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Excerpts

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[fig:24]Fig. 24 is usually known as Argand’s diagram.
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When $y = 0$ we say that *$z$ is real*, when $x = 0$ that *$z$ is purely imaginary*.
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Two numbers $x + yi$, $x - yi$ which differ only in the signs of their imaginary parts, we call *conjugate*.
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There are other special values of $n$ for which such constructions are possible, the most interesting being $n = 17$.
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Common sense at once suggests that we should define the sum of two displacements as the displacement which is the result of the successive application of the two given displacements.
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In other words, *addition of displacements obeys the commutative law* expressed in ordinary algebra by the equation $a + b = b + a$.
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The required definition is therefore [x, y] [x’, y’] = [xx’ - yy’, xy’ + yx’]. (6)
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We conclude that a quadratic equation with real coefficients has exactly two roots.
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The application of any of the ordinary algebraical operations to complex numbers will yield only complex numbers.
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One most important property of real numbers is that known as *the factor theorem*, which asserts that *the product of two numbers cannot be zero unless one of the two is itself zero*.
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All such theorems as these are true whether $a$, $b$, … $\alpha$, $\beta$, … are real or complex.
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we call $z$ the *complex variable*.
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It must be observed that $\theta$ or $\am z$ is a many-valued function of $x$ and $y$, having an infinity of values, which are angles differing by multiples of $2\pi$.
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Hence *De Moivre’s Theorem holds for all integral values of $n$, positive or negative*.
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The four points are said to be *harmonic* or *harmonically related* if any one of these is equal to $-1$.
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in this notation, suggested by Profs. Harkness and Morley, De Moivre’s theorem is expressed by the equation $(\Cis\theta)^{n} = \Cis n\theta$.
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These definitions do not prejudge the question as to whether there are or are not more than one (or any) roots of the equation.
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That these $n$ roots are in reality all distinct is easily seen by plotting them on Argand’s diagram.
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This is the analytical equivalent of the geometrical theorem that, if $M$ is the middle point of $PQ$, then $OP^{2} + OQ^{2} = 2OM^{2} + 2MP^{2}$.
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[The amplitudes have not necessarily their principal values.]
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Thus we define $\sqrt[n]{a}$ or $a^{1/n}$, where $n$ is a positive integer, as a number $z$ which satisfies the equation $z^{n} = a$; and $a^{m/n}$, where $m$ is an integer, as $(a^{1/n})^{m}$. These definitions do not prejudge the question as to whether there are or are not more than one (or any) roots of the equation.
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The particular root [n](/n) + i(/n) is called the *principal value* of $\sqrt[n]{a}$.
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These numbers are called the $n$th roots of unity; the principal value is unity itself.
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Euclid gives constructions for $n = 3$, $4$, $5$, $6$, $8$, $10$, $12$, and $15$. It is evident that the construction is possible for any value of $n$ which can be found from these by multiplication by any power of $2$. There are other special values of $n$ for which such constructions are possible, the most interesting being $n = 17$.
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If cricket were a mathematical science, it would be very important to distinguish between the *motion* of the batsman between the wickets, the *run* which he scores, and the *mark* which is put down in the score-book.
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To specify a displacement completely three things are needed, its *magnitude*, its *sense* forwards or backwards along the line, and what may be called its *point of application*, *i.e.* the original position $P$ of the particle.
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In the first place our definition would be futile. We should only be introducing a new method of expressing something which we can perfectly well express without it.
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For the present the reader must regard $x + yi$ as *simply another way of writing $[x, y]$*. The expression $x + yi$ is called a *complex number*.
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The reader will now easily satisfy himself that the upshot of the rules for addition and multiplication of complex numbers is this, that *we operate with complex numbers in exactly the same way as with real numbers, treating the symbol $i$ as itself a number, but replacing the product $ii = i^{2}$ by $-1$ whenever it occurs*.
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In other words, *multiplication of a complex number by $i$ turns the corresponding displacement through a right angle*.
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It cannot, however, be too strongly impressed upon the reader that an ‘imaginary number’ is no more ‘imaginary’, in any ordinary sense of the word, than a ‘real’ number; and that it is not a number at all, in the sense in which the ‘real’ numbers are numbers, but, as should be clear from the preceding discussion, *a pair of numbers $(x, y)$*, united symbolically, for purposes of technical convenience, in the form $x + yi$. Such a pair of numbers is no less ‘real’ than any ordinary number such as $\frac{1}{2}$, or than the paper on which this is printed, or than the Solar System.
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We can only attach a meaning to $3 - 7$ if we admit *negative* numbers, or to $\frac{3}{7}$ if we admit *rational fractions*.
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It should be observed that it is not always true that the principal value of $\am(zz')$ is the sum of the principal values of $\am z$ and $\am z'$. For example, if $z = z' = -1 + i$, then the principal values of the amplitudes of $z$ and $z'$ are each $\frac{3}{4}\pi$. But $zz' = -2i$, and the principal value of $\am(zz')$ is $-\frac{1}{2}\pi$ and not $\frac{3}{2}\pi$.
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A line originally lying along $OX$ will, if turned through any of these angles, come to lie along $OP$.
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It will be observed that the sum $2x$ of two conjugate numbers and their product $x^{2} + y^{2}$ are both real, that they have the same modulus $\sqrtp{x^{2} + y^{2}}$ and that their product is equal to the square of the modulus of either.
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Thus the modulus of the reciprocal of $z$ is the reciprocal of the modulus of $z$, and the amplitude of the reciprocal is the negative of the amplitude of $z$.
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The length $OU$ is the modulus of the sum of the complex numbers, whereas the sum of their moduli is the total length of the broken line $OPQR\dots U$, which is not less than $OU$.
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This theorem is sometimes stated as follows: *in an equation with real coefficients complex roots occur in conjugate pairs*. It should be compared with the result of viii. 7, which may be stated as follows: *in an equation with rational coefficients irrational roots occur in conjugate pairs*.
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Thus *the general linear transformation is equivalent to the combination of a translation, a magnification, and a rotation*.
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The general bilinear transformation is the most general type of transformation for which one and only one value of $z$ corresponds to each value of $Z$, and conversely.
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The only possible value of $r$ is $\sqrt[n]{\rho}$, the ordinary arithmetical $n$th root of $\rho$; and in order that the last two equations should be satisfied it is necessary and sufficient that $n\theta = \phi + 2k\pi$, where $k$ is an integer, or = (+ 2k)/n.
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Raising each of these expressions to the power $p$ (where $p$ is any integer positive or negative), we obtain the theorem that one of the values of $(\cos\theta + i\sin\theta)^{p/q}$ is $\cos(p\theta/q) + i\sin(p\theta/q)$, or that *if $\alpha$ is any rational number then one of the values of $(\cos\theta + i\sin\theta)^{\alpha}$ is* + i. This is a generalised form of De Moivre’s Theorem ([§]45).
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The problem of finding the accurate value of $\omega_{n}$ in a numerical form involving square roots only, as in the formula $\omega_{3} = \frac{1}{2}(-1 + i\sqrt{3})$, is the algebraical equivalent of the geometrical problem of inscribing a regular polygon of $n$ sides in a circle of unit radius by Euclidean methods, *i.e.* by ruler and compasses.
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Prove that $|a + b|^{2} + |a - b|^{2} = 2\{|a|^{2} + |b|^{2}\}$. [This is the analytical equivalent of the geometrical theorem that, if $M$ is the middle point of $PQ$, then $OP^{2} + OQ^{2} = 2OM^{2} + 2MP^{2}$.]
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The aggregate of pairs of real or complex values of $x$ and $y$ which satisfy the equation is called an *imaginary straight line*; the pairs of values are called *imaginary points*, and are said *to lie on the line*.

Equations

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x + yi = x' + y'i
Two complex numbers are equal (equivalent) exactly when their real parts and their coefficients of i agree.
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(x + yi) + (x' + y'i) = (x + x') + (y + y')i
The sum of two complex numbers is the complex number whose real and imaginary coefficients are the sums of the corresponding parts.
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(x + yi) (x' + y'i) = xx' - yy' + (xy' + yx')i
The product of two complex numbers is the complex number with real part xx' - yy' and imaginary coefficient xy' + yx'.
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(x + yi) (x' + y'i) = (x' + y'i) (x + yi)
Multiplication of complex numbers obeys the commutative law.
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i^{2} = ii = (0 + 1i) (0 + 1i) = (0 · 0 - 1 · 1) + (0 · 1 + 1 · 0)i = -1
The imaginary unit i, multiplied by itself, gives -1, so i and -i both satisfy x^2 = -1.
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(x + yi)i = -y + xi
Multiplying a complex number by i turns its displacement through a positive right angle.
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(x + yi)(x - yi) = x^{2} + y^{2}
The product of a complex number and its conjugate is the real number x^2 + y^2.
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az^{2} + 2bz + c = 0
The general quadratic equation with real coefficients a, b, c, whose roots may be real or complex.
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\{z + (b/a)\}^{2} = -(ac - b^{2})/a^{2}
The quadratic az^2 + 2bz + c = 0 written in completed-square form, so that its roots are z = (-b ± i√(ac - b^2))/a when b^2 < ac.
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\alpha + \beta = -(2b/a)
The sum of the two roots of az^2 + 2bz + c = 0 equals -2b/a, and this holds for complex as well as real roots.
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\alpha\beta = (c/a)
The product of the two roots of az^2 + 2bz + c = 0 equals c/a.
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\alpha + \beta + \gamma = -(3b/a)
The sum of the three roots of az^3 + 3bz^2 + 3cz + d = 0 equals -3b/a.
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\beta\gamma + \gamma\alpha + \alpha\beta = (3c/a)
The sum of the pairwise products of the three roots of the cubic az^3 + 3bz^2 + 3cz + d = 0 equals 3c/a.
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\alpha\beta\gamma = -(d/a)
The product of the three roots of the cubic az^3 + 3bz^2 + 3cz + d = 0 equals -d/a.
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f(z) = A(z - a_{1}) (z - a_{2}) \dots (z - a_{n})
A polynomial of degree n with roots a_1,...,a_n factors as A times the product of (z - a_k), where A is its leading coefficient.
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x' \xi - y' \eta = x
The quotient (x + yi)/(x' + y'i) is the complex number xi + eta i satisfying this real part condition of the product equation.
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x' \eta + y' \xi = y
The quotient (x + yi)/(x' + y'i) is the complex number xi + eta i satisfying this imaginary part condition of the product equation.
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\xi = \frac{xx' + yy'}{x'^{2} + y'^{2}}
The real part of the quotient of two complex numbers, obtained by solving the division equations; it fails when x' + y'i = 0.
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\eta = \frac{yx' - xy'}{x'^{2} + y'^{2}}
The imaginary coefficient of the quotient of two complex numbers, obtained by solving the division equations.
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x = \rho\cos\theta
The real part of a complex number written in polar form with modulus rho and angle theta.
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y = \rho\sin\theta
The imaginary coefficient of a complex number written in polar form with modulus rho and angle theta.
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[x, y] = [x, 0] + [0, y]
A displacement [x, y] is the sum of its components [x, 0] along OX and [0, y] along OY.
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[x, y] + [x', y'] = [x + x', y + y']
The sum of two displacements has coordinates equal to the sums of their coordinates.
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\alpha[x, y] = [\alpha x, \alpha y]
Multiplying a displacement by a real number multiplies each of its coordinates by that number.
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[x, y] - [x', y'] = [x, y] + (-[x', y'])
Subtraction of displacements is defined as adding the reversed displacement.
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[0, 0] = 0
The zero displacement, which leaves the particle where it was, is written as the number 0.
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[x, y] [x', y'] = [xx' - yy', xy' + yx']
The product of two displacements is the displacement obtained by similar-triangle construction, with coordinates xx' - yy' and xy' + yx'.
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[x, 0] [x', y'] = [xx', xy']
Multiplying a displacement along OX by a displacement agrees with ordinary multiplication by the real number x.
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1/z = (\cos\theta - i\sin\theta)/r
The reciprocal of z has modulus 1/r and amplitude minus theta, where z = r(cos theta + i sin theta).
This equation is in FUNCTIONS OF REAL VARIABLES (FUNCTIONS OF REAL VARIABLES)
This equation is in FUNCTIONS OF REAL VARIABLES (FUNCTIONS OF REAL VARIABLES)
This equation is in FUNCTIONS OF REAL VARIABLES (FUNCTIONS OF REAL VARIABLES)
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x = \Real(z)
The real part x of the complex number z is its real component.
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y = \Imag(z)
The imaginary part y of the complex number z is its imaginary component.
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r = |z|
The modulus r of z is the absolute value of z.
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\theta = \am z
The angle theta is the amplitude of z.
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(\cos\theta + i\sin\theta)^{n} = \cos n\theta + i\sin n\theta
The n-th power of cos theta plus i sin theta equals cos n theta plus i sin n theta, for any positive integer n.
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r(\cos\theta + i\sin\theta) × \rho(\cos\phi + i\sin\phi) = r\rho\{\cos(\theta + \phi) + i\sin(\theta + \phi)\}
The product of two complex numbers has modulus equal to the product of the moduli and amplitude equal to the sum of the amplitudes.
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\frac{(z_{1} - z_{3}) (z_{2} - z_{4})}{(z_{1} - z_{4}) (z_{2} - z_{3})} = -1
The four points z1, z2, z3, z4 are harmonic when this cross ratio equals -1.
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a_{0}z^{n} + a_{1}z^{n-1} + \dots + a_{n} = 0
A polynomial equation of degree n in the complex variable z with coefficients a_0, a_1, ..., a_n.
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z^{2} + 2(b + Bi)z + (c + Ci) = 0
The standard form of the quadratic equation with complex coefficients, obtained after dividing by a + iA.
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x^{2} - y^{2} + 2(bx - By) + c = 0
The real part of the quadratic equation with complex coefficients, after substituting z = x + yi.
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2xy + 2(by + Bx) + C = 0
The imaginary part of the quadratic equation with complex coefficients, after substituting z = x + yi.
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\xi^{2} - \eta^{2} = h
The shifted real part of the quadratic equation, with xi = x + b and eta = y + B.
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2\xi\eta = k
The shifted imaginary part of the quadratic equation.
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\xi^{2} + \eta^{2} = \sqrtp{h^{2} + k^{2}}
Squaring and adding the two shifted equations gives the sum of the squares equal to the square root of h squared plus k squared.
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c + Ci = (b + Bi)^{2}
The two roots of the quadratic are equal exactly when the constant term is the square of the middle coefficient, i.e. the left side is a perfect square.
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C^{2} - 4bBC + 4cB^{2} = 0
Condition on the coefficients for the quadratic to have a real root.
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C^{2} - 4bBC - 4b^{2}c = 0
Condition on the coefficients for the quadratic to have a purely imaginary root.
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z^{3} + 3Hz + G = 0
The cubic equation with complex coefficients in the reduced form studied in Example 15.
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\sigma^{3} + 27\lambda\mu^{2}\sigma - 27\mu^{3}\rho = 0
Condition for the cubic z^3 + 3Hz + G = 0 to have a real root when mu is not zero.
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\rho^{3} - 27\lambda\mu^{2}\rho - 27\mu^{3}\sigma = 0
Condition for the cubic z^3 + 3Hz + G = 0 to have a purely imaginary root when mu is not zero.
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y^{2} - 3x^{2} = 3H
Relation between the real part x and imaginary part y of a complex-pair root of the cubic and H.
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2x(x^{2} + y^{2}) = G
Relation between the real part x and imaginary part y of a complex-pair root of the cubic and G.
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\alpha z + \beta = 0
The general linear equation with complex coefficients, which has one solution unless alpha is zero.
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z = -(\beta/\alpha)
The unique solution of the general linear equation with complex coefficients, when alpha is not zero.
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aB - bA = 0
Consistency condition for the two real equations ax + b = 0 and Ax + B = 0 to have a common real root.
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8\alpha^{3} + 6\alpha H - G = 0
The real part alpha of the complex roots of z^3 + 3Hz + G = 0 is a root of this real-coefficient cubic.
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\left|\frac{z - b}{z - a}\right| = \lambda
The locus of P is a circle when the ratio of distances PA to PB is a constant lambda.
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z = Z + a
The translation: z is Z shifted by the complex number a, so figures are moved without change of size or orientation.
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z = \rho Z
The magnification: z is Z multiplied by the positive real rho, scaling figures by rho.
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z = (\cos\phi + i \sin\phi)Z
The rotation: z is Z multiplied by a unit complex number, turning the figure through angle phi about the origin.
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z = aZ + b
The general linear transformation, equivalent to a translation, a magnification and a rotation combined.
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z = 1/Z
The inversion transformation: modulus becomes 1/R and amplitude becomes minus Theta, followed by reflection in the axis ox.
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z = \frac{aZ + b}{cZ + d}
The general bilinear transformation, the most general one-to-one transformation between z and Z in the plane.
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Z = \frac{dz - b}{cz - a}
The inverse of the general bilinear transformation, solving for Z in terms of z.
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ac' + a'c - 2bb' = 0
Condition under which OA1 and OA2 are equally inclined to A3A4 with OA1·OA2 = OA3^2 = OA4^2.
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\cot\omega = \cot A + \cot B + \cot C
For the point P inside a triangle with equal angles omega at the vertices, cot omega equals the sum of the cotangents of the triangle's angles.
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z = \frac{1}{3}(\alpha + \beta + \gamma)
The centre of gravity of a triangle with complex vertices alpha, beta, gamma is one third of their sum.
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z = c + \rho\left(\frac{1 + ti}{1 - ti}\right)
As the real parameter t varies, z describes the circle with centre c and radius rho.
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z = a + 2bt + ct^{2}
As the real parameter t varies, z describes a parabola in general, and a straight line if b/c is real.
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\am\left(\frac{a - b}{c - d}\right) = ±\tfrac{1}{2} \pi
The lines joining z=a to z=b and z=c to z=d are perpendicular when this amplitude is plus or minus pi/2.
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\tan\phi_{m+n} &= \tan\phi_{m} \sec\phi_{n} &&+ \sec\phi_{m} \tan\phi_{n}
The analogue of De Moivre's theorem for the tangent: tan of phi(m+n) equals tan phi_m sec phi_n plus sec phi_m tan phi_n.
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\sec\phi_{m+n} &= \sec\phi_{m} \sec\phi_{n} &&+ \tan\phi_{m} \tan\phi_{n}
The analogue of De Moivre's theorem for the secant: sec of phi(m+n) equals sec phi_m sec phi_n plus tan phi_m tan phi_n.
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\tan\phi_{m} + \sec\phi_{m} = (\tan\phi_{1} + \sec\phi_{1})^{m}
The sum of tan and sec of phi_m equals the m-th power of tan phi_1 plus sec phi_1.
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z^{n} = a
A number z is an n-th root of a when its n-th power equals a; the book uses this to define the symbol a^{1/n} for complex a.
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a = \rho(\cos\phi + i\sin\phi)
A non-zero complex number a is written in modulus-amplitude form, with positive modulus rho and angle phi.
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z = r(\cos\theta + i\sin\theta)
The unknown complex number z is written in modulus-amplitude form with modulus r and angle theta.
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r^{n} = \rho
The modulus of an n-th root z of a must satisfy r^n equal to the modulus rho of a.
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n\theta = \phi + 2k\pi
The angle of an n-th root of a satisfies n theta equal to phi plus a whole number of turns 2k pi, where k is an integer.
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r = \sqrt[n]{\rho}
The only possible modulus of an n-th root of a is the ordinary positive n-th root of rho.
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\theta = (\phi + 2q\pi)/n
The angles of the n distinct roots of a are (phi + 2q pi)/n for q = 0, 1, ..., n-1.
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z^{n} = a = \rho(\cos\phi + i\sin\phi)
The equation z^n = a, with a in modulus-amplitude form, has exactly n roots, all of which are given by the formulas for r and theta.
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\cos\alpha\theta + i\sin\alpha\theta
For rational alpha, one of the values of (cos theta + i sin theta)^alpha is cos(alpha theta) + i sin(alpha theta).

Problems

Exercise XXII

Exercise XXII, problem 1, p. 99
The two square roots of $1$ are $1$, $-1$; the three cube roots are $1$, $\frac{1}{2}(-1 + i\sqrt{3})$, $\frac{1}{2}(-1 - i\sqrt{3})$; the four fourth roots are $1$, $i$, $-1$, $-i$; and the five fifth roots are alignat*4 1,0pt minus 3pt&14 [ &&5 - 1 + i10 + 25],0pt minus 3pt && 14 [-&&5 - 1 + i10 - 25], &14 [-&&5 - 1 - i10 - 25],0pt minus 3pt && 14 [ &&5 - 1 - i10 + 25]. alignat*
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Exercise XXII, problem 10, p. 99
The problem of finding the accurate value of $\omega_{n}$ in a numerical form involving square roots only, as in the formula $\omega_{3} = \frac{1}{2}(-1 + i\sqrt{3})$, is the algebraical equivalent of the geometrical problem of inscribing a regular polygon of $n$ sides in a circle of unit radius by Euclidean methods, *i.e.* by ruler and compasses. For this construction will be possible if and only if we can construct lengths measured by $\cos(2\pi/n)$ and $\sin(2\pi/n)$; and this is possible (Ch.II, [misc:II]Misc. Exs. 22) if and only if these numbers are expressible in a form involving square roots only. Euclid gives constructions for $n = 3$, $4$, $5$, $6$, $8$, $10$, $12$, and $15$. It is evident that the construction is possible for any value of $n$ which can be found from these by multiplication by any power of $2$. There are other special values of $n$ for which such constructions are possible, the most interesting being $n = 17$.
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Exercise XXII, problem 2, p. 99
Prove that 1 + _n + _n^2 + …+ _n^n-1 = 0.
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Exercise XXII, problem 3, p. 99
Prove that (x + y_3 + z_3^2) (x + y_3^2 + z_3) = x^2 + y^2 + z^2 - yz - zx - xy.
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Exercise XXII, problem 4, p. 99
The $n$th roots of $a$ are the products of the $n$th roots of unity by the principal value of $\sqrt[n]{a}$.
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Exercise XXII, problem 5, p. 99
It follows from xxi. 14 that the roots of z^2 = + i are ± 12 ^2 + ^2 + ± i12 ^2 + ^2 - , like or unlike signs being chosen according as $\beta$ is positive or negative. Show that this result agrees with the result of [§]48.
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Exercise XXII, problem 6, p. 99
Show that $(x^{2m} - a^{2m})/(x^{2} - a^{2})$ is equal to (x^2 - 2axm + a^2) (x^2 - 2ax2m + a^2) …(x^2 - 2ax(m - 1)m + a^2). [The factors of $x^{2m} - a^{2m}$ are (x - a),0pt minus 3pt(x - a_2m),0pt minus 3pt(x - a_2m^2), …0pt minus 3pt(x - a_2m^2m-1). The factor $x - a\omega_{2m}^{m}$ is $x + a$. The factors $(x - a\omega_{2m}^{s})$, $(x - a\omega_{2m}^{2m-s})$ taken together give a factor $x^{2} - 2ax \cos(s\pi/m) + a^{2}$.]
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Exercise XXII, problem 7, p. 99
Resolve $x^{2m+1} - a^{2m+1}$, $x^{2m} + a^{2m}$, and $x^{2m+1} + a^{2m+1}$ into factors in a similar way.
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Exercise XXII, problem 8, p. 99
Show that $x^{2n} - 2x^{n}a^{n} \cos\theta + a^{2n}$ is equal to multline* (x^2 - 2xan + a^2) (x^2 - 2xa+ 2n + a^2) … …(x^2 - 2xa+ 2(n - 1)n + a^2). multline* [Use the formula x^2n - 2x^na^n + a^2n = x^n - a^n(+ i) x^n - a^n(- i), and split up each of the last two expressions into $n$ factors.]
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Exercise XXII, problem 9, p. 99
Find all the roots of the equation $x^{6} - 2x^{3} + 2 = 0$. % [0]% (*Math. Trip.* 1910.)% [1]%
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Exercise Misc-III

Exercise Misc-III, problem 1, p. 101
The condition that a triangle $(xyz)$ should be equilateral is that x^2 + y^2 + z^2 - yz - zx - xy = 0.
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Exercise Misc-III, problem 10, p. 101
Show that the necessary and sufficient conditions that both the roots of the equation $z^{2} + az + b = 0$ should be of unit modulus are |a| 2,0pt minus 3pt|b| = 1,0pt minus 3ptb = 2a.
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Exercise Misc-III, problem 11, p. 101
If $x^{4} + 4a_{1}x^{3} + 6a_{2}x^{2} + 4a_{3}x + a_{4} = 0$ is an equation with real coefficients and has two real and two complex roots, concyclic in the Argand diagram, then a_3^2 + a_1^2a_4 + a_2^3 - a_2a_4 - 2a_1a_2a_3 = 0.
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Exercise Misc-III, problem 12, p. 101
The four roots of $a_{0}x^{4} + 4a_{1}x^{3} + 6a_{2}x^{2} + 4a_{3}x + a_{4} = 0$ will be harmonically related if a_0a_3^2 + a_1^2a_4 + a_2^3 - a_0a_2a_4 - 2a_1a_2a_3 = 0.
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Exercise Misc-III, problem 13, p. 101
**points and straight lines.** Let $ax + by + c = 0$ be an equation with complex coefficients (which of course may be real in special cases). If we give $x$ any particular real or complex value, we can find the corresponding value of $y$. The aggregate of pairs of real or complex values of $x$ and $y$ which satisfy the equation is called an *imaginary straight line*; the pairs of values are called *imaginary points*, and are said *to lie on the line*. The values of $x$ and $y$ are called the *coordinates* of the point $(x, y)$. When $x$ and $y$ are real, the point is called a *real point*: when $a$, $b$, $c$ are all real (or can be made all real by division by a common factor), the line is called a *real line*. The points $x = \alpha + \beta i$, $y = \gamma + \delta i$ and $x = \alpha - \beta i$, $y = \gamma - \delta i$ are said to be *conjugate*; and so are the lines (A + A’i)x + (B + B’i)y + C + C’i = 0,0pt minus 3pt(A - A’i)x + (B - B’i)y + C - C’i = 0. Verify the following assertions:---every real line contains infinitely many pairs of conjugate imaginary points; an imaginary line in general contains one and only one real point; an imaginary line cannot contain a pair of conjugate imaginary points:---and find the conditions (*a*) that the line joining two given imaginary points should be real, and (*b*) that the point of intersection of two imaginary lines should be real.
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Exercise Misc-III, problem 14, p. 101
Prove the identities gather* (x + y + z) (x + y_3 + z_3^2) (x + y_3^2 + z_3) = x^3 + y^3 + z^3 - 3xyz, (x + y + z) (x + y_5 + z_5^4) (x + y_5^2 + z_5^3) (x + y_5^3 + z_5^2) (x + y_5^4 + z_5) = x^5 + y^5 + z^5 - 5x^3yz + 5xy^2z^2. gather*
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Exercise Misc-III, problem 15a, p. 101
Solve the equations x^3 - 3ax + (a^3 + 1) = 0,0pt minus 3ptx^5 - 5ax^3 + 5a^2x + (a^5 + 1) = 0.
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Exercise Misc-III, problem 15b, p. 101
Solve the equations x^3 - 3ax + (a^3 + 1) = 0,0pt minus 3ptx^5 - 5ax^3 + 5a^2x + (a^5 + 1) = 0.
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Exercise Misc-III, problem 16, p. 101
If $f(x) = a_{0} + a_{1}x + \dots + a_{k}x^{k}$, then f(x) + f(x) + …+ f(^n-1x)/n = a_0 + a_nx^n + a_2nx^2n + …+ a_nx^n, $\omega$ being any root of $x^{n} = 1$ (except $x = 1$), and $\lambda n$ the greatest multiple of $n$ contained in $k$. Find a similar formula for $a_{\mu} + a_{\mu+n}x^{n} + a_{\mu+2n}x^{2n} + \dots$.
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Exercise Misc-III, problem 17, p. 101
If (1 + x)^n = p_0 + p_1x + p_2x^2 + …, $n$ being a positive integer, then p_0 - p_2 + p_4 - …= 2^12 n 14n,0pt minus 3ptp_1 - p_3 + p_5 - …= 2^12 n 14n.
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Exercise Misc-III, problem 18, p. 101
Sum the series x2! n - 2!(n - 2)! + x^25! n - 5!(n - 5)! + x^38! n - 8!(n - 8)! + … + x^n/3n - 1!(n - 1)!, $n$ being a multiple of $3$. % [0]% (*Math. Trip.* 1899.)% [1]%
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Exercise Misc-III, problem 19, p. 101
0.375em plus 0.75em minus 0.25emIf $t$ is a complex number such that $|t| = 1$, then the point $x = (at + b)/(t - c)$ describes a circle as $t$ varies, unless $|c| = 1$, when it describes a straight line.
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Exercise Misc-III, problem 2, p. 101
If $XYZ$, $X'Y'Z'$ are two triangles, and YZP’ · Y’Z’P’ = ZXP’ · Z’X’P’ = XYP’ · X’Y’P’, then both triangles are equilateral.
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Exercise Misc-III, problem 20, p. 101
If $t$ varies as in the last example then the point $x = \frac{1}{2}\{at + (b/t)\}$ in general describes an ellipse whose foci are given by $x^{2} = ab$, and whose axes are $|a| + |b|$ and $|a| - |b|$. But if $|a| = |b|$ then $x$ describes the finite straight line joining the points $-\sqrtp{ab}$, $\sqrtp{ab}$.
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Exercise Misc-III, problem 21, p. 101
Prove that if $t$ is real and $z = t^{2} - 1 + \sqrtp{t^{4} - t^{2}}$, then, when $t^{2} < 1$, $z$ is represented by a point which lies on the circle $x^{2} + y^{2} + x = 0$. Assuming that, when $t^{2} > 1$, $\sqrtp{t^{4} - t^{2}}$ denotes the positive square root of $t^{4} - t^{2}$, discuss the motion of the point which represents $z$, as $t$ diminishes from a large positive value to a large negative value. % [0]% (*Math. Trip.* 1912.)% [1]%
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Exercise Misc-III, problem 22, p. 101
The coefficients of the transformation $z = (aZ + b)/(cZ + d)$ are subject to the condition $ad - bc = 1$. Show that, if $c \neq 0$, there are two *fixed points* $\alpha$, $\beta$, *i.e.* points unaltered by the transformation, except when $(a + d)^{2} = 4$, when there is only one fixed point $\alpha$; and that in these two cases the transformation may be expressed in the forms z - z - = KZ - Z - ,0pt minus 3pt1z - = 1Z - + K. Show further that, if $c = 0$, there will be one fixed point $\alpha$ unless $a = d$, and that in these two cases the transformation may be expressed in the forms z - = K(Z - ),0pt minus 3ptz = Z + K. Finally, if $a$, $b$, $c$, $d$ are further restricted to positive integral values (including zero), show that the only transformations with less than two fixed points are of the forms $(1/z) = (1/Z) + K$, $z = Z + K$. % [0]% (*Math. Trip.* 1911.)% [1]%
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Exercise Misc-III, problem 23, p. 101
Prove that the relation $z = (1 + Zi)/(Z + i)$ transforms the part of the axis of $x$ between the points $z = 1$ and $z = -1$ into a semicircle passing through the points $Z = 1$ and $Z = -1$. Find all the figures that can be obtained from the originally selected part of the axis of $x$ by successive applications of the transformation. % [0]% (*Math. Trip.* 1912.)% [1]%
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Exercise Misc-III, problem 24, p. 101
If $z = 2Z + Z^{2}$ then the circle $|Z| = 1$ corresponds to a cardioid in the plane of $z$.
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Exercise Misc-III, problem 25, p. 101
Discuss the transformation $z = \frac{1}{2}\{Z + (1/Z)\}$, showing in particular that to the circles $X^{2} + Y^{2} = \alpha^{2}$ correspond the confocal ellipses x^212(+ 1)^2 + y^212(- 1)^2 = 1.
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Exercise Misc-III, problem 26, p. 101
If $(z + 1)^{2} = 4/Z$ then the unit circle in the $z$-plane corresponds to the parabola $R\cos^{2} \frac{1}{2}\Theta = 1$ in the $Z$-plane, and the inside of the circle to the outside of the parabola.
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Exercise Misc-III, problem 27, p. 101
Show that, by means of the transformation $z = \{(Z - ci)/(Z + ci)\}^{2}$, the upper half of the $z$-plane may be made to correspond to the interior of a certain semicircle in the $Z$-plane.
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Exercise Misc-III, problem 28, p. 101
If $z = Z^{2} - 1$, then as $z$ describes the circle $|z| = \kappa$, the two corresponding positions of $Z$ each describe the Cassinian oval $\rho_{1}\rho_{2} = \kappa$, where $\rho_{1}$, $\rho_{2}$ are the distances of $Z$ from the points $-1$, $1$. Trace the ovals for different values of $\kappa$.
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Exercise Misc-III, problem 29, p. 101
Consider the relation $az^{2} + 2hzZ + bZ^{2} + 2gz + 2fZ + c = 0$. Show that there are two values of $Z$ for which the corresponding values of $z$ are equal, and *vice versa*. We call these the *branch points* in the $Z$ and $z$-planes respectively. Show that, if $z$ describes an ellipse whose foci are the branch points, then so does $Z$.
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Exercise Misc-III, problem 3, p. 101
Similar triangles $BCX$, $CAY$, $ABZ$ are described on the sides of a triangle $ABC$. Show that the centres of gravity of $ABC$, $XYZ$ are coincident.
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Exercise Misc-III, problem 30, p. 101
If $z = aZ^{m} + bZ^{n}$, where $m$, $n$ are positive integers and $a$, $b$ real, then as $Z$ describes the unit circle, $z$ describes a hypo- or epi-cycloid.
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Exercise Misc-III, problem 31, p. 101
Show that the transformation z = (a + di)Z_0 + bcZ_0 - (a - di), where $a$, $b$, $c$, $d$ are real and $a^{2} + d^{2} + bc > 0$, and $Z_{0}$ denotes the conjugate of $Z$, is equivalent to an inversion with respect to the circle c(x^2 + y^2) - 2ax - 2dy - b = 0. What is the geometrical interpretation of the transformation when a^2 + d^2 + bc < 0?
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Exercise Misc-III, problem 32, p. 101
The transformation 1 - z1 + z = (1 - Z1 + Z)^c, where $c$ is rational and $0 < c < 1$, transforms the circle $|z| = 1$ into the boundary of a circular lune of angle $\pi/c$.
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Exercise Misc-III, problem 4, p. 101
If $X$, $Y$, $Z$ are points on the sides of the triangle $ABC$, such that BX/XC = CY/YA = AZ/ZB = r, and if $ABC$, $XYZ$ are similar, then either $r = 1$ or both triangles are equilateral.
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Exercise Misc-III, problem 5, p. 101
If $A$, $B$, $C$, $D$ are four points in a plane, then AD · BC BD · CA + CD · AB.
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Exercise Misc-III, problem 6, p. 101
Deduce Ptolemy’s Theorem concerning cyclic quadrilaterals from the fact that the cross ratios of four concyclic points are real.
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Exercise Misc-III, problem 7, p. 101
If $z^{2} + z'^{2} = 1$, then the points $z$, $z'$ are ends of conjugate diameters of an ellipse whose foci are the points $1$, $-1$.
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Exercise Misc-III, problem 8, p. 101
Prove that $|a + b|^{2} + |a - b|^{2} = 2\{|a|^{2} + |b|^{2}\}$.
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Exercise Misc-III, problem 9, p. 101
Deduce from Ex. 8 that |a + a^2 - b^2| + |a - a^2 - b^2| = |a + b| + |a - b|.
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Exercise XX

Exercise XX, problem 1a, p. 73
Prove that 0pt minus 3pt% [2.25em][l](i)% [2.25em][l](i)% % $\alpha [\beta x, \beta y] = \beta [\alpha x, \alpha y] = [\alpha \beta x, \alpha \beta y]$, 0pt minus 3pt% [2.25em][l](ii)% [2.25em][l](ii)% % $([x, y] + [x', y']) + [x'', y''] = [x, y] + ([x', y'] + [x'', y''])$, 0pt minus 3pt% [2.25em][l](iii)% [2.25em][l](iii)% % $[x, y] + [x', y'] = [x', y'] + [x, y]$, 0pt minus 3pt% [2.25em][l](iv)% [2.25em][l](iv)% % $(\alpha + \beta) [x, y] = \alpha [x, y] + \beta [x, y]$, 0pt minus 3pt% [2.25em][l](v)% [2.25em][l](v)% % $\alpha \{[x, y] + [x', y']\} = \alpha [x, y] + \alpha [x', y']$. [We have already proved (iii). The remaining equations follow with equal ease from the definitions. The reader should in each case consider the geometrical significance of the equation, as we did above in the case of (iii).]
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Exercise XX, problem 1b, p. 73
Prove that 0pt minus 3pt% [2.25em][l](i)% [2.25em][l](i)% % $\alpha [\beta x, \beta y] = \beta [\alpha x, \alpha y] = [\alpha \beta x, \alpha \beta y]$, 0pt minus 3pt% [2.25em][l](ii)% [2.25em][l](ii)% % $([x, y] + [x', y']) + [x'', y''] = [x, y] + ([x', y'] + [x'', y''])$, 0pt minus 3pt% [2.25em][l](iii)% [2.25em][l](iii)% % $[x, y] + [x', y'] = [x', y'] + [x, y]$, 0pt minus 3pt% [2.25em][l](iv)% [2.25em][l](iv)% % $(\alpha + \beta) [x, y] = \alpha [x, y] + \beta [x, y]$, 0pt minus 3pt% [2.25em][l](v)% [2.25em][l](v)% % $\alpha \{[x, y] + [x', y']\} = \alpha [x, y] + \alpha [x', y']$. [We have already proved (iii). The remaining equations follow with equal ease from the definitions. The reader should in each case consider the geometrical significance of the equation, as we did above in the case of (iii).]
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Exercise XX, problem 1c, p. 73
Prove that 0pt minus 3pt% [2.25em][l](i)% [2.25em][l](i)% % $\alpha [\beta x, \beta y] = \beta [\alpha x, \alpha y] = [\alpha \beta x, \alpha \beta y]$, 0pt minus 3pt% [2.25em][l](ii)% [2.25em][l](ii)% % $([x, y] + [x', y']) + [x'', y''] = [x, y] + ([x', y'] + [x'', y''])$, 0pt minus 3pt% [2.25em][l](iii)% [2.25em][l](iii)% % $[x, y] + [x', y'] = [x', y'] + [x, y]$, 0pt minus 3pt% [2.25em][l](iv)% [2.25em][l](iv)% % $(\alpha + \beta) [x, y] = \alpha [x, y] + \beta [x, y]$, 0pt minus 3pt% [2.25em][l](v)% [2.25em][l](v)% % $\alpha \{[x, y] + [x', y']\} = \alpha [x, y] + \alpha [x', y']$. [We have already proved (iii). The remaining equations follow with equal ease from the definitions. The reader should in each case consider the geometrical significance of the equation, as we did above in the case of (iii).]
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Exercise XX, problem 1d, p. 73
Prove that 0pt minus 3pt% [2.25em][l](i)% [2.25em][l](i)% % $\alpha [\beta x, \beta y] = \beta [\alpha x, \alpha y] = [\alpha \beta x, \alpha \beta y]$, 0pt minus 3pt% [2.25em][l](ii)% [2.25em][l](ii)% % $([x, y] + [x', y']) + [x'', y''] = [x, y] + ([x', y'] + [x'', y''])$, 0pt minus 3pt% [2.25em][l](iii)% [2.25em][l](iii)% % $[x, y] + [x', y'] = [x', y'] + [x, y]$, 0pt minus 3pt% [2.25em][l](iv)% [2.25em][l](iv)% % $(\alpha + \beta) [x, y] = \alpha [x, y] + \beta [x, y]$, 0pt minus 3pt% [2.25em][l](v)% [2.25em][l](v)% % $\alpha \{[x, y] + [x', y']\} = \alpha [x, y] + \alpha [x', y']$. [We have already proved (iii). The remaining equations follow with equal ease from the definitions. The reader should in each case consider the geometrical significance of the equation, as we did above in the case of (iii).]
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Exercise XX, problem 1e, p. 73
Prove that 0pt minus 3pt% [2.25em][l](i)% [2.25em][l](i)% % $\alpha [\beta x, \beta y] = \beta [\alpha x, \alpha y] = [\alpha \beta x, \alpha \beta y]$, 0pt minus 3pt% [2.25em][l](ii)% [2.25em][l](ii)% % $([x, y] + [x', y']) + [x'', y''] = [x, y] + ([x', y'] + [x'', y''])$, 0pt minus 3pt% [2.25em][l](iii)% [2.25em][l](iii)% % $[x, y] + [x', y'] = [x', y'] + [x, y]$, 0pt minus 3pt% [2.25em][l](iv)% [2.25em][l](iv)% % $(\alpha + \beta) [x, y] = \alpha [x, y] + \beta [x, y]$, 0pt minus 3pt% [2.25em][l](v)% [2.25em][l](v)% % $\alpha \{[x, y] + [x', y']\} = \alpha [x, y] + \alpha [x', y']$. [We have already proved (iii). The remaining equations follow with equal ease from the definitions. The reader should in each case consider the geometrical significance of the equation, as we did above in the case of (iii).]
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Exercise XX, problem 2, p. 73
If $M$ is the middle point of $PQ$, then $\Seg{OM} = \frac{1}{2}(\Seg{OP} + \Seg{OQ})$. More generally, if $M$ divides $PQ$ in the ratio $\mu : \lambda$, then OMP’ = + OPP’ + + OQP’.
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Exercise XX, problem 3, p. 73
If $G$ is the centre of mass of equal particles at $P_{1}$, $P_{2}$, …, $P_{n}$, then OGP’ = (OP_1P’ + OP_2P’ + …+ OP_nP’)/n.
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Exercise XX, problem 4, p. 73
If $P$, $Q$, $R$ are collinear points in the plane, then it is possible to find real numbers $\alpha$, $\beta$, $\gamma$, not all zero, and such that · OPP’ + · OQP’ + · ORP’ = 0; and conversely. [This is really only another way of stating Ex. 2.]
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Exercise XX, problem 5, p. 73
If $\Seg{AB}$ and $\Seg{AC}$ are two displacements not in the same straight line, and · ABP’ + · ACP’ = · ABP’ + · ACP’, then $\alpha = \gamma$ and $\beta = \delta$. [Take $AB_{1} = \alpha · AB$, $AC_{1} = \beta · AC$. Complete the parallelogram $AB_{1}P_{1}C_{1}$. Then $\Seg{AP_{1}} = \alpha · \Seg{AB} + \beta · \Seg{AC}$. It is evident that $\Seg{AP_{1}}$ can only be expressed in this form in one way, whence the theorem follows.]
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Exercise XX, problem 6, p. 73
$ABCD$ is a parallelogram. Through $Q$, a point inside the parallelogram, $RQS$ and $TQU$ are drawn parallel to the sides. Show that $RU$, $TS$ intersect on $AC$. %[Illustration: Fig. 21.] [2.75in]21p074 [Let the ratios $AT:AB$, $AR:AD$ be denoted by $\alpha$, $\beta$. Then gather* ATP’ = · ABP’,0pt minus 3ptARP’ = · ADP’, AUP’ = · ABP’ + ADP’,0pt minus 3ptASP’ = ABP’ + · ADP’. gather* Let $RU$ meet $AC$ in $P$. Then, since $R$, $U$, $P$ are collinear, APP’ = + ARP’ + + AUP’, where $\mu/\lambda$ is the ratio in which $P$ divides $RU$. That is to say APP’ = + ABP’ + + + ADP’. But since $P$ lies on $AC$, $\Seg{AP}$ is a numerical multiple of $\Seg{AC}$; say APP’ = k · ACP’ = k · ABP’ + k · ADP’. Hence (Ex. 5) $\alpha\mu = \beta\lambda + \mu = (\lambda + \mu)k$, from which we deduce k = + - 1. The symmetry of this result shows that a similar argument would also give AP’P’ = + - 1 ACP’, if $P'$ is the point where $TS$ meets $AC$. Hence $P$ and $P'$ are the same point.]
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Exercise XX, problem 7, p. 73
$ABCD$ is a parallelogram, and $M$ the middle point of $AB$. Show that $DM$ trisects and is trisected by $AC$. The two preceding examples are taken from Willard Gibbs’ *Vector Analysis*.
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Exercise XXI

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