真实之路 宇宙法则的完全指南 (彭罗斯)4神奇的复数.pdf

4 Magical complex numbers 4 1The magic number i How is it that 1 can have a square root The square of a positive number is always positive and the square of a negative number is again positive and the square of 0 is just 0 again so that is hardly of use to us here It seems impossible that we can Wnd a number whose square is actually negative Yet this is the kind of situation that we have seen before when we ascertained that 2 has no square root within the system of rational numbers In that case we resolved the situation by extending our system of numbers from the rationals to a larger system and we settled on the system of reals Perhaps the same trick will work again Indeed it will In fact what we have to do is something much easier and far less drastic than the passage from the rationals to the reals Raphael Bombelli introduced the procedure in 1572 in his work L Algebra following Gerolamo Cardano s original encounters with complex numbers in his Ars Magna of 1545 All we need do is introduce a single quantity called i which is to square to 1 and adjoin it to the system of reals allowing combinations of i with real numbers to form expressions such as a ib where a and b are arbitrary real numbers Any such combination is called a complex number It is easy to see how to add complex numbers a ib c id a c i b d which is of the same form as before with the real numbers a c and b d taking the place of the a and b that we had in our original expression What about multiplication This is almost as easy Let us Wnd the product of a ib with c id We Wrst simply multiply these factors expanding the expression using the ordinary rules of algebra 1 a ib c id ac ibc aid ibid ac i bc ad i2bd 71 But i2 1 so we can rewrite this as a ib c id ac bd i bc ad which is again of the same form as our original a ib but with ac bd taking the place of a and bc ad taking the place of b It is easy enough to subtract two complex numbers but what about division Recall that in the ordinary arithmetic we are allowed to divide by any real number that is not zero Now let us try to divide the complex number a ib by the complex number c id We must take the latter to be non zero which means that the real numbers c and d cannot both be zero Hence c2 d2 0 and therefore c2 d26 0 so we are allowed to divide by c2 d2 It is a direct exercise 4 1 to check multiplying both sides of the expression below by c id that a ib c id ac bd c2 d2 i bc ad c2 d2 This is of the same general form as before so it is again a complex number When we get used to playing with these complex numbers we cease to think of a ib as a pair of things namely the two real numbers a and b but we think of a ib as an entire thing on its own and we could use a single letter say z to denote the whole complex number z a ib It may be checked that all the normal rules of algebra are satisWed by complex numbers 4 2 In fact all this is a good deal more straightforward than checking everything for real numbers For that check we imagine that we had previously convinced ourselves that the rules of algebra are satisWed for fractions and then we have to use Dedekind s cuts to show that the rules still work for real numbers From this point of view it seems rather extraordinary that complex numbers were viewed with suspicion for so long whereas the much more complicated extension from the rationals to the reals had after ancient Greek times been generally accepted without question Presumably this suspicion arose because people could not see the complex numbers as being presented to them in any obvious way by the physical world In the case of the real numbers it had seemed that distances times and other physical quantities were providing the reality that such numbers required yet the complex numbers had appeared to be merely invented entities called forth from the imaginations of mathemat 4 1 Do this 4 2 Check this the relevant rules being w z z w w u z w u z wz zw w uz wu z w u z wu wz w 0 w w1 w 4 1CHAPTER 4 72 icians who desired numbers with a greater scope than the ones that they had known before But we should recall from 3 3 that the connection the mathematical real numbers have with those physical concepts of length or time is not as clear as we had imagined it to be We cannot directly see the minute details of a Dedekind cut nor is it clear that arbitrarily great or arbitrarily tiny times or lengths actually exist in nature One could say that the so called real numbers are as much a product of mathematicians imaginations as are the complex numbers Yet we shall Wnd that complex numbers as much as reals and perhaps even more Wnd a unity with nature that is truly remarkable It is as though Nature herself is as impressed by the scope and consistency of the complex number system as we are ourselves and has entrusted to these numbers the precise operations of her world at its minutest scales In Chapters 21 23 we shall be seeing in detail how this works Moreover to refer just to the scope and to the consistency of complex numbers does not do justice to this system There is something more which in my view can only be referred to as magic In the remainder of this chapter and in the next two I shall endeavour to convey to the reader something of the Xavour of this magic Then in Chapters 7 9 we shall again witness this complex number magic in some of its most striking and unexpected manifestations Over the four centuries that complex numbers have been known a great many magical qualities have been gradually revealed Yet this is a magic that had been perceived to lie within mathematics and it indeed provided a utility and a depth of mathematical insight that could not be achieved by use of the reals alone There had not been any reason to expect that the physical world should be concerned with it And for some 350 years from the time that these numbers were introduced through the works of Car dano and Bombelli it was purely through their mathematical role that the magic of the complex number system was perceived It would no doubt have come as a great surprise to all those who had voiced their suspicion of complex numbers to Wnd that according to the physics of the latter three quarters of the 20th century the laws governing the behaviour of the world at its tiniest scales is fundamentally governed by the complex number system These matters will be central to some of the later parts of this book particularly in Chapters 21 23 For the moment let us concentrate on some of the mathematical magic of complex numbers leaving their phys ical magic until later Recall that all we have done is to demand that 1 have a square root together with demanding that the normal laws of arithmetic be retained and we have ascertained that these demands can be satisWed consistently This seems like a fairly simple thing to have done But now for the magic Magical complex numbers 4 1 73 4 2Solving equations with complex numbers In what follows I shall Wnd it necessary to introduce somewhat more mathematical notation than previously I apologize for this However it is hardly possible to convey serious mathematical ideas without the use of a certain amount of notation I appreciate that there will be many readers who are uncomfortable with these things My advice to such readers is basically just to read the words and not to bother too much about trying to understand the equations At least just skim over the various formulae and press on There will indeed be quite a number of serious mathemat ical expressions scattered about this book particularly in some of the later chapters My guess is that certain aspects of understanding will eventually begin to come through even if you make little attempt to understand what all the expressions actually mean in detail I hope so because the magic of complex numbers in particular is a miracle well worth appreciating If you can cope with the mathematical notation then so much the better First of all we may ask whether other numbers have square roots What about 2 for example That s easy The complex number i ffi ffiffi 2 p certainly squares to 2 and so also does i ffi ffiffi 2 p Moreover for any positive real number a the complex number i ffi ffiffi a p squares to a and i ffi ffiffi a p does also There is no real magic here But what about the general complex number a ib where a and b are real We Wnd that the complex number ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 1 2 a ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi a2 b2 p r i ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi 1 2 a ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi a2 b2 p r squares to a ib and so does its negative 4 3 Thus we see that even though we only adjoined a square root for a single quantity namely 1 we Wnd that every number in the resulting system now automatically has a square root This is quite diVerent from what happened in the passage from the rationals to the reals In that case the mere introduction of the quantity ffi ffiffi 2 p into the system of rationals would have got us almost no where But this is just the very beginning We can ask about cube roots Wfth roots 999th roots pth roots or even i th roots We Wnd miraculously that whatever complex root we choose and whatever complex number we apply it to excluding 0 there is always a complex number solution to this problem In fact there will normally be a number of diVerent solutions to the problem as we shall be seeing shortly We noted above that for square roots we get two solutions the negative of the square root of a complex number z being also a square root of z For higher roots there are more solutions see 5 4 4 3 Check this 4 2CHAPTER 4 74 Wearestillbarelyscratchingthesurfaceofcomplex numbermagic What Ihavejustassertedaboveisreallyquitesimpletoestablish oncewehavethe notionofalogarithmofacomplexnumber asweshallshortly inChapter5 Somewhat more remarkable is the so called fundamental theorem of alge bra which in eVect asserts that any polynomial equation such as 1 z z4 0 or p iz ffi ffi ffi ffi ffi ffi ffiffi 417 p z3 z999 0 must have complex number solutions More explicitly there will always be a solution normally several diVerent ones to any equation of the form a0 a1z a2z2 a3z3 anzn 0 wherea0 a1 a2 a3 anaregivencomplexnumberswiththeantakenas non zero 2 Here n can be any positive integer that we care to choose as big as we like For comparison we may recall that i was introduced in eVect simply to provide a solution to the one particular equation 1 z2 0 We get all the rest free Before proceeding further it is worth mentioning the problem that Car danohadbeenconcernedwith fromaround1539 whenheWrstencountered complex numbers and caught a hint of another aspect of their attendant magicalproperties Thisproblemwas ineVect toWndanexpressionforthe generalsolution ofa real cubicequation i e n 3in the above Cardano found that the general cubic could be reduced to the form x3 3px 2q by a simple transformation Here p and q are to be real numbers and I have reverted to the use of x in the equation rather than z to indicate that we are now concerned with real number solutions rather than complex ones Cardano s complete solution as published in his 1545 book Ars Magna seems to have been developed from an earlier partial solution that he had learnt in 1539 from Niccolo Fontana Tartaglia although this partial solution and perhaps even the complete solution had been found earlier before 1526 by Scipione del Ferro 3The del Ferro Cardano solution was essentially the following written in modern notation x q w 1 3 q w 1 3 where Magical complex numbers 4 2 75 w q2 p3 1 2 Now this equation presents no fundamental problem within the system of real numbers if q2 p3 In this case there is just one real solution x to the equation and it is indeed correctly given by the del Ferro Cardano formula as given above But if q2 p3 the so called irreducible case then although there are now three real solu tions the formula involves the square root of the negative number q2 p3 and so it cannot be used without bringing in complex numbers In fact as Bombellilatershowed inChapter2ofhisL Algebraof1572 ifwedoallow ourselvestoadmitcomplexnumbers thenallthreerealsolutionsareindeed correctly expressed by the formula 4 This makes sense because the expres sionprovidesuswithtwocomplexnumbersaddedtogether wheretheparts involving i cancel out in the sum giving a real number answer 5 What is mysterious about this is that even though it would seem that the problem has nothing to do with complex numbers the equation having real coeY cients and all its solutions being real in this irreducible case we need to journeythroughthisseeminglyalienterritoryofthecomplex numberworld in order that the formula may allow us to return with our purely real number solutions Had we restricted ourselves to the straight and narrow real path we should have returned empty handed Ironically complex solutions to the original equation can only come about in those cases when the formula does not necessarily involve this complex journey 4 3Convergence of power series Despite these remarkable facts we have still not got very far into complex number magic There is much more to come For example one area where complex numbers are invaluable is in providing an understanding of the behaviour of what are called power series A power series is an inWnite sum of the form a0 a1x a2x2 a3x3 Because this sum involves an inWnite number of terms it may be the case that the series diverges which is to say that it does not settle down to a particular Wnite value as we add up more and more of its terms For an example consider the series 1 x2 x4 x6 x8 4 3CHAPTER 4 76 whereIhavetakena0 1 a1 0 a2 1 a3 0 a4 1 a5 0 a6 1 If we put x 1 then adding the terms successively we get 1 1 1 2 1 1 1 3 1 1 1 1 4 1 1 1 1 1 5 etc and we see that the series has no chance of settling down to a particular Wnite value that is it is divergent Things are even worse if we try x 2 for example since now the individual terms are getting bigger and adding terms successively we get 1 1 4 5 1 4 16 21 1 4 16 64 85 etc which clearly diverges On the other hand if we put x 1 2 then we get 1 1 1 4 5 4 1 1 4 1 16 21 16 1 1 4 1 16 1 64 85 64 etc and it turns out that these numbers become closer and closer to the limiting value 4 3 so the series is now convergent With this series it is not hard to appreciate in a sense an underlying reason why the series cannot help but diverge for x 1 and x 2 while converging for x 1 2to give the answer 4 3 For we can explicitly write down the answer to the sum of the entire series Wnding 4 4 1 x2 x4 x6 x8 1 x2 1 When we substitute x 1 we Wnd that this answer is 1 12 1 0 1 which is inWnity 6and this provides us with an understanding of why the series has to diverge for that value of x When we substitute x 1 2 the answer is 1 1 4 1 4 3 and the series actually converges to this particular value as stated above This all seems very sensible But what about x 2 Now there is an answer givenbytheexplicitformula namely 1 4 1 1 3 althoughwe do not seem to get this value simply by adding up the terms of the series We could hardly get this answer because we are just adding together positive quantities whereas 1 3 is negative The reason that the series diverges is that when x 2 each term is actually bigger than the corresponding term was when x 1 so that divergence for x 2 follows logically from the divergence for x 1 In the case of x 2 it is not that the answer is really inWnite but that we cannot reach this answer by attempting to sum the series directly In Fig 4 1 I have plotted the partial sums of the series i e the sums up to some Wnite number of terms successively up to terms together with the answer 1 x2 1 4 4 Can you see how to check this expression Magical complex numbers 4 3 77 x y Not accessed by series Fig 4 1The respective partial sums 1 1 x2 1 x2 x4 1 x2 x4 x6of the series for 1 x2 1are plotted illustrating the convergence of the series to 1 x2 1for jxj 1 and we can see that provided x lies strictly7between the values 1 and 1 the curves depicting these partial sums do indeed converge on this answer namely 1 x2 1 as we expect But outside this range the series simply diverges and does not actually reach any Wnite value at all As a slight digression it will be helpful to address a certain issue here that will be of importance to us later Let us ask the following question does the equation that we obtain by putting x 2 in the above expression namely 1 22 24 26 28 1 22 1 1 3 actually make any sense The great 18th century mathematician Leonhard Euler often wrote down equations like this and it has become fashionable to poke gentle fun at him for holding to such absurdities while one might excuse him on the grounds that in those early days nothing was properly understood about matters of convergence of series and the like Indeed it is true that the rigorous mathematical treatment of series did not come about until the late 18th and early 19th century through the work of Augustin Cauchy and others Moreover according to this rigorous treat ment the above equation would be oYcially classiWed as nonsense Yet I think that it is important to appreciate that in the appropriate sense Euler really knew what he was doing when he wrote down apparent absurdities of this nature and that there are senses according to which the above equation must be regarded as correct 4 3CHAPTER 4 78 In mathematics it is indeed imperative to be absolutely clear that one s equations make strict and accurate sense However it is equally important not to be insensitive to things going on behind the scenes which may ultimately lead to deeper insights It is easy to lose sight of such things by adhering too rigidly to what appears