REAL VARIABLES

Excerpts

REAL VARIABLES
The use of geometrical illustrations in this way does not, of course, imply that analysis has any sort of dependence upon geometry: they are illustrations and nothing more, and are employed merely for the sake of clearness of exposition.
REAL VARIABLES
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, *e.g.* for $n = 100,000$ or $100,000,000$.
REAL VARIABLES
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.
REAL VARIABLES
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.
REAL VARIABLES
A section of the rational numbers, in which both classes exist and the lower class has no greatest member, is called a **number**, or simply a ****.
REAL VARIABLES
What is essential in mathematics is that its symbols should be capable of *some* interpretation; generally they are capable of *many*, and then, so far as mathematics is concerned, it does not matter which we adopt.
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Moreover, for a beginner, the chief difficulty in the elements of analysis is that of learning to attach precise senses to phrases containing the word ‘infinity’; and experience seems to show that he is likely to be confused by any addition to their number.
REAL VARIABLES
A system of real numbers, or of the points on a straight line corresponding to them, defined in any way whatever, is called an **** or **** of numbers or points.
REAL VARIABLES
Let us suppose that (ii) is true. Then any interval $\DPmod{(\xi - \delta, \xi + \delta)}{[\xi - \delta, \xi + \delta]}$, however small its length, contains at least one point $\xi_{1}$ which belongs to $S$ and does not coincide with $\xi$; and this whether $\xi$ itself be a member of $S$ or not. In this case we shall say that $\xi$ is a **of accumulation** of $S$.
REAL VARIABLES
Suppose, for example, that $S$ consists of the points corresponding to all the positive integers. If $\xi$ is itself a positive integer, we can take $\delta$ to be any number less than $1$, and (i) will be true; or, if $\xi$ is halfway between two positive integers, we can take $\delta$ to be any number less than $\frac{1}{2}$. On the other hand, if $S$ consists of all the rational points, then, whatever the value of $\xi$, (ii) is true; for any interval whatever contains an infinity of rational points.
REAL VARIABLES
This point may of course coincide with $\alpha$ or $\beta$, as for instance when $\alpha = 0$, $\beta = 1$, and $S$ consists of the points $1$, $\frac{1}{2}$, $\frac{1}{3}, \dots$. In this case $0$ is the sole point of accumulation.
REAL VARIABLES
The general theory of sets of points is of the utmost interest and importance in the higher branches of analysis; but it is for the most part too difficult to be included in a book such as this.
REAL VARIABLES
It is clear that we may repeat this argument until we have replaced each of $a_{1}$, $a_{2}$, …, $a_{n}$ by $G$; at most $n$ repetitions will be necessary. As the final value of the arithmetic mean is $G$, the initial value cannot have been less.
REAL VARIABLES
When $p = q = 1$ in (1), or $p = 2$ in (4), the inequalities are merely different forms of the inequality $a_{1}^{2} + a_{2}^{2} \geq 2a_{1} a_{2}$, which expresses the fact that the arithmetic mean of two positive numbers is not less than their geometric mean.
REAL VARIABLES
Such irrational numbers are called *algebraical* numbers: all other irrational numbers, such as $\pi$ ([§]15), are called *transcendental* numbers.
REAL VARIABLES
Mr Bertrand Russell has said that ‘mathematics is the science in which we do not know what we are talking about, and do not care whether what we say about it is true’, a remark which is expressed in the form of a paradox but which in reality embodies a number of important truths.
REAL VARIABLES
A fraction $r = p/q$, where $p$ and $q$ are positive or negative integers, is called a *rational number*.
REAL VARIABLES
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.
REAL VARIABLES
But it is easy to see that *there is no rational number such that its square is $2$*.
REAL VARIABLES
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 $2$.
REAL VARIABLES
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 $1/n$.
REAL VARIABLES
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. % [examples:iii]Ex. iii%. 5).
REAL VARIABLES
then there is a number $\alpha$, which has the property that all the numbers less than it belong to $L$ and all the numbers greater than it to $R$.
REAL VARIABLES
A number of the form $±\sqrt{a}$, where $a$ is a positive rational number which is not the square of another rational number, is called a *pure quadratic surd*.
REAL VARIABLES
Two pure quadratic surds are said to be *similar* if they can be expressed as rational multiples of the same surd, and otherwise to be *dissimilar*.
REAL VARIABLES
It is important to observe that a pair of properties which suffice to define a section of the rational numbers may not suffice to define one of the real numbers. This is so, for example, with the pair ‘$x < \sqrt{2}$’ and ‘$x > \sqrt{2}$’ or (if we confine ourselves to positive numbers) with ‘$x^{2} < 2$’ and ‘$x^{2} > 2$’.
REAL VARIABLES
In the theory of decimals, for instance, we may denote by $x$ any figure in the expression of any number as a decimal. Then $x$ is a variable, but a variable which has only ten different values, viz. $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$.
REAL VARIABLES
He will find interesting examples in ordinary life: policeman $x$, the driver of cab $x$, the year $x$, the $x$th day of the week.
REAL VARIABLES
In this case we shall say that $\xi$ is a **of accumulation** of $S$.
REAL VARIABLES
On the other hand, if $S$ consists of all the rational points, then, whatever the value of $\xi$, (ii) is true; for any interval whatever contains an infinity of rational points.
REAL VARIABLES
If a set $S$ contains infinitely many points, and is entirely situated in an interval $\DPmod{(\alpha, \beta)}{[\alpha, \beta]}$, then at least one point of the interval is a point of accumulation of $S$.
REAL VARIABLES
As a corollary, if $a + b\sqrt[3]{2} + c\sqrt[3]{4} = d + e\sqrt[3]{2} + f\sqrt[3]{4}$, then $a = d$, $b = e$, $c = f$.
REAL VARIABLES
We have to show that these ideas can be applied to the new numbers, and that, when this extension of them is made, all the ordinary laws of algebra retain their validity, so that we can operate with real numbers in general in exactly the same way as with the rational numbers of [§]1.
REAL VARIABLES
Of the two numbers $\alpha$ and $-\alpha$ one is always positive (unless $\alpha = 0$). The one which is positive we denote by $|\alpha|$ and call the *modulus* of $\alpha$.
REAL VARIABLES
The $x$ which occurs in propositions such as these is called *the continuous real variable*: and the individual numbers are called the *values* of the variable.
REAL VARIABLES
The reader should think of other examples of variables with different fields of variation. He will find interesting examples in ordinary life: policeman $x$, the driver of cab $x$, the year $x$, the $x$th day of the week. The values of these variables are naturally not numbers.
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We add a few further examples to show how very special these particular classes of numbers are, and how, to put it roughly, they comprise only a minute fraction of the infinite variety of numbers which constitute the continuum.
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It can in fact be proved (though the proof is difficult) that it is *generally* impossible to find such an expression for the root of an equation of higher degree than $4$.
REAL VARIABLES
And this number $\pi$ is no isolated or exceptional case. Any number of other examples can be constructed. In fact it is only special classes of irrational numbers which are roots of equations of this kind, just as it is only a still smaller class which can be expressed by means of surds.
REAL VARIABLES
This conclusion is of very great importance; for it shows that the consideration of sections of all the real numbers does not lead to any further generalisation of our idea of number.

Equations

REAL VARIABLES
(-p)/(-q) = p/q
A fraction with both numerator and denominator negated equals the original fraction.
REAL VARIABLES
A_{0}A_{r}/A_{0}A_{1} = r
The point A_r for a positive rational r is chosen so that the ratio of the segment A_0A_r to A_0A_1 equals r.
REAL VARIABLES
AB = -BA
Length is signed, so reversing the direction of a segment changes the sign of its length.
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A_{0}A_{-s} = -A_{-s}A_{0}
The point representing the negative rational -s is placed so that its signed distance from the origin is minus the distance of A_s.
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A_{0}A_{r} = r · A_{0}A_{1}
The signed length from the origin to the point for r equals r times the unit length A_0A_1.
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r = p/q
A rational number r is defined as the fraction p/q, with p and q integers and q positive.
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p/(-q) = (-p)/q
Moving a minus sign from the denominator to the numerator leaves a fraction unchanged.
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A_{0}A_{r} = r
With the unit segment A_0A_1 taken as 1, the length from the origin to the point for r is the number r.
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k · BC > 1
For any segment BC there is a positive integer k with k times BC greater than the unit length (the Axiom of Archimedes, assumed).
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x^{2} = 1
The equation whose two rational roots are 1 and -1.
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x^{2} = 2
The equation with no rational root, whose solution is the irrational number sqrt(2).
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a + b = b + a
Addition of lengths (numbers) is commutative.
REAL VARIABLES
a + (b + c) = (a + b) + c
Addition of lengths (numbers) is associative.
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ab = ba
Multiplication of lengths (numbers) is commutative.
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a(bc) = (ab)c
Multiplication of lengths (numbers) is associative.
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a(b + c) = ab + ac
Multiplication distributes over addition for lengths (numbers).
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A_{0}P = x
The required point P on the line is at distance x from the origin, where x is the length whose square is 2.
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x = \sqrt{2}
The number x whose square is 2 is denoted by the symbol sqrt(2); it is not rational.
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x^{q} = n
The q-th root of n is the number x whose q-th power is n, written n^{1/q} or the radical form.
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n^{p/q} = (n^{1/q})^{p}
A fractional power is defined as the p-th power of the q-th root of n.
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n^{p/q} n^{-p/q} = 1
A number raised to a rational power times the same number raised to the negative of that power equals 1.
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n^{r} × n^{s} = n^{r+s}
Multiplying powers of the same base adds their exponents, extended to rational exponents.
REAL VARIABLES
(n^{r})^{s} = n^{rs}
Raising a power to a further power multiplies the exponents, extended to rational exponents.
REAL VARIABLES
x^{n} + p_{1}x^{n-1} + p_{2}x^{n-2} + \dots + p_{n} = 0
An algebraic equation with integral coefficients cannot have a rational but non-integral root.
REAL VARIABLES
y^{2} - x^{2} = (y - x)(y + x)
The difference of two squares factorises as the product of the difference and the sum of the numbers.
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x^{2} = N
For an integer N that is not a perfect square, the same argument shows that the root x of this equation is irrational.
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-(-\alpha) = \alpha
The negative of the negative of a real number is the number itself.
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\gamma = \alpha + \beta
The sum of two real numbers is defined as the real number given by the section of all sums of their lower and upper classes.
REAL VARIABLES
\alpha - \beta = \alpha + (-\beta)
Subtraction of real numbers is defined as adding the negative of the second number.
REAL VARIABLES
(-\alpha)\beta = -\alpha\beta
For positive alpha and beta, a negative times a positive gives the negative of the product.
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\alpha(-\beta) = -\alpha\beta
For positive alpha and beta, a positive times a negative gives the negative of the product.
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(-\alpha)(-\beta) = \alpha\beta
For positive alpha and beta, the product of two negatives is the product of the positive numbers.
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1/(-\alpha) = -(1/\alpha)
The reciprocal of a negative number is the negative of the reciprocal of its positive counterpart.
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\alpha/\beta = \alpha × (1/\beta)
Division of real numbers is defined as multiplication by the reciprocal of the divisor.
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(\sqrt{2})^{2} = \sqrt{2}\sqrt{2} = 2.
The square of the number sqrt(2), defined as a section of the rational numbers, equals 2.
REAL VARIABLES
x^{2} - 2ax + a^{2} - b = 0
The quadratic equation whose roots are a + sqrt(b) and a - sqrt(b), for rational a and b.
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ax^{2} + 2bx + c = 0
The general quadratic equation with rational coefficients, whose real roots are {-b ± sqrt(b^2 - ac)}/a when b^2 - ac > 0.
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\sqrt{8} = 2\sqrt{2}
The surd sqrt(8) is a rational multiple of sqrt(2), so the two surds are similar.
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A + \sqrt{B} = C + \sqrt{D}
If A, B, C, D are rational and this holds, then either A = C and B = D, or B and D are both squares of rational numbers.
REAL VARIABLES
A - \sqrt{B} = C - \sqrt{D}
Corollary: if A + sqrt(B) = C + sqrt(D), then A - sqrt(B) = C - sqrt(D), unless sqrt(B) and sqrt(D) are both rational.
REAL VARIABLES
z = \sqrtp[3]{4 + \sqrt{15}} + \sqrtp[3]{4 - \sqrt{15}}
Defines z as the sum of the real cube roots of 4 + sqrt(15) and 4 - sqrt(15); z is shown to satisfy z^3 = 3z + 8.
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z^{3} = 3z + 8
The unique real number z satisfying this cubic equation exists, is positive, and is not rational.
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x^{5} = x + 16\DPtypo{.}{,}
This quintic has a unique positive real root, which cannot in general be expressed by surds.
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\pi^{5} = \pi + n
pi is not the root of any algebraic equation with integer coefficients, such as this one, where n is an integer.
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\beta \leq x \leq \gamma
The closed interval of real numbers x lying from beta to gamma inclusive.
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n \tsum{a^{p+q}} \geq \tsum a^{p} \tsum a^{q}
Summing the two-number inequality over all pairs of n positive numbers gives n times the sum of their (p+q)th powers is at least the product of the sums of their pth and qth powers (Miscellaneous Example 8, equation (5)).
REAL VARIABLES
a_{1}^{p+q} + a_{2}^{p+q} \geq a_{1}^{p} a_{2}^{q} + a_{1}^{q} a_{2}^{p}
For positive numbers a_1, a_2 and positive integers p, q, the sum of the (p+q)th powers is at least the sum of the mixed products a_1^p a_2^q + a_1^q a_2^p (derived in the text of Miscellaneous Example 7, equation (1)).
REAL VARIABLES
\frac{a_{1}^{p+q} + a_{2}^{p+q}}{2} \geq \left(\frac{a_{1}^{p} + a_{2}^{p}}{2}\right) \left(\frac{a_{1}^{q} + a_{2}^{q}}{2}\right)
The mean of the (p+q)th powers of two positive numbers is at least the product of the means of their pth and qth powers (Miscellaneous Example 7, equation (2)).
REAL VARIABLES
\frac{a_{1}^{p} + a_{2}^{p}}{2} \geq \left(\frac{a_{1} + a_{2}}{2}\right)^{p}
The arithmetic mean of the pth powers of two positive numbers is at least the pth power of their arithmetic mean (Miscellaneous Example 7, equation (4)).
REAL VARIABLES
\left(\tsum a^{p}\right)/n \geq \left\{\left(\tsum a\right)/n\right\}^{p}
For n positive numbers, the arithmetic mean of their pth powers is at least the pth power of their arithmetic mean (Miscellaneous Example 8, equation (7)).
REAL VARIABLES
a_{r}' + a_{s}' - a_{r} - a_{s} = (a_{r} - G)(a_{s} - G)/G
Replacing a_r and a_s by G and a_r a_s/G changes their sum by the product (a_r - G)(a_s - G)/G, used in the proof that the arithmetic mean is not less than the geometric mean (Miscellaneous Example 9).
REAL VARIABLES
\left(\tsum a_{r} b_{r}\right)^{2} = \tsum a_{r}^{2} \tsum a_{s}^{2} - \tsum (a_{r} b_{s} - a_{s} b_{r})^{2}
The square of the sum of products a_r b_r equals the product of the sums of squares minus a sum of squared cross-differences; this identity is used to prove Schwarz's inequality (Miscellaneous Example 10).
REAL VARIABLES
\left(\tsum a_{r} b_{r}\right)^{2} \leq \tsum a_{r}^{2} \tsum b_{r}^{2}
The square of the sum of products of two sets of n numbers is at most the product of the sums of their squares; the inequality is usually known as Schwarz's, though due originally to Cauchy (Miscellaneous Example 10).
REAL VARIABLES
a_{0}x^{n} + a_{1}x^{n-1} + \dots + a_{n} = 0
An irrational number that is a root of a polynomial equation with integer coefficients is called an algebraical number; all other irrational numbers are transcendental (Miscellaneous Example 32, the defining equation).

Problems

Exercise I

Exercise I, problem 1, p. 1
If $r$ and $s$ are rational numbers, then $r + s$, $r - s$, $rs$, and $r/s$ are rational numbers, unless in the last case $s = 0$ (when $r/s$ is of course meaningless).
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Exercise I, problem 2, p. 1
0.375em plus 0.75em minus 0.25emIf $\lambda$, $m$, and $n$ are positive rational numbers, and $m > n$, then $\lambda(m^{2} - n^{2})$, $2\lambda mn$, and $\lambda(m^{2} + n^{2})$ are positive rational numbers. Hence show how to determine any number of right-angled triangles the lengths of all of whose sides are rational.
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Exercise I, problem 3, p. 1
Any terminated decimal represents a rational number whose denominator contains no factors other than $2$ or $5$. Conversely, any such rational number can be expressed, and in one way only, as a terminated decimal.
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Exercise I, problem 4, p. 1
The positive rational numbers may be arranged in the form of a simple series as follows: 11,0pt minus 3pt21,0pt minus 3pt12,0pt minus 3pt31,0pt minus 3pt22,0pt minus 3pt13,0pt minus 3pt41,0pt minus 3pt32,0pt minus 3pt23,0pt minus 3pt14, …. Show that $p/q$ is the $[\frac{1}{2}(p + q - 1)(p + q - 2) + q]$th term of the series.
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Exercise II

Exercise II, problem 1, p. 6
Show that no rational number can have its cube equal to $2$.
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Exercise II, problem 2, p. 6
Prove generally that a rational fraction $p/q$ in its lowest terms cannot be the cube of a rational number unless $p$ and $q$ are both perfect cubes.
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Exercise II, problem 3, p. 6
A more general proposition, which is due to Gauss and includes those which precede as particular cases, is the following: *an algebraical equation x^n + p_1x^n-1 + p_2x^n-2 + …+ p_n = 0, with integral coefficients, cannot have a rational but non-integral root*. [For suppose that the equation has a root $a/b$, where $a$ and $b$ are integers [pg]7 without a common factor, and $b$ is positive. Writing $a/b$ for $x$, and multiplying by $b^{n-1}$, we obtain -a^nb = p_1a^n-1 + p_2a^n-2b + …+ p_nb^n-1, a fraction in its lowest terms equal to an integer, which is absurd. Thus $b = 1$, and the root is $a$. It is evident that $a$ must be a divisor of $p_{n}$.]
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Exercise II, problem 4, p. 6
Show that if $p_{n} = 1$ and neither of 1 + p_1 + p_2 + p_3 + …,0pt minus 3pt1 - p_1 + p_2 - p_3 + … is zero, then the equation cannot have a rational root.
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Exercise II, problem 5, p. 6
Find the rational roots (if any) of x^4 - 4x^3 - 8x^2 + 13x + 10 = 0. [The roots can only be integral, and so $±1$, $±2$, $±5$, $±10$ are the only possibilities: whether these are roots can be determined by trial. It is clear that we can in this way determine the rational roots of any such equation.]
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Exercise III

Exercise III, problem 1, p. 11
Find the difference between $2$ and the squares of the decimals given in [§]4 as approximations to $\sqrt{2}$.
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Exercise III, problem 2, p. 11
Find the differences between $2$ and the squares of 11,0pt minus 3pt32,0pt minus 3pt75,0pt minus 3pt1712,0pt minus 3pt4129,0pt minus 3pt9970.
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Exercise III, problem 3, p. 11
Show that if $m/n$ is a good approximation to $\sqrt{2}$, then $(m + 2n)/(m + n)$ is a better one, and that the errors in the two cases are in opposite directions. Apply this result to continue the series of approximations in the last example.
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Exercise III, problem 4, p. 11
If $x$ and $y$ are approximations to $\sqrt{2}$, by defect and by excess respectively, and $2 - x^{2} < \delta$, $y^{2} - 2 < \delta$, then $y - x < \delta$.
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Exercise III, problem 5, p. 11
The equation $x^{2} = 4$ is satisfied by $x = 2$. Examine how far the argument of the preceding sections applies to this equation (writing $4$ for $2$ throughout). [If we define the classes $L$, $R$ as before, they do not include *all* rational numbers. The rational number $2$ is an exception, since $2^{2}$ is neither less than or greater than $4$.]
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Exercise IV

Exercise IV, problem 1, p. 16
Prove that $0 = -0$.
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Exercise IV, problem 2, p. 16
Prove that $\beta = \alpha$, $\beta < \alpha$, or $\beta > \alpha$ according as $\alpha = \beta$, $\alpha > \beta$, or $\alpha < \beta$.
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Exercise IV, problem 3, p. 16
If $\alpha = \beta$ and $\beta = \gamma$, then $\alpha = \gamma$.
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Exercise IV, problem 4, p. 16
If $\alpha \leq \beta$, $\beta < \gamma$, or $\alpha < \beta$, $\beta \leq \gamma$, then $\alpha < \gamma$.
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Exercise IV, problem 5, p. 16
Prove that $-\beta = -\alpha$, $-\beta < -\alpha$, or $-\beta > -\alpha$, according as $\alpha = \beta$, $\alpha < \beta$, or $\alpha > \beta$.
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Exercise IV, problem 6, p. 16
Prove that $\alpha > 0$ if $\alpha$ is positive, and $\alpha < 0$ if $\alpha$ is negative.
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Exercise IV, problem 7, p. 16
Prove that $\alpha \leq |\alpha|$.
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Exercise IV, problem 8, p. 16
Prove that $1 < \sqrt{2} < \sqrt{3} < 2$.
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Exercise IV, problem 9, p. 16
Prove that, if $\alpha$ and $\beta$ are two different real numbers, we can always find an infinity of rational numbers lying between $\alpha$ and $\beta$.
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Exercise V

Exercise V, problem 1, p. 17
Prove that $\alpha + (-\alpha) = 0$.
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Exercise V, problem 10, p. 17
Prove that ||| - ||| |± | || + ||.
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Exercise V, problem 2, p. 17
Prove that $\alpha + 0 = 0 + \alpha = \alpha$.
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Exercise V, problem 3, p. 17
Prove that $\alpha + \beta = \beta + \alpha$. [This follows at once from the fact that the classes $(a + b)$ and $(b + a)$, or $(A + B)$ and $(B + A)$, are the same, since, *e.g.*, $a + b = b + a$ when $a$ and $b$ are rational.]
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Exercise V, problem 4, p. 17
Prove that $\alpha + (\beta + \gamma) = (\alpha + \beta) + \gamma$.
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Exercise V, problem 5, p. 17
Prove that $\alpha - \alpha = 0$.
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Exercise V, problem 6, p. 17
Prove that $\alpha - \beta = -(\beta - \alpha)$.
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Exercise V, problem 7, p. 17
From the definition of subtraction, and Exs. 4, 1, and 2 above, it follows that (- ) + = + (-) + = + (-) + = + 0 = . We might therefore define the difference $\alpha - \beta = \gamma$ by the equation $\gamma + \beta = \alpha$.
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Exercise V, problem 8, p. 17
Prove that $\alpha - (\beta - \gamma) = \alpha - \beta + \gamma$.
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Exercise V, problem 9, p. 17
Give a definition of subtraction which does not depend upon a previous definition of addition. [To define $\gamma = \alpha - \beta$, form the classes $(c)$, $(C)$ for which $c = a - B$, $C = A - b$. It is easy to show that this definition is equivalent to that which we adopted in the text.]
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Exercise VI

Exercise VI, problem 1, p. 19
$\alpha × 0 = 0 × \alpha = 0$.
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Exercise VI, problem 2, p. 19
$\alpha × 1 = 1 × \alpha = \alpha$.
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Exercise VI, problem 3, p. 19
$\alpha × (1/\alpha) = 1$.
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Exercise VI, problem 4, p. 19
$\alpha\beta = \beta\alpha$.
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Exercise VI, problem 5, p. 19
$\alpha(\beta\gamma) = (\alpha\beta)\gamma$.
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Exercise VI, problem 6, p. 19
$\alpha(\beta + \gamma) = \alpha\beta + \alpha\gamma$.
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Exercise VI, problem 7, p. 19
$(\alpha + \beta)\gamma = \alpha\gamma + \beta\gamma$.
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Exercise VI, problem 8, p. 19
$|\alpha\beta| = |\alpha|\, |\beta|$.
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Exercise VII

Exercise VII, problem 1, p. 20
Give geometrical constructions for 2,0pt minus 3pt2 + 2,0pt minus 3pt2 + 2 + 2.
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Exercise VII, problem 2, p. 20
The quadratic equation $ax^{2} + 2bx + c = 0$ has two real roots *I.e.* there are two values of $x$ for which $ax^{2} + 2bx + c = 0$. If $b^{2} - ac < 0$ there are no such values of $x$. The reader will remember that in books on elementary algebra the equation is said to have two ‘complex’ roots. The meaning to be attached to this statement will be explained in Ch.III. When $b^{2} = ac$ the equation has only one root. For the sake of uniformity it is generally said in this case to have ‘two equal’ roots, but this is a mere convention. if $b^{2} - ac > 0$. Suppose $a$, $b$, $c$ rational. Nothing is lost by taking all three to be integers, for we can multiply the equation by the least common multiple of their denominators. The reader will remember that the roots are $\{-b ± \sqrtp{b^{2} - ac}\}/a$. It is easy to construct these lengths geometrically, first constructing $\sqrtp{b^{2} - ac}$. A much more elegant, though less straightforward, construction is the following. [pg]21 Construction Draw a circle of unit radius, a diameter $PQ$, and the tangents at the ends of the diameters. %[Illustration: Fig. 5.] [0.7]5p021 Take $PP' = -2a/b$ and $QQ' = -c/2b$, having regard to sign. The figure is drawn to suit the case in which $b$ and $c$ have the same and $a$ the opposite sign. The reader should draw figures for other cases. Join $P'Q'$, cutting the circle in $M$ and $N$. Draw $PM$ and $PN$, cutting $QQ'$ in $X$ and $Y$. Then $QX$ and $QY$ are the roots of the equation with their proper signs. I have taken this construction from Klein’s *Leçons sur certaines questions de géométrie élémentaire* (French translation by J. Griess, Paris, 1896). Construction The proof is simple and we leave it as an exercise to the reader. Another, perhaps even simpler, construction is the following. Construction[]Take a line $AB$ of unit length. Draw $BC = -2b/a$ perpendicular to $AB$, and $CD = c/a$ perpendicular to $BC$ and in the same direction as $BA$. On $AD$ as diameter describe a circle cutting $BC$ in $X$ and $Y$. Then $BX$ and $BY$ are the roots. Construction
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Exercise VII, problem 3, p. 20
If $ac$ is positive $PP'$ and $QQ'$ will be drawn in the same direction. Verify that $P'Q'$ will not meet the circle if $b^{2} < ac$, while if $b^{2} = ac$ it will be a tangent. Verify also that if $b^{2} = ac$ the circle in the second construction will touch $BC$.
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Exercise VII, problem 4, p. 20
Prove that pq = p × q,0pt minus 3ptp^2q = pq.
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Exercise VIII

Exercise VIII, problem 1, p. 22
Prove *ab initio* that $\sqrt{2}$ and $\sqrt{3}$ are not similar surds.
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Exercise VIII, problem 10, p. 22
If $p$, $q$, and $p^{2} - q$ are positive, we can express $\sqrtp{p + \sqrt{q}}$ in the form $\sqrt{x} + \sqrt{y}$, where x = 12p + p^2 - q,0pt minus 3pty = 12p - p^2 - q.
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Exercise VIII, problem 11, p. 22
Determine the conditions that it may be possible to express $\sqrtp{p + \sqrt{q}}$, where $p$ and $q$ are rational, in the form $\sqrt{x} + \sqrt{y}$, where $x$ and $y$ are rational.
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Exercise VIII, problem 12, p. 22
If $a^{2} - b$ is positive, the necessary and sufficient conditions that a + b + a - b should be rational are that $a^{2} - b$ and $\frac{1}{2}\{a + \sqrtp{a^{2} - b}\}$ should both be squares of rational numbers.
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Exercise VIII, problem 2, p. 22
Prove that $\sqrt{a}$ and $\sqrtp{1/a}$, where $a$ is rational, are similar surds (unless both are rational).
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Exercise VIII, problem 3, p. 22
If $a$ and $b$ are rational, then $\sqrt{a} + \sqrt{b}$ cannot be rational unless $\sqrt{a}$ and $\sqrt{b}$ are rational. The same is true of $\sqrt{a}- \sqrt{b}$, unless $a = b$.
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Exercise VIII, problem 4, p. 22
If A + B = C + D, then either (*a*) $A = C$ and $B = D$, or (*b*) $A = D$ and $B = C$, or (*c*) $\sqrt{A}$, $\sqrt{B}$, $\sqrt{C}$, $\sqrt{D}$ are all rational or all similar surds. [Square the given equation and apply the theorem above.]
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Exercise VIII, problem 5, p. 22
Neither $(a + \sqrt{b})^{3}$ nor $(a - \sqrt{b})^{3}$ can be rational unless $\sqrt{b}$ is rational.
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Exercise VIII, problem 6, p. 22
Prove that if $x = p + \sqrt{q}$, where $p$ and $q$ are rational, then $x^{m}$, where $m$ is any integer, can be expressed in the form $P + Q \sqrt{q}$, where $P$ and $Q$ are rational. For example, (p + q)^2 = p^2 + q + 2pq,0pt minus 3pt(p + q)^3 = p^3 + 3pq + (3p^2 + q)q. Deduce that any polynomial in $x$ with rational coefficients (*i.e.* any expression of the form a_0x^n + a_1x^n-1 + …+ a_n, where $a_{0}$, … $a_{n}$ are rational numbers) can be expressed in the form $P + Q\sqrt{q}$.
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Exercise VIII, problem 7, p. 22
If $a + \sqrt{b}$, where $b$ is not a perfect square, is the root of an algebraical equation with rational coefficients, then $a - \sqrt{b}$ is another root of the same equation.
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Exercise VIII, problem 8, p. 22
Express $1/(p + \sqrt{q})$ in the form prescribed in Ex. 6. [Multiply numerator and denominator by $p - \sqrt{q}$.]
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Exercise VIII, problem 9, p. 22
Deduce from Exs. 6 and 8 that any expression of the form $G(x)/H(x)$, where $G(x)$ and $H(x)$ are polynomials in $x$ with rational coefficients, can be expressed in the form $P + Q\sqrt{q}$, where $P$ and $Q$ are rational.
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Exercise IX

Exercise IX, problem 1, p. 30
If $S$ consists of the points corresponding to the positive integers, or all the integers, there are no points of accumulation.
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Exercise IX, problem 2, p. 30
If $S$ consists of all the rational points, every point of the line is a point of accumulation.
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Exercise IX, problem 3, p. 30
If $S$ consists of the points $1$, $\frac{1}{2}$, $\frac{1}{3}, \dots$, there is one point of accumulation, viz. the origin.
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Exercise IX, problem 4, p. 30
If $S$ consists of all the positive rational points, the points of accumulation are the origin and all positive points of the line.
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Exercise Misc-I

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