Chapter III
Complex Numbers

34. Displacements along a line and in a plane. The ‘real number’ $x$, with which we have been concerned in the two preceding chapters, may be regarded from many different points of view. It may be regarded as a pure number, destitute of geometrical significance, or a geometrical significance may be attached to it in at least three different ways. It may be regarded as the measure of a length, viz. the length $A_{0}P$ along the line $\Lambda$ of Chap. I. It may be regarded as the mark of a point, viz. the point $P$ whose distance from $A_0$ is $x$. Or it may be regarded as the measure of a displacement or change of position on the line $\Lambda$. It is on this last point of view that we shall now concentrate our attention.

Imagine a small particle placed at $P$ on the line $\Lambda$ and then displaced to $Q$. We shall call the displacement or change of position which is needed to transfer the particle from $P$ to $Q$ the displacement $\overline{PQ}$. 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. But, when we are thinking merely of the change of position produced by the displacement, it is natural to disregard the point of application and to consider all displacements as equivalent whose lengths and senses are the same. Then the displacement is completely specified by the length $PQ=x$, the sense of the displacement being fixed by the sign of $x$. We may therefore, without ambiguity, speak of the displacement ,²⁶ and we may write .

We use the square bracket to distinguish the displacement  from the length or number $x$.²⁷ If the coordinate of $P$ is $a$, that of $Q$ will be $a+x$; the displacement  therefore transfers a particle from the point $a$ to the point $a+x$.

We come now to consider displacements in a plane. We may define the displacement $\overline{PQ}$ as before. But now more data are required in order to specify it completely. We require to know: (i) the magnitude of the displacement, i.e. the length of the straight line $PQ$; (ii) the direction of the displacement, which is determined by the angle which $PQ$ makes with some fixed line in the plane; (iii) the sense of the displacement; and (iv) its point of application. Of these requirements we may disregard the fourth, if we consider two displacements as equivalent if they are

the same in magnitude, direction, and sense. In other words, if $PQ$ and $RS$ are equal and parallel, and the sense of motion from $P$ to $Q$ is the same as that of motion from $R$ to $S$, we regard the displacements $\overline{PQ}$ and $\overline{RS}$ as equivalent, and write

Now let us take any pair of coordinate axes in the plane (such as $OX$, $OY$ in Fig. 19). Draw a line $OA$ equal and parallel to $PQ$, the sense of motion from $O$ to $A$ being the same as that from $P$ to $Q$. Then $\overline{PQ}$ and $\overline{OA}$ are equivalent displacements. Let $x$ and $y$ be the coordinates of $A$. Then it is evident that $\overline{OA}$ is completely specified if $x$ and $y$ are given. We call $\overline{OA}$ the displacement and write

35. Equivalence of displacements. Multiplication of displacements by numbers. If $\xi$ and $\eta$ are the coordinates of $P$, and $\xi \prime$ and $\eta \prime$ those of $Q$, it is evident that

It is clear that two displacements , are equivalent if, and only if, $x=x\prime$, $y=y\prime$. Thus if and only if

The reverse displacement $\overline{QP}$ would be , and it is natural to agree that

these equations being really definitions of the meaning of the symbols , $-\overline{PQ}$. Having thus agreed that

where $\alpha$ is any real number, positive or negative. Thus (Fig. 19) if $OB=-\frac{1}{2}OA$ then

The equations (1) and (2) define the first two important ideas connected with displacements, viz. equivalence of displacements, and multiplication of displacements by numbers.

36. Addition of displacements. We have not yet given any definition which enables us to attach any meaning to the expressions

Let us consider the consequences of adopting this definition. If the coordinates of $B$ are $x\prime$, $y\prime$, then those of the middle point of $AB$ are , , and those of $C$ are $x+x\prime$, $y+y\prime$. Hence

which may be regarded as the symbolic definition of addition of displacements. We observe that

In other words, addition of displacements obeys the commutative law expressed in ordinary algebra by the equation $a+b=b+a$. This law expresses the obvious geometrical fact that if we move from $P$ first through a distance $P{Q}_2$ equal and parallel to $P\prime Q\prime$, and then through a distance equal and parallel to $PQ$, we shall arrive at the same point $Q_1$ as before.

In particular

When we have once defined addition of two displacements, there is no further difficulty in the way of defining addition of any number. Thus, by definition,

We define subtraction of displacements by the equation

which is the same thing as or as . In particular

The displacement  leaves the particle where it was; it is the zero displacement, and we agree to write .

Examples XX. 1. Prove that

(i) ,

(ii) ,

(iii) ,

(iv) ,

(v) .

[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).]

2. If $M$ is the middle point of $PQ$, then . More generally, if $M$ divides $PQ$ in the ratio $\mu :\lambda$, then

3. If $G$ is the centre of mass of equal particles at $P_1$, $P_2$, …, $P_n$, then

4. 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

5. If $\overline{AB}$ and $\overline{AC}$ are two displacements not in the same straight line, and

[Take $A{B}_1=\alpha \cdot AB$, $A{C}_1=\beta \cdot AC$. Complete the parallelogram $A{B}_1{P}_1{C}_1$. Then $\overline{A{P}_1}=\alpha \cdot \overline{AB}+\beta \cdot \overline{AC}$. It is evident that $\overline{A{P}_1}$ can only be expressed in this form in one way, whence the theorem follows.]

6. $ABCD$ 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 $AC$.

[Let the ratios $AT:AB$, $AR:AD$ be denoted by $\alpha$, $\beta$. Then

Let $RU$ meet $AC$ in $P$. Then, since $R$, $U$, $P$ are collinear,

But since $P$ lies on $AC$, $\overline{AP}$ is a numerical multiple of $\overline{AC}$; say

7. $ABCD$ is a parallelogram, and $M$ the middle point of $AB$. Show that $DM$ trisects and is trisected by $AC$.²⁸

37. Multiplication of displacements. So far we have made no attempt to attach any meaning whatever to the notion of the product of two displacements. The only kind of multiplication which we have considered is that in which a displacement is multiplied by a number. The expression

We might, for example, define it as being equal to

A more reasonable definition might appear to be

In fact, it is by no means obvious what is the best meaning to attach to the product . All that is clear is (1) that, if our definition is to be of any use, this product must itself be a displacement whose coordinates depend on $x$ and $y$, or in other words that we must have

38. The right definition to take is suggested as follows. We know that, if $OAB$, $OCD$ are two similar triangles, the angles corresponding in the order in which they are written, then

Now let

If the polar coordinates of $B$ and $C$ are and , so that

The required definition is therefore

We observe (1) that if $y=0$, then $X=xx\prime$, $Y=xy\prime$, as we desired; (2) that the right-hand side is not altered if we interchange $x$ and $x\prime$, and $y$ and $y\prime$, so that

Similarly we can verify that all the equations at the end of § 37 are satisfied. Thus the definition (6) fulfils all the requirements which we made of it in § 37.

Example. Show directly from the geometrical definition given above that multiplication of displacements obeys the commutative and distributive laws. [Take the commutative law for example. The product $\overline{OB}\cdot \overline{OC}$ is $\overline{OD}$ (Fig. 22), $COD$ being similar to $AOB$. To construct the product $\overline{OC}\cdot \overline{OB}$ we should have to construct on $OB$ a triangle $BO{D}_1$ similar to $AOC$; and so what we want to prove is that $D$ and $D_1$ coincide, or that $BOD$ is similar to $AOC$. This is an easy piece of elementary geometry.]

39. Complex numbers. Just as to a displacement  along $OX$ correspond a point  and a real number $x$, so to a displacement  in the plane correspond a point  and a pair of real numbers $x$, $y$.

We shall find it convenient to denote this pair of real numbers $x$, $y$ by the symbol

We proceed next to define equivalence, addition, and multiplication of complex numbers. To every complex number corresponds a displacement. Two complex numbers are equivalent if the corresponding displacements are equivalent. The sum or product of two complex numbers is the complex number which corresponds to the sum or product of the two corresponding displacements. Thus

if and only if $x=x\prime$, $y=y\prime$;

In particular we have, as special cases of (2) and (3),

and these equations suggest that there will be no danger of confusion if, when dealing with complex numbers, we write $x$ for $x+0i$ and $yi$ for $0+yi$, as we shall henceforth.

Positive integral powers and polynomials of complex numbers are then defined as in ordinary algebra. Thus, by putting $x=x\prime$, $y=y\prime$ in (3), we obtain

The reader will easily verify for himself that addition and multiplication of complex numbers obey the laws of algebra expressed by the equations

the proofs of these equations being practically the same as those of the corresponding equations for the corresponding displacements.

Subtraction and division of complex numbers are defined as in ordinary algebra. Thus we may define as

Solving these equations for $\xi$ and $\eta$, we obtain

Examples. (1) From a geometrical point of view, the problem of the division of the displacement $\overline{OB}$ by $\overline{OC}$ is that of finding $D$ so that the triangles $COB$, $AOD$ are similar, and this is evidently possible (and the solution unique) unless $C$ coincides with $0$, or $\overline{OC}=0$.

(2) The numbers $x+yi$, $x-yi$ are said to be conjugate. Verify that

40. 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. To prove that this is also true of complex numbers we put $x=0$, $y=0$ in the equations (4) of the preceding section. Then

41. The equation $i^2=-1$. We agreed to simplify our notation by writing $x$ instead of $x+0i$ and $yi$ instead of $0+yi$. The particular complex number $1i$ we shall denote simply by $i$. It is the number which corresponds to a unit displacement along $OY$. Also

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. Thus, for example,

42. The geometrical interpretation of multiplication by $i$. Since

We might have developed the whole theory of complex numbers from this point of view. Starting with the ideas of $x$ as representing a displacement along $OX$, and of $i$ as a symbol of operation equivalent to turning $x$ through a right angle, we should have been led to regard $yi$ as a displacement of magnitude $y$ along $OY$. It would then have been natural to define $x+yi$ as in §§ 37 and 40, and would have represented the displacement obtained by turning $x+yi$ through a right angle, i.e. $-y+xi$. Finally, we should naturally have defined as $xx\prime +yx\prime i$, as $-yy\prime +xy\prime i$, and as the sum of these displacements, i.e. as

43. The equations $z^2+1=0$, $a{z}^2+2bz+c=0$. There is no real number $z$ such that $z^2+1=0$; this is expressed by saying that the equation has no real roots. But, as we have just seen, the two complex numbers $i$ and $-i$ satisfy this equation. We express this by saying that the equation has the two complex roots $i$ and $-i$. Since $i$ satisfies $z^2=-1$, it is sometimes written in the form $\sqrt[]{-1}$.

Complex numbers are sometimes called imaginary.²⁹ The expression is by no means a happily chosen one, but it is firmly established and has to be accepted. 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 , 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. Thus

Now let us consider the more general equation

If we agree as a matter of convention to say that when $b^2=ac$ (in which case the equation is satisfied by one value of $x$ only, viz. $-b/a$), the equation has two equal roots, we can say that a quadratic equation with real coefficients has two roots in all cases, either two distinct real roots, or two equal real roots, or two distinct complex roots.

The question is naturally suggested whether a quadratic equation may not, when complex roots are once admitted, have more than two roots. It is easy to see that this is not possible. Its impossibility may in fact be proved by precisely the same chain of reasoning as is used in elementary algebra to prove that an equation of the $n$th degree cannot have more than $n$ real roots. Let us denote the complex number $x+yi$ by the single letter $z$, a convention which we may express by writing $z=x+yi$. Let  denote any polynomial in $z$, with real or complex coefficients. Then we prove in succession:

(1) that the remainder, when  is divided by $z-a$, $a$ being any real or complex number, is ;

(2) that if $a$ is a root of the equation , then  is divisible by $z-a$;

(3) that if  is of the $n$th degree, and has the $n$ roots $a_1$, $a_2$, …, $a_n$, then

We conclude that a quadratic equation with real coefficients has exactly two roots. We shall see later on that a similar theorem is true for an equation of any degree and with either real or complex coefficients: an equation of the $n$th degree has exactly $n$ roots. The only point in the proof which presents any difficulty is the first, viz. the proof that any equation must have at least one root. This we must postpone for the present.³¹ We may, however, at once call attention to one very interesting result of this theorem. In the theory of number we start from the positive integers and from the ideas of addition and multiplication and the converse operations of subtraction and division. We find that these operations are not always possible unless we admit new kinds of numbers. We can only attach a meaning to $3-7$ if we admit negative numbers, or to $\frac{3}{7}$ if we admit rational fractions. When we extend our list of arithmetical operations so as to include root extraction and the solution of equations, we find that some of them, such as that of the extraction of the square root of a number which (like $2$) is not a perfect square, are not possible unless we widen our conception of a number, and admit the irrational numbers of Chap. I.

Others, such as the extraction of the square root of $-1$, are not possible unless we go still further, and admit the complex numbers of this chapter. And it would not be unnatural to suppose that, when we come to consider equations of higher degree, some might prove to be insoluble even by the aid of complex numbers, and that thus we might be led to the considerations of higher and higher types of, so to say, hyper-complex numbers. The fact that the roots of any algebraical equation whatever are ordinary complex numbers shows that this is not the case. The application of any of the ordinary algebraical operations to complex numbers will yield only complex numbers. In technical language ‘the field of the complex numbers is closed for algebraical operations’.

Before we pass on to other matters, let us add that all theorems of elementary algebra which are proved merely by the application of the rules of addition and multiplication are true whether the numbers which occur in them are real or complex, since the rules referred to apply to complex as well as real numbers. For example, we know that, if $\alpha$ and $\beta$ are the roots of

Similarly, if $\alpha$, $\beta$, $\gamma$ are the roots of

44. Argand’s diagram. Let $P$ (Fig. 24) be the point , $r$ the length $OP$, and $\theta$ the angle $XOP$, so that

We denote the complex number $x+yi$ by $z$, as in § 43, and we call $z$ the complex variable.

We call $P$ the point $z$, or the point corresponding to $z$; $z$ the argument of $P$, $x$ the real part, $y$ the imaginary part, $r$ the modulus, and $\theta$ the amplitude of $z$; and we write

When $y=0$ we say that $z$ is real, when $x=0$ that $z$ is purely imaginary. Two numbers $x+yi$, $x-yi$ which differ only in the signs of their imaginary parts, we call conjugate. 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 $\sqrt[]{x^2+y^2}$ and that their product is equal to the square of the modulus of either. The roots of a quadratic with real coefficients, for example, are conjugate, when not real.

It must be observed that $\theta$ or $amz$ is a many-valued function of $x$ and $y$, having an infinity of values, which are angles differing by multiples of $2\pi$.³² A line originally lying along $OX$ will, if turned through any of these angles, come to lie along $OP$. We shall describe that one of these angles which lies between $-\pi$ and $\pi$ as the principal value of the amplitude of $z$. This definition is unambiguous except when one of the values is $\pi$, in which case $-\pi$ is also a value. In this case we must make some special provision as to which value is to be regarded as the principal value. In general, when we speak of the amplitude of $z$ we shall, unless the contrary is stated, mean the principal value of the amplitude.

Fig. 24 is usually known as Argand’s diagram.

45. De Moivre’s Theorem. The following statements follow immediately from the definitions of addition and multiplication.

(1) The real (or imaginary) part of the sum of two complex numbers is equal to the sum of their real (or imaginary) parts.

(2) The modulus of the product of two complex numbers is equal to the product of their moduli.

(3) The amplitude of the product of two complex numbers is either equal to the sum of their amplitudes, or differs from it by $2\pi$.

It should be observed that it is not always true that the principal value of  is the sum of the principal values of $amz$ and $amz\prime$. For example, if $z=z\prime =-1+i$, then the principal values of the amplitudes of $z$ and $z\prime$ are each $\frac{3}{4}\pi$. But $zz\prime =-2i$, and the principal value of  is $-\frac{1}{2}\pi$ and not $\frac{3}{2}\pi$.

The two last theorems may be expressed in the equation

A particularly interesting case is that in which

We then obtain the equation

Again, if

(4) The modulus of the quotient of two complex numbers is equal to the quotient of their moduli.

(5) The amplitude of the quotient of two complex numbers either is equal to the difference of their amplitudes, or differs from it by $2\pi$.

Again

Hence De Moivre’s Theorem holds for all integral values of $n$, positive or negative.

To the theorems (1)–(5) we may add the following theorem, which is also of very great importance.

(6) The modulus of the sum of any number of complex numbers is not greater than the sum of their moduli.

Let $\overline{OP}$, $\overline{OP\prime }$, … be the displacements corresponding to the various complex numbers. Draw $PQ$ equal and parallel to $OP\prime$, $QR$ equal and parallel to $OP\prime \prime$, and so on. Finally we reach a point $U$, such that

A purely arithmetical proof of this theorem is outlined in Exs. XXI. 1.

46. We add some theorems concerning rational functions of complex numbers. A rational function of the complex variable $z$ is defined exactly as is a rational function of a real variable $x$, viz. as the quotient of two polynomials in $z$.

THEOREM 1. Any rational function  can be reduced to the form $X+Yi$, where $X$ and $Y$ are rational functions of $x$ and $y$ with real coefficients.

In the first place it is evident that any polynomial can be reduced, in virtue of the definitions of addition and multiplication, to the form $A+Bi$, where $A$ and $B$ are polynomials in $x$ and $y$ with real coefficients. Similarly can be reduced to the form $C+Di$. Hence

which proves the theorem.

THEOREM 2. If , $R$ denoting a rational function as before, but with real coefficients, then .

In the first place this is easily verified for a power by actual expansion. It follows by addition that the theorem is true for any polynomial with real coefficients. Hence, in the notation used above,

THEOREM 3. The roots of an equation

For it follows from Theorem 2 that if $x+yi$ is a root then so is $x-yi$. A particular case of this theorem is the result (§ 43) that the roots of a quadratic equation with real coefficients are either real or conjugate.

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 Exs. VIII. 7, which may be stated as follows: in an equation with rational coefficients irrational roots occur in conjugate pairs.³⁴

Examples XXI. 1. Prove theorem (6) of §45 directly from the definitions and without the aid of geometrical considerations.

[First, to prove that $\mid z+z\prime \mid \leqq \mid z\mid +\mid z\prime \mid$ is to prove that

2. The one and only case in which

3. The modulus of the sum of any number of complex numbers is not less than the sum of their real (or imaginary) parts.

4. If the sum and product of two complex numbers are both real, then the two numbers must either be real or conjugate.

5. If

6. Express the following numbers in the form $A+Bi$, where $A$ and $B$ are real numbers:

7. Express the following functions of $z=x+yi$ in the form $X+Yi$, where $X$ and $Y$ are real functions of $x$ and $y$: $z^2$, $z^3$, $z^n$, $1/z$, , , where $\alpha$, $\beta$, $\gamma$, $\delta$ are real numbers.

8. Find the moduli of the numbers and functions in the two preceding examples.

9. The two lines joining the points $z=a$, $z=b$ and $z=c$, $z=d$ will be perpendicular if

10. The three angular points of a triangle are given by $z=\alpha$, $z=\beta$, $z=\gamma$, where $\alpha$, $\beta$, $\gamma$ are complex numbers. Establish the following propositions:

(i) the centre of gravity is given by ;

(ii) the circum-centre is given by $\mid z-\alpha \mid =\mid z-\beta \mid =\mid z-\gamma \mid$;

(iii) the three perpendiculars from the angular points on the opposite sides meet in a point given by

(iv) there is a point $P$ inside the triangle such that

[To prove (iii) we observe that if $A$, $B$, $C$ are the vertices, and $P$ any point $z$, then the condition that $AP$ should be perpendicular to $BC$ is (Ex. 9) that should be purely imaginary, or that

To prove (iv), take $BC$ parallel to the positive direction of the axis of $x$. Then³⁵

We have to determine $z$ and $\omega$ from the equations

Adding the numerators and denominators of the three equal fractions, and using the equation

To determine $z$, we multiply the numerators and denominators of the equal fractions by , , , and add to form a new fraction. It will be found that

11. The two triangles whose vertices are the points $a$, $b$, $c$ and $x$, $y$, $z$ respectively will be similar if

[The condition required is that $\overline{AB}/\overline{AC}=\overline{XY}/\overline{XZ}$ (large letters denoting the points whose arguments are the corresponding small letters), or , which is the same as the condition given.]

12. Deduce from the last example that if the points $x$, $y$, $z$ are collinear then we can find real numbers $\alpha$, $\beta$, $\gamma$ such that $\alpha +\beta +\gamma =0$ and $\alpha x+\beta y+\gamma z=0$, and conversely (cf. Exs. XX. 4). [Use the fact that in this case the triangle formed by $x$, $y$, $z$ is similar to a certain line-triangle on the axis $OX$, and apply the result of the last example.]

13. The general linear equation with complex coefficients. The equation $\alpha z+\beta =0$ has the one solution , unless $\alpha =0$. If we put

14. The general quadratic equation with complex coefficients. This equation is

Unless $a$ and $A$ are both zero we can divide through by $a+iA$. Hence we may consider