of a line, and more, are involved in the presuppositions of our elementary geometry.

Now it is very easy to see that the idea of a straight line as composed of a series of points, each corresponding to a rational number, cannot possibly satisfy all these requirements. There are various elementary geometrical constructions, for example, which purport to construct a length $x$ such that $x^2=2$. For instance, we may construct an isosceles right-angled triangle $ABC$ such that $AB=AC=1$. Then if $BC=x$, $x^2=2$. Or we may determine the length $x$ by means of Euclid’s construction (Euc. vi. 13) for a mean proportional to $1$ and $2$, as indicated in the figure.

Our requirements therefore involve the existence of a length measured by a number $x$, and a point $P$ on $\mathrm{\Lambda }$ such that $$A_{0}P=x,x^2=2.$$ But it is easy to see that there is no rational number such that its square is $2$. In fact we may go further and say that there is no rational number whose square is $m/n$, where $m/n$ is any positive fraction in its lowest terms, unless $m$ and $n$ are both perfect squares.

For suppose, if possible, that $$p^2/q^2=m/n,$$ $p$ having no factor in common with $q$, and $m$ no factor in common with $n$. Then $n{p}^2=m{q}^2$. Every factor of $q^2$ must divide $n{p}^2$, and as $p$ and $q$ have no common factor, every factor of $q^2$ must divide $n$. Hence $n=\lambda q^2$, where $\lambda$ is an integer. But this involves $m=\lambda p^2$: and as $m$ and $n$ have no common factor, $\lambda$ must be unity. Thus $m=p^2$, $n=q^2$, as was to be proved. In particular it follows, by taking $n=1$, that an integer cannot be the square of a rational number, unless that rational number is itself integral.

It appears then that our requirements involve the existence of a number $x$ and a point $P$, not one of the rational points already constructed, such that $A_{0}P=x$, $x^2=2$; and (as the reader will remember from elementary algebra) we write $x=\sqrt{2}$.

The following alternative proof that no rational number can have its square equal to $2$ is interesting.

Suppose, if possible, that $p/q$ is a positive fraction, in its lowest terms, such that $(p/q{)}^2=2$ or $p^2=2q^2$. It is easy to see that this involves $(2q–p{)}^2=2(p–q{)}^2$; and so $(2q–p)/(p–q)$ is another fraction having the same property. But clearly $q<p<2q$, and so $p–q<q$. Hence there is another fraction equal to $p/q$ and having a smaller denominator, which contradicts the assumption that $p/q$ is in its lowest terms.

4. Irrational numbers (continued)

The result of our geometrical representation of the rational numbers is therefore to suggest the desirability of enlarging our conception of ‘number’ by the introduction of further numbers of a new kind.

The same conclusion might have been reached without the use of geometrical language. One of the central problems of algebra is that of the solution of equations, such as $$x^2=1,x^2=2.$$ The first equation has the two rational roots $1$ and $-1$. But, if our conception of number is to be limited to the rational numbers, we can only say that the second equation has no roots; and the same is the case with such equations as $x^3=2$, $x^4=7$. These facts are plainly sufficient to make some generalisation of our idea of number desirable, if it should prove to be possible.

Let us consider more closely the equation $x^2=2$.

We have already seen that there is no rational number $x$ which satisfies this equation. The square of any rational number is either less than or greater than $2$. We can therefore divide the rational numbers into two classes, one containing the numbers whose squares are less than $2$, and the other those whose squares are greater than $2$. We shall confine our attention to the positive rational numbers, and we shall call these two classes the class $L$, or the lower class, or the left-hand class, and the class $R$, or the upper class, or the right-hand class. It is obvious that every member of $R$ is greater than all the members of $L$. Moreover it is easy to convince ourselves that we can find a member of the class $L$ whose square, though less than $2$, differs from $2$ by as little as we please, and a member of $R$ whose square, though greater than $2$, also differs from $2$ by as little as we please. In fact, if we carry out the ordinary arithmetical process for the extraction of the square root of $2$, we obtain a series of rational numbers, viz. $$1,1.4,1.41,1.414,1.4142,\dots$$ whose squares $$1,1.96,1.9881,1.999396,1.99996164,\dots$$ are all less than $2$, but approach nearer and nearer to it; and by taking a sufficient number of the figures given by the process we can obtain as close an approximation as we want. And if we increase the last figure, in each of the approximations given above, by unity, we obtain a series of rational numbers $$2,1.5,1.42,1.415,1.4143,\dots$$ whose squares $$4,2.25,2.0164,2.002225,2.00024449,\dots$$ are all greater than $2$ but approximate to $2$ as closely as we please.

The reasoning which precedes, although it will probably convince the reader, is hardly of the precise character required by modern mathematics. We can supply a formal proof as follows. In the first place, we can find a member of $L$ and a member of $R$, differing by as little as we please. For we saw in §3 that, given any two rational numbers $a$ and $b$, we can construct a chain of rational numbers, of which $a$ and $b$ are the first and last, and in which any two consecutive numbers differ by as little as we please. Let us then take a member $x$ of $L$ and a member $y$ of $R$, and interpolate between them a chain of rational numbers of which $x$ is the first and $y$ the last, and in which any two consecutive numbers differ by less than $\delta$, $\delta$ being any positive rational number as small as we please, such as $.01$ or $.0001$ or $.000001$. In this chain there must be a last which belongs to $L$ and a first which belongs to $R$, and these two numbers differ by less than $\delta$.

We can now prove that an $x$ can be found in $L$ and a $y$ in $R$ such that $2–x^2$ and $y^2–2$ are as small as we please, say less than $\delta$. Substituting $\frac{1}{4}\delta$ for $\delta$ in the argument which precedes, we see that we can choose $x$ and $y$ so that $y–x<\frac{1}{4}\delta$; and we may plainly suppose that both $x$ and $y$ are less than $2$. Thus $$y+x<4,y^2–x^2=(y–x)(y+x)<4(y–x)<\delta ;$$ and since $x^2<2$ and $y^2>2$ it follows a fortiori that $2–x^2$ and $y^2–2$ are each less than $\delta$.

It follows also that there can be no largest member of $L$ or smallest member of $R$. For if $x$ is any member of $L$, then $x^2<2$. Suppose that $x^2=2–\delta$. Then we can find a member $x_1$ of $L$ such that $x_1^2$ differs from $2$ by less than $\delta$, and so $x_1^2>x^2$ or $x_1>x$. Thus there are larger members of $L$ than $x$; and as $x$ is any member of $L$, it follows that no member of $L$ can be larger than all the rest. Hence $L$ has no largest member, and similarly $R$ has no smallest.

5. Irrational numbers (continued)

We have thus divided the positive rational numbers into two classes, $L$ and $R$, such that (i) every member of $R$ is greater than every member of $L$, (ii) we can find a member of $L$ and a member of $R$ whose difference is as small as we please, (iii) $L$ has no greatest and $R$ no least member. Our common-sense notion of the attributes of a straight line, the requirements of our elementary geometry and our elementary algebra, alike demand the existence of a number $x$ greater than all the members of $L$ and less than all the members of $R$, and of a corresponding point $P$ on $\mathrm{\Lambda }$ such that $P$ divides the points which correspond to members of $L$ from those which correspond to members of $R$.

Let us suppose for a moment that there is such a number $x$, and that it may be operated upon in accordance with the laws of algebra, so that, for example, $x^2$ has a definite meaning. Then $x^2$ cannot be either less than or greater than $2$. For suppose, for example, that $x^2$ is less than $2$. Then it follows from what precedes that we can find a positive rational number $\xi$ such that ${\xi }^2$ lies between $x^2$ and $2$. That is to say, we can find a member of $L$ greater than $x$; and this contradicts the supposition that $x$ divides the members of $L$ from those of $R$. Thus $x^2$ cannot be less than $2$, and similarly it cannot be greater than $2$. We are therefore driven to the conclusion that $x^2=2$, and that $x$ is the number which in algebra we denote by $\sqrt{2}$. And of course this number $\sqrt{2}$ is not rational, for no rational number has its square equal to $2$. It is the simplest example of what is called an number.

But the preceding argument may be applied to equations other than $x^2=2$, almost word for word; for example to $x^2=N$, where $N$ is any integer which is not a perfect square, or to $$x^3=3,x^3=7,x^4=23,$$ or, as we shall see later on, to $x^3=3x+8$. We are thus led to believe in the existence of irrational numbers $x$ and points $P$ on $\mathrm{\Lambda }$ such that $x$ satisfies equations such as these, even when these lengths cannot (as $\sqrt{2}$ can) be constructed by means of elementary geometrical methods.

The reader will no doubt remember that in treatises on elementary algebra the root of such an equation as $x^q=n$ is denoted by $\sqrt[q]{n}$ or $n^{1/q}$, and that a meaning is attached to such symbols as $$n^{p/q},n^{-p/q}$$ by means of the equations $$n^{p/q}=(n^{1/q}{)}^p,n^{p/q}{n}^{-p/q}=1.$$ And he will remember how, in virtue of these definitions, the ‘laws of indices’ such as $$n^r\times n^s=n^{r+s},(n^r{)}^s=n^{rs}$$ are extended so as to cover the case in which $r$ and $s$ are any rational numbers whatever.

The reader may now follow one or other of two alternative courses. He may, if he pleases, be content to assume that ‘irrational numbers’ such as $\sqrt{2}$, $\sqrt[3]{3},\dots$ exist and are amenable to the algebraical laws with which he is familiar.¹ If he does this he will be able to avoid the more abstract discussions of the next few sections, and may pass on at once to §§13 et seq.

If, on the other hand, he is not disposed to adopt so naive an attitude, he will be well advised to pay careful attention to the sections which follow, in which these questions receive fuller consideration.²

6. Irrational numbers (continued)

In §4 we discussed a special mode of division of the positive rational numbers $x$ into two classes, such that $x^2<2$ for the members of one class and $x^2>2$ for those of the others. Such a mode of division is called a of the numbers in question. It is plain that we could equally well construct a section in which the numbers of the two classes were characterised by the inequalities $x^3<2$ and $x^3>2$, or $x^4<7$ and $x^4>7$. Let us now attempt to state the principles of the construction of such ‘sections’ of the positive rational numbers in quite general terms.

Suppose that $P$ and $Q$ stand for two properties which are mutually exclusive and one of which must be possessed by every positive rational number. Further, suppose that every such number which possesses $P$ is less than any such number which possesses $Q$. Thus $P$ might be the property ‘$x^2<2$’ and $Q$ the property ‘$x^2>2$.’ Then we call the numbers which possess $P$ the lower or left-hand class $L$ and those which possess $Q$ the upper or right-hand class $R$. In general both classes will exist; but it may happen in special cases that one is non-existent and that every number belongs to the other. This would obviously happen, for example, if $P$ (or $Q$) were the property of being rational, or of being positive. For the present, however, we shall confine ourselves to cases in which both classes do exist; and then it follows, as in §4, that we can find a member of $L$ and a member of $R$ whose difference is as small as we please.

In the particular case which we considered in §4, $L$ had no greatest member and $R$ no least. This question of the existence of greatest or least members of the classes is of the utmost importance. We observe first that it is impossible in any case that $L$ should have a greatest member and $R$ a least. For if $l$ were the greatest member of $L$, and $r$ the least of $R$, so that $l<r$, then $\frac{1}{2}(l+r)$ would be a positive rational number lying between $l$ and $r$, and so could belong neither to $L$ nor to $R$; and this contradicts our assumption that every such number belongs to one class or to the other. This being so, there are but three possibilities, which are mutually exclusive. Either (i) $L$ has a greatest member $l$, or (ii) $R$ has a least member $r$, or (iii) $L$ has no greatest member and $R$ no least.

The section of §4 gives an example of the last possibility. An example of the first is obtained by taking $P$ to be ‘$x^2\le 1$’ and $Q$ to be ‘$x^2>1$’; here $l=1$. If $P$ is ‘$x^2<1$’ and $Q$ is ‘$x^2\ge 1$’, we have an example of the second possibility, with $r=1$. It should be observed that we do not obtain a section at all by taking $P$ to be ‘$x^2<1$’ and $Q$ to be ‘$x^2>1$’; for the special number $1$ escapes classification (cf. Ex. III. 5).

7. Irrational numbers (continued)

In the first two cases we say that the section corresponds to a positive rational number $a$, which is $l$ in the one case and $r$ in the other. Conversely, it is clear that to any such number $a$ corresponds a section which we shall denote by $\alpha$.³ For we might take $P$ and $Q$ to be the properties expressed by $$x\le a,x>a$$ respectively, or by $x<a$ and $x\ge a$. In the first case $a$ would be the greatest member of $L$, and in the second case the least member of $R$. There are in fact just two sections corresponding to any positive rational number. In order to avoid ambiguity we select one of them; let us select that in which the number itself belongs to the upper class. In other words, let us agree that we will consider only sections in which the lower class $L$ has no greatest number.

There being this correspondence between the positive rational numbers and the sections defined by means of them, it would be perfectly legitimate, for mathematical purposes, to replace the numbers by the sections, and to regard the symbols which occur in our formulae as standing for the sections instead of for the numbers. Thus, for example, $\alpha >{\alpha }^{\prime }$ would mean the same as $a>a^{\prime }$, if $\alpha$ and ${\alpha }^{\prime }$ are the sections which correspond to $a$ and $a^{\prime }$.

But when we have in this way substituted sections of rational numbers for the rational numbers themselves, we are almost forced to a generalisation of our number system. For there are sections (such as that of §4) which do not correspond to any rational number. The aggregate of sections is a larger aggregate than that of the positive rational numbers; it includes sections corresponding to all these numbers, and more besides. It is this fact which we make the basis of our generalisation of the idea of number. We accordingly frame the following definitions, which will however be modified in the next section, and must therefore be regarded as temporary and provisional.

A section of the positive rational numbers, in which both classes exist and the lower class has no greatest member, is called a positive real number.

A positive real number which does not correspond to a positive rational number is called a positive irrational number.