Real and complex numbers

It is tempting to cut through all this fretting about particular equations and simply declare that we already know what the real numbers are–they are the collection of all possible decimal expansions, both positive and negative. These are very familiar, in practice we know how to use them, and so we feel on safe ground.At least until we ask some verybasic questions. The main feature of numbers is that you can add, subtract, multiply, and divide. But,for example, how are you supposed to multiply two infinite non-recurring decimals? We depend on decimals being finite in length so that you‘start from the right-hand end’, but there is no such thing with an infinite decimal expansion. It can be done, but it is complicated both in theory and in practice. A number system where you struggle to explain how to add and multiply does not seem satisfactory.

You maynd the foundational questions raised above interesting or you may grow impatient with all the introspection as we seem to be making trouble for ourselves when previously all was smooth sailing. There is a serious point, however. Mathematicians appreciate that, whenever new mathematical objects are introduced, it important to construct them from known mathematical objects, the way, for instance, fractions can be thought of as pairs of ordinary integers. In this way, we may carefully build up the rules that govern the new extended system and know where we stand. If we neglect foundations completely, it will come back to haunt us later. For example, the rapid development of calculus, which was born out of the study of motion, led to spectacular results, such as prediction of the movement of the planets. However, manipulation of innite things as if they werenite sometimes provided amazing insights and at other times patent nonsense. By putting your mathematical systems on arm foundation, we can learn how to tell the difference. In practice, mathematicians often indulge in ‘formal’manipulations in order to see ifsome sparkling new theorem is in the offing. Ifthe outcome is worthy of attention, the result can be proved rigorously by going back to basics and by invoking results that have been properly established earlier.

This is why Julius Dedekind(1831-1916)took the trouble of formally constructing the real number system based on his idea that is now referred to as Dedekind cuts of the real line. The first mathematician, however, to successfully deal with the dilemma caused by the existence of irrational numbers was Eudoxus of Cnidus(fl 380 BC)whose Theory of Proportions allowed Archimedes to use the so-calledMethod of Exhaustion to rigorously derive results on areas and volumes ofcurved shapes before the advent of calculus some 1,900 years later.

The final piece of the numberjigsaw–the imaginary unit

13. Addition ofcomplexnumbers by adding directedline segments

The arithmetic ofcomplex numbers presents itselfvery nicely in the complexplane. We think of the complex number a+bi as being represented by the point(a, b)in the coordinate plane.When we add two complex numbers z=(a, b)and w=(c, d), we simply add their first and second entries together, to give us z+w=(a+c, b+d). Ifwe make use of the symbol i, we have for example(2+i)+(1+3i)=3+4i.

This corresponds to what is known as vector addition in the plane,where directed line segments(vectors)are added together, top to tail(see Figure 13). We begin at the origin, which has coordinates of(0,0), and in this example we lay down our first arrow from there to the point(2,1). To add the number represented by(1,3),we go to the point(2,1), and draw an arrow that represents moving 1 unit right in the horizontal direction(that is the direction of the real axis), and 3 units up in the direction of the vertical(the imaginary axis). We end up at the point with coordinates(3,4). In much the same way, we can define subtraction ofcomplex numbers by subtracting the real and imaginary parts so that, for example,(11+7i)-(2+5i)=9+2i.This can be pictured as starting with the vector(11,7), and subtracting the vector(2,5), to finish at the point(9,2).

Multiplication is another matter. Formally it is easy to do:we multiply two complex numbers together by multiplying out the brackets, remembering that i2=-1. Assuming the Distributive Law continues to hold, which is the algebraic rule that allows us to expand the brackets in the usual way, then multiplication proceeds as follows:

(a+bi)(c+di)=a(c+di)+bi(c+di)=

ac+adi+bci+bdi2=(ac-bd)+(ad+bc)i

By using general rather than specific complex numbers we can, in the same way,find the outcome of a general division of complex numbers in terms of their real and imaginary parts as we have done above for general complex multiplication. However, as long as the technique is understood, there is no pressing need to produce and to memorize the resulting formula.

14. The position of a complexnumberin polar coordinates

Multiplication has a geometric interpretation that is revealed if we alter our coordinate system from the ordinary rectangular coordinates topolar coordinates. In this system, a point z is once again specified by an ordered pair of numbers, which we shall write as(r,θ). The number r is the distance of our point z from the origin O(called in this context thepole). Therefore r is a non-negative quantity and all points with the same value of r form a circle of radius r centred at the pole. We use the second coordinate θ to specify z on this circle by taking θto be the angle,measured in an anti-clockwise direction, from the real axis to the line Oz. The number r is called the modulus(plural moduli)of z,while the angle θ is called the argument of z.

Suppose nowthat we have two complex numbers, z and w, whose polar coordinates are(r1,θ1)and(r2,θ2)respectively. It turns out that the polar coordinates of their product zw take on a simple and pleasing form. The rule of combination can be expressed neatly in ordinary language:the modulus of the product zw is the product of the moduli of z and w, while the argument of zw is the sum of the arguments of z and w. In symbols, zw has polar coordinates(r1r2,θ1+θ2). The multiplication of the real numbers is subsumed under this more general way of looking at things:a positive real number r, for instance, has polar coordinates(r,0), and ifwe multiply by another(s,0), the result is the expected(rs,0),corresponding to the real number rs.

Much more of the character of the multiplication of complex numbers is revealed through this representation. The polar coordinates of the complex unit i are given by(1,90°).(Normally,angles are not measured in degrees in such circumstances but in the natural mathematical unit of the radian:there are 2π radians in a circle, so that a turn of one radian corresponds to moving one unit along the circumference of the unit circle, centred at the origin. One radian is about 57.3°.)Ifwe now take any complex number z=(r,θ)and multiply by i=(1,90°), we find that zi=(r,θ+90°). That is to say, multiplication by i corresponds to rotation through a right angle about the centre of the complex plane. In other words, the right angle, that most fundamental geometric idea, can be represented as a number.

Indeed, the effect of adding or multiplying by a complex number z on all the points in a given region of the complex plane can be pictured geometrically. Imagine any region you fancy in the plane.If we addz to every point inside your region, we simply move each point the same distance and direction determined by the arrow, or vector as we often call it, represented by z. That is to saywe translate the region to some other position in the plane so that the shape and size are exactly maintained, as is its attitude, by which we mean the region has not undergone any rotation or reflection.Multiplying every point in your region by z=(r,θ)has two effects,however, one caused by r and the other by θ. The modulus of each point in the region is increased by a factor r, so all the dimensions of the region are increased by a factor of r also(so its area is multiplied by a factor of r2). Of course, if r〈1 then this‘expansion’ is better described as a contraction as the new region will be smaller than the original. The region will, however,maintain its shape-for instance, a triangle is mapped on to a similar triangle with the same angles as before. The effect of θ, as we have explained above, is to rotate the region through an angleθ, anticlockwise about the pole. The net effect then in multiplying all points of your region by z is to expand and rotate your region about the pole. The new region will still have the same shape as before but will be of a different size, determined by r, and will be lying in a different attitude as determined by the rotation angle θ.

Further consequences

There are a host of applications ofcomplex numbers, even at the elementary level. The interplay between rectangular and polar representations brings trigonometry into play in a surprising and advantageous way. For instance, a standard exercise for students is the derivation ofimportant identities that now arise very naturally by taking arbitrary complex numbers ofunit modulus(i.e. r=1),and calculating powers using both rectangular and then polar coordinates. Equating the two forms of the answer then reveals a trigonometric equation.

while the same in polar coordinates gives:

equating the real and imaginary parts of the two versions of this one product then painlessly yields the standard angle sum formulas of trigonometry:

Alternatively, the polar form for complex multiplication can be derived using these trigonometric formulas. Indeed, the rule that we have stated here, without proof, for multiplication in polar form is usuallyrst derived from the rectangular form by using trigonometric formulas.

Much more comes quite easily now as the use of complex numbers reveals a connection between the exponential or power function,and the seemingly unrelated trigonometric functions. Without passing through the portal offered by the square root of minus one, the connection may be glimpsed, but not understood. The so-called hyperbolic functions arise from taking what are known as the even and odd parts of the exponential function. To every trigonometric identity there corresponds one of identical form,except perhaps for sign, involving these hyperbolic functions. This can be veried easily in any particular case, but then the question remains as to why it should happen at all. Why should the behaviour of one class of functions be so closely mirrored in another class, dened in so different a manner, and of such different character? Resolution of the mystery is by way of the formula eiθ=cosθ+i sinθ, which shows that the exponential and trigonometric functions are intimately linked, but only through use of the imaginary unit i. Once this is revealed(for it is surprising and is by no means obvious), it becomes clear that results along the lines described are inevitable through performing calculations using the two alternative representations offered by this equation and then equating real and imaginary parts. Without the formula, however, it all remains a mystery.

Complex numbers and matrices

Let us examine some consequences of the revelation that multiplication by i represents a rotation through a right angle about the centre of the coordinate plane. If z=x+iy, we have through expanding the brackets and reordering multiplications that i(x+iy)=-y+ix, so that the point(x，y)is taken to(-y，x)under this rotation;see Figure 15. In this way, multiplication by i can be regarded as operating on points in the plane. This operation enjoys the special property that for any two points z and w and any real number a, we have i(z+w)=iz+iw, and i(aw)=a(iw).

Moreover, if we multiply a real number a by a complex number x+iy,wegeta(x+iy)=ax+i(ay). In terms of points in the complex plane, we have that(x, y)is moved to(ax, ay), or to write it another way, a(x, y)=(ax, ay).

15. Multiplication by i rotates a complex number by a right angle

The kinds ofoperations that enjoy these two properties are known as linear and are ofparamount importance throughout all mathematics. Here, I wish only to draw to your attention to the fact that the effect ofsuch an operation L is determined by its action on the two points(1,0)and(0,1), for let us suppose that L(1,0)=(a, b)and L(0,1)=(c, d). Then for any point(x, y)we have(x, y)=x(1,0)+y(0,1), and so using the properties ofa linear operation we obtain:

L(x, y)=L(x(1,0)+y(0,1))=xL(1,0)+yL(0,1)=

=x(a, b)+y(c, d)=(ax, bx)+(cy, dy)=(ax+cy, bx+dy).

This information maybe summarized bywhat is known as a matrix equation:

Here we have drawn out an example ofmatrix multiplication,which indicates how that operation is carried out in general.A matrix isjust a rectangular array ofrows and columns of numbers. Matrices, however, represent another kind of two-dimensional numerical object and, what is more, they pervade nearly all ofhigher mathematics, both pure and applied.They represent awhole corpus ofalgebra, and much ofmodern mathematics strives to represent itselfthrough matrices, so useful have they proved to be. Two matrices with the same number of rows and the same number ofcolumns as each other are added entry-to-entry:for example, to find the entry in the second row and third column of the sum of two matrices, we simply add the correspondingly placed entries in the two matrices in question. It is matrix multiplication, however, that gives the subject a new and important character, and how it is conducted has emerged ofits own accord in the previous example–each entry in the product matrix is formed by taking the dotproduct ofa row of the first matrix with a column of the second, meaning that the entry is the sum of the corresponding products when the row of the first matrix is placed on top of the column of the second.

Matrices follow all the usual laws ofalgebra except commutativity ofmultiplication, meaning that for two matrices A and B it is not generally true that AB=BA. However, matrix multiplication is associative, meaning that products ofany length may be written unambiguously without the need for bracketing.

Linear transformations of the plane are typically rotations about the origin, reflections in lines through the origin, enlargments and contractions about the origin, and so called shears(or slanting),which move points parallel to a fixed axis by an amount proportional to their distance from that axis in a manner similar to the way the pages ofa book can slide past one another. Any sequence of these transformations can be effected by multiplying all of the relevant matrices together to reveal a single matrix that has the same net effect as all those transformations acting in turn.The rows of the resultant matrix are simply the images of the two points(1,0)and(0,1), as we saw above, known as basis vectors.

It is now natural to look at the matrix J that represents an anticlockwise rotation ofa right angle about the origin as it should mimic the behaviour we see when we multiplyby the imaginary unit i. Since the point(1,0)is taken onto the point(0,1)by the rotation and similarly the point(1,0)moves to(-1,0), these two vectors form the rows ofour matrix J. The result ofsquaring J will be a matrix that has the geometric effect ofrotating points through 2×90°=180°about the origin. We calculate this below by matrix multiplication. To find, for example, the bottom right entry of J2 we take the dot product of the second row and second column, which gives(-1)×1+0×0=-1+0=-1. The complete calculation has the following outcome:

The matrix I with rows(10)and(01)is the identity matrix, so called as it acts like the number 1 in that when multiplied by another matrix A the result is A. The matrix-I, which represents a full half turn rotation about the origin, does behave like-1 in that(-I)2=I. The upshot of all this is that the matrices aI+bJ,where a and b are real numbers, faithfully mimic the complex numbers a+bi with respect to addition and multiplication, and so give a matrix representation of the complex numbereld. The matrix corresponding to the typical complex number a+bi is

The matrices that represent the complex numbers do commute with one another but, as was mentioned above, this does not rebs generally apply to all matrix products and another way in which uNm matrices can misbehave is that not all of them can be‘inverted’.For most square matrices A(a matrix with equal numbers of rows and columns), we maynd a unique inverse matrix B such that AB=BA=I, the identity matrix. The existence of the inverse matrix however depends upon a single number associated with a square matrix known as its determinant. In general, this is a certain sum of signed products formed by taking one entry from each row and column of the array. For the typical 22 matrix array as introduced on page 118, the determinant is the number△=adbc. Determinants have many uses and agreeable properties. For instance,△ represents the area scale factor of the corresponding matrix transformation:a shape of area a will be transformed into one of area△a when undergoing a transformation by that matrix(and if△ is negative, the shape also undergoes a reection, reversing the original orientation). What is more, the determinant of the product of two square matrices is the product of the determinants of those matrices. A square matrix A will have an inverse B except in the case where△=0, in which case it will not. A zero determinant corresponds geometrically to a degenerate transformation where areas are collapsed by the matrix togures of zero area such as a line segment or even a single point.

For the matrix of a complex number z=a+bi, we note that△=a2+b2, which is never zero except when z=0–but of course the number 0 never had a reciprocal before, and that remains the case in the wider arena of the complex numbers. This does conrm however that every non-zero complex number possesses a multiplicative inverse.

We stand here on the edge of the vast worlds of linear algebra,representation theory, and applications to multi-dimensional calculus, and this is not the place to go further. However, the reader should be aware that matrices apply to three dimensions and indeed to n-dimensional space, typically through n×n matrices. Although the arrays become larger and more complicated, the matrices themselves yet remain two-dimensional numerical objects.

Numbers beyond the complex plane

Thefield C of all complex numbers is complete in two important ways. An innite sequence of complex numbers in which the terms cluster into ever smaller circles of radius that approaches 0 is called convergent. Any convergent sequence of complex numbers approaches a limiting complex number. This is also true of the real numbers, but not of the rationals–the successive decimal approximations to any irrational number represent a sequence of rational numbers that approach a limit outside of the rationals.Moreover, C is complete(or closed)in the algebraic sense that it can be shown that any polynomial equation p(z)=a+bz+cz2+…+zn=0 has n(complex)solutions,z1,z2…zn, which then allows p(z)itself to be fully factorized as p(z)=(z-z1)(z-z2)…(z-zn).

This and other stunning successes of the complex numbers largely obviate the need to expand the number system further beyond the complex plane. Indeed, it is not possible to construct an augmented number system that contains C and also retains all the normal laws ofalgebra. Moreover, there are only two extended systems that retain much algebraic structure at all, these being the quaternions and the octonions. Although their use is not nearly so widespread as that of the complex numbers, the quaternions are put to work, for example, in three-dimensional computer graphics.The octonions, which can be thought ofas pairs ofquaternions,lack not only the commutative propertybut also the associative property ofmultiplication.

A quaternion is a number of the form z=a+bi+cj+dk, where the first part a+bi is an ordinary complex number and the two quaternion units j and k also satisfy j2=k2=-1. In order to do multiplication with quaternions, we need to know how the units multiply with one another and this is determined by the rules ij=k, jk=i, ki=j but the reversed products carry the opposite sign, so that, for example, ji=-k(indeed, all these products may be derived from the single additional equation:ijk=-1). The quaternions then form an enhanced algebraic system that satisfies all the laws ofalgebra except for commutativity ofmultiplication,due to the sign changes mentioned above in the reversed products.The consistency of the system can also be demonstrated through representation by 2×2 matrices, but this time we allow complex rather thanjust real entries. The number 1 is once more identified with I, the identity matrix but the units i, j, and khave as their matrix counterparts:

while the typical quaternion z has as its matrix:

This representation of the quaternions by matrices is not unique,however, and indeed the representation of the complex numbers by matrices also has equivalent alternatives. Moreover, it is possible to represent the quaternions without employing complex numbers but only at the expense ofusing larger matrix arrays:the quaternions can be represented by certain 4×4 matrices with only real number entries.

Newkinds of numbers and the extensions of old systems have come about through the need to perform calculations the outcome of which could not be accommodated bythe number system as it stood. Every civilization begins with the counting numbers, but calculations involving fragments lead to fractions, those involving debt lead to negatives, and as Pythagoras discovered, those involving lengths lead to irrational numbers. Although a very ancient revelation, the fact that not all numerical matters could be dealt with using whole numbers and their ratios was a subtle discovery of a deeper kind. As science became more sophisticated,the number systems required have needed to mature in order to deal with these advances. Scientists do not generally look to create new numbers systems in a whimsical fashion. On the contrary,they are introduced often reluctantly and hesitatingly at first, to deal with research problems. For example, although first introduced in the 19th century, matrices arose irresistibly in quantum mechanics in the early 20th century when physicists encountered a quantity of the form q=AB-BA that was nevertheless not zero. In any commutative system of numbers,q would of course be 0, so the numerical objects needed here were not of a kind they had met before:theywere matrices.

It seems nowthat the world of mathematics and physics has enough number types. Although there are kinds of numbers not mentioned in this book, the number types that are commonly used throughout mathematics and science have not needed to change a great deal since the first half of the 20th century.

These observations, however, bring our mathematical balloon ride to its conclusion. We began at ground level and have ascended to where I hope the reader can gaze down upon a view of the rich and mysterious world of numbers.