Realandcomplexnumbers

The construction of the complex numbers is much simpler and goes much more smoothly than the construction of the real numbers. Thefirst stage in producing the reals is development of the rationals, at which point we have to explain what is meant by a fraction. A fraction, such as 23 is just a pair of integers, which we represent in this familiar but peculiar manner. The idea of fractional parts is not difficult to understand, although the corresponding arithmetic takes real effort to master. Along the way your teachers explain in passing that such fractions as 23,46,69etc. are‘equal’–they are not the same number pairs but they do represent equal slices of pie. This is not hard to accept but it does draw our attention to the fact that a rational number is in reality an infinite set of equivalent fractions, each represented by a pair of integers. This sounds intimidating and we might prefer not to think too much about this, for the prospect of manipulating infinite collections of pairs of integers might leave us feeling uneasy. There is one saving grace in that any fraction has a unique reduced representation where the numerator and denominator are coprime, which can be got by cancelling any common factors in the fraction with which you originally began. Nonetheless, once you are familiar with the properties of fractions and the rules for using them, nothing should go wrong even though closer examination reveals that, as you do your sums, you are implicitly manipulating infinite collections of integer pairs.

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 very basic 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 beingfinite 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 mayfind 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 infinite things as if they werefinite sometimes provided amazing insights and at other times patent nonsense. By putting your mathematical systems on afirm foundation, we can learn how to tell the difference. In practice, mathematicians often indulge in‘formal’manipulations in order to see if some sparkling new theorem is in the offing. If the 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. Thefirst mathematician, however, to successfully deal with the dilemma caused by the existence of irrational numbers was Eudoxus of Cnidus(fl380 BC)whose Theory of Proportions allowed Archimedes to use the so-called Method of Exhaustion to rigorously derive results on areas and volumes of curved shapes before the advent of calculus some 1,900 years later.

Thefinal piece of the number jigsaw–the imaginary unit

13. Addition of complex numbers by adding directed line segments

The arithmetic of complex numbers presents itself very nicely in the complex plane. 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)andw=(c,d), we simply add theirfirst and second entries together, to give us z+w=(a+c,b+d). If we make use of the symbol i,wehavefor 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 ourfirst 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 of complex 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), tofinish 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 complex number in polarcoordinates

Further consequences

There are a host of applications of complex 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 of important identities that now arise very naturally by taking arbitrary complex numbers of unit 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:

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 usuallyfirst derived from the rectangular form by using trigonometric formulas.

Complexnumbersand 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,wehave through expanding the brackets and reordering multiplications that i(x+iy)=-y+ix, so that the point(x, y)istakento(-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,wehavei(z+w)=iz+iw,andi(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)ismovedto(ax,ay), or to write it another way, a(x, y)=(ax,ay).

15. Multiplication by i rotates a complex number by a rightangle

The kinds of operations that enjoy these two properties are known as linear and are of paramount importance throughout all mathematics. Here, I wish only to draw to your attention to the fact that the effect of such 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)andL(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 of a 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 may be summarized by what is known as a matrix equation:

Here we have drawn out an example of matrix multiplication,which indicates how that operation is carried out in general.A matrix is just a rectangular array of rows and columns of numbers. Matrices, however, represent another kind of two-dimensional numerical object and, what is more, they pervade nearly all of higher mathematics, both pure and applied.They represent a whole corpus of algebra, and much of modern mathematics strives to represent itself through matrices, so useful have they proved to be. Two matrices with the same number of rows and the same number of columns as each other are added entry-to-entry:for example, tofind 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 of its own accord in the previous example–each entry in the product matrix is formed by taking the dot product of a row of thefirst matrix with a column of the second, meaning that the entry is the sum of the corresponding products when the row of thefirst matrix is placed on top of the column of the second.

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

Linear transformations of the plane are typically rotations about the origin, reflkections in lines through the origin, enlargments and contractions about the origin, and so called shears(or slanting),which move points parallel to afixed axis by an amount proportional to their distance from that axis in a manner similar to the way the pages of a 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 of a right angle about the origin as it should mimic the behaviour we see when we multiply by 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 of our matrix J. The result of squaring J will be a matrix that has the geometric effect of rotating points through 2×90°=180°about the origin. We calculate this below by matrix multiplication. Tofind, 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.Thematrix-I, which represents a full half turn rotation about the origin, does behave like-1in 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 numberfield. 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 generally apply to all matrix products and another way in which 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 mayfind 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 2×2matrix array as introduced on page 118, the determinant is the number=ad-bc. 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 reflection, 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,inwhich case it will not. A zero determinant corresponds geometrically to a degenerate transformation where areas are collapsed by the matrix tofigures of zero area such as a line segment or even a single point.

For the matrix of a complex number z=a+bi,wenotethat=a2+b2,whichisneverzeroexceptwhenz=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 confirm 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.

Numbersbeyondthecomplexplane

Thefield C of all complex numbers is complete in two important ways. An infinite 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=0hasn(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 of algebra. 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 of as pairs of quaternions,lack not only the commutative property but also the associative property of multiplication.

A quaternion is a number of the form z=a+bi+cj+dk,where thefirst part a+bi is an ordinary complex number and the two quaternion units j and kalso 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 of algebra except for commutativity of multiplication,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 than just real entries. The number 1 is once more identified with I, the identity matrix but the units i, j,andk have as their matrix counterparts:

while the typical quaternion z hasasitsmatrix:

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 of using larger matrix arrays:the quaternions can be represented by certain 4×4matriceswith only real number entries.

New kinds 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 by the 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 atfirst, to deal with research problems. For example, althoughfirst 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-BAthat 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:they were matrices.

It seems now that 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 thefirst 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.