Lecture 2: The real umbers The purpose of this lecture is for us to develop the real umber system. This might seem like a very strage thig for us to be doig. It must seem to you that you have bee studyig real umbers most of your life. However, some itrospectio is likely to reveal that ot everythig you have bee told about the real umbers is etirely believable. (As a example, a recet 7th grade textbook explais that to add ad multiply ratioal umbers, you should follow a set of rules you have bee give. To add ad multiply reals, you should plug them ito your calculator.) Because the real umbers will be the cetral focus of iquiry i this course, we will take this momet to specify exactly what they are. The cetral teet of mathematics is that oe must always tell the truth, ad oe ca t be sure that oe is doig this about real umbers, uless oe is sure exactly what they are. There is more tha oe possible approach to doig this. Most mathematicias treatmet of this (see Apostol s book, or Diakar Ramakrisha s otes) focus o what oe expects to do with real umbers. Oe is give a set of axioms to cover this. It should be possible to perform basic arithmetic o the reals (the field axioms), there should be a way of comparig the size of two real umbers (the total orderig axiom) ad a lot of limits should exist (the least upper boud property). After oe has writte dow these axioms, oe is a good positio to start provig theorems about the real umbers. But oe might be quite cofused about what is goig o? Are these really the same real umbers I ve always heard about? Are the real umbers the oly set of umbers satisfyig these axioms? What are idividual real umbers like? It is possible with some work to proceed from these axioms to aswer that questio, but the work is o-trivial. We will take a slightly differet approach. We will describe the real umbers i much the way they were described to you i grade school, as decimal expasios. (Mathematicias ted ot to like this because there are arbitrary choices like the choice of the base te.) The because we d like to use the real umbers, we will check that they satisfy the axioms allowig us to order them, take limits, ad do arithmetic. It will tur out that doig arithmetic is the hardest part. (There s a reaso you eed a calculator!) While you have bee traied that oe ca do arithmetic i real umbers sice log before you had Calculus, i order to actually be able to perform ay arithmetic operatio o geeral real umbers, you have to take limits. Before we start, perhaps a few words are required about the usefuless of this. You are right to be cocered. As scietists ad egieers pursuig practical objectives, you will ot ecouter ay typical real umber. Sure, you might collect some data. But it will come to you as floatig poit umbers with implicit error itervals. Why the should we study somethig so abstract, so idealized, dare I say it so ureal as the real umbers? The aswer is that quite happily, the processes which we use to draw coclusios about real umbers ad especially to study limits of them (the mai subject of this course) are exactly the same as those used to rigorously study floatig poit umbers with error itervals. It 1
might be wise to take this viewpoit about the whole course. But this requires thikig differetly tha oe is used to about what are the mai questios. Now we begi formally. What is a real umber? It will be a expressio of the form ±a 1 a 2 a 3... a m.b 1 b 2 b 3.... Here the ± represets a choice betwee plus ad mius. The digit a 1 is a iteger betwee 0 ad 9 iclusive (uless m is differet from 1 i which case it is restricted to beig betwee 1 ad 9, sice it is the leadig digit.) All other digits are itegers betwee 0 ad 9 iclusive. Oe detail is that some real umber have two such represetatios. Namely a termiatig decimal ±a 1 a 2... a m.b 1 b 2... b 000..., where here b is differet from 0, is the same as ±a 1 a 2... a m.b 1 b 2... (b 1)999..., a decimal with repeatig 9 s. (Note that the repeatig 9 s could start to the left of the decimal place just as well as to the right.) The set of real umbers, we will ivariably refer to as R. Hopefully, we have ow described the real umbers as you have see them sice grade school. It is ofte poited out that they ca be visualized as populatig a lie. You ca do this by first markig of the itegers at equal distaces o the lie. The the iterval betwee ay two cosecutive itegers ca be cut ito te equal subitervals. The value of the first digit after the decimal describes which iterval the real umber lies i. Oe cotiues the process, subdividig each of those te itervals ito te equal subitervals ad so o. Whe dealig with the real umbers i practice, we very ofte approximate to a few decimal places. Stragely, there is o stadard otatio for this, so we itroduce some. Give a real umber x = ±a 1 a 2... a m.b 1 b 2... b b +1 b +2..., we defie t (x), the trucatio to decimal places, as t (x) = ±a 1 a 2... a m.b 1 b 2... b. I order for t to be a well defied fuctio o the reals, we must specify how it acts o reals with two decimal represetatios (the case of repeatig zeroes ad repeatig ies). We specify that to apply t, we always take the represetatio with repeatig zeroes. Thus give ay real umber, we uiquely map it with t to a termiatig decimal, which we ca also view as a ratioal umber with deomiator. We ote that as icreases with x fixed, the trucatio t (x) icreases. We are ow ready to defie iequalities amog real umbers. Give two real umbers x ad y, we say that x y if x = y or there is some for which t (x) > t (y). (Ask yourself 2
why we eed the iequality betwee the trucatios to be strict.) Whe preseted with a ew defiitio, it is ofte valuable to thik about it i terms of algorithms. How do we check if the umber x is greater tha or equal to the umber y. If x is actually greater, the we ll fid out i a fiite umber of steps as we fid a decimal place where the trucatio of x is actually bigger. If x ad y are equal, we ll ever fid out, because we have to check all the trucatios. While at first, this seems a uhappy state of affairs, it actually agrees with our itutio about approximatios ad error itervals. If two approximatios are far apart so that their error itervals are disjoit, we ca tell which oe is bigger. Otherwise, we re ot sure. Already, we see that i this way, that the real umbers which are a idealizatio, model reality well. We ow state as a propositio, that ay two real umbers ca be ordered. Propositio 1 Give two real umbers x ad y, the x y or y x. Proof: If for some, we have t (x) > t (y) or t (x) < t (y), the we re doe. The oly case remaiig is that t (x) = t (y) for all. I this case, x ad y have the same decimal expasio ad are therefore the same umber. I this case, both x y ad y x hold. Thus we have completed oe third of our project for defiig the real umbers. They are ordered. Decimals are i fact quite helpful i the orderig which is basically alphabetical. (A more techial term for this kid of orderig is lexicographic.) We are ow prepared to establish the least upper boud property for the real umbers. Give a set A of real umbers, we say that a real umber x is a upper boud for A if for every y A, we have that x y. We say that x is the least upper boud for A if for every other upper boud z for A, we have that x z. We are iterested i least upper bouds as a kid of upper limit of the real umbers i the set A. A upper boud might miss beig i A by a great deal. A least upper boud is just outside of A. Propositio 2 Ay oempty set of real umbers A which has a real upper boud, has a least upper boud i the reals. Proof We are give that A is oempty. Let z be a elemet of it. We are give that it has a upper boud y. Now we are goig to fid the least upper boud x by costructig its decimal expasio. Sice y is a upper boud, so is a = t (y) + 1. The umber a is a upper boud which also has a decimal expasio which termiates at the th place. Moreover a > t (z). I fact (a t (z)) is a positive atural umber. We let B be the set of all atural umbers of the form (c t (z)) with c a upper boud for A with decimal expasio termiatig at or before the th place. This set B is a oempty set of atural umbers which serves as a proxy for the set of upper bouds for A which termiate at decimal places. To the set B, we may apply the Well Orderig Priciple which we proved i the first lecture. The set B has a smallest elemet, b. Thus b + t (z) is the smallest upper boud for A with a -place decimal expasio. We 3
defie x = b + t (z). Thus x just misses beig a upper boud. If after some fiite, all x m with m > are the same, we let x be this x m. Otherwise, we let x be the real umber so that t (x) = x. (We used the well orderig priciple to costruct the decimal expasio for x. (Questio for the reader: Why did we treat the case of x with a termiatig expasio separately. Hit: it was because of our defiitio for t.) The above proof may be a little hard to digest. To uderstad it better, let us cosider a example. Ofte, it is touted that oe of the virtues of the real umber system is that it cotais 2. (You should be a little cocered that we have t defied multiplicatio yet, but this example ca be viewed as motivatio for the defiitio.) How do we see that the real umbers cotai 2? We fid a least upper boud for all umbers whose square is less tha 2. We do this i the spirit of the above proof. First, we fid the small umber with oe decimal place whose square is more tha 2. It is 1.5. We subtract.1 ad record 1.4. The we fid the smallest umber with two decimal places whose square is larger tha 2. It is 1.42. We subtract.01 ad record 1.41. Gradually, we build up the decimal expasio for 2, which begis 1.414213562. Our algorithm ever termiates but we get a arbitrarily log decimal expasio with a fiite umber of steps. The least upper boud property, while it is easy to prove usig the decimal system, is a pretty sophisticated piece of mathematics. It is a rudimetary tool for takig limits, somethig we do t cosider i school util we take Calculus. Addig ad multiplyig, though, is oe of the first thigs we thik of doig to real umbers. Perhaps we wat a calculator to hadle it but we imagie that othig facier is goig o tha our usual algorithms for addig ad muliplyig. Let s cosider how this works. Let s say I wat to add two typical real umbers. I write out their decimal expasios oe above the other. The I start at the right. Oops. The umbers have ifiite decimal expasios, so I ca ever get to the right. This problem is ot easily waved away. Through the process of carryig, quite isigificat digits of the summads ca affect quite sigificat digits of the sum. I order to calculate, as a practical matter, a arithmetic operatio performed o two umbers, we have to take a limit. Luckily, we have established the least upper boud property. We ca use it to defie the arithmetic operatios o the reals. Let x ad y be two oegative real umbers. We let A = {t (x) + t (y)} be the set of sums of trucatios of x ad y. We let M = {t (x)t (y)} be the set of products of trucatios of x ad y. Note that both sets have upper bouds. (We ca use t (x) + t (y) + 2 as a upper boud for A (why?) ad (t (x) + 1 )(t (y) + 1 ) as a upper boud for M. (Why?) Now we apply the least upper boud property to see that A ad M have least upper bouds. We defie x + y to be the least upper boud for A ad xy to be the least upper boud for M. We restricted to x ad y positive, so that the expressios t (x) + t (y) ad t (x)t (y) are icreasig i so that the least upper bouds are really what we wat. Sice we have so far oly defied additio ad multiplicatio for positive umbers, defiig subtractio of positive umbers seems a high priority. Agai give x ad y oegative real umbers. We defie S = {t (x) t (y) 1 }. We subtracted 1 4
from the th elemet so that while we are replacig x by a uderestimate t (x), we are replacig y by a overestimate t (y)+ 1 ad whe we subtract, we have a uderestimate for the differece. We defie x y to be the least upper boud of S. What about divisio? Let x ad y be oegative real umbers. Let D = {z : x yz}. Thus D cosists of real umbers we ca multiply by y to get less tha x. These are the uderestimates of the quotiet. We defie x y to be the least upper boud of D. So how are we doig? We have defied the real umbers i a way that we recogize them from grade school. We have show that this set of real umbers has a order, that it satisfies the least upper boud property ad that we may perform arithmetic operatios. Mathematicias might still ot be etirely satisfied as these arithmetic operatios still must be prove to satisfy the laws they should iherit from the ratioal umbers. This is ot quite as easy as it looks. For istace, let s say we wat to prove the distributive law. Thus if x, y, ad z are oegative real umbers, we would like to show that {t (x + y)t (z)} has the same least upper boud as {t (xz) + t (yz)}. It is true ad it ca be doe. But to do it, it really helps to deal carefully with somethig we have completely set aside thus far. It helps to have estimates o how far away a trucated versio of (x + y)z actually is from the least upper boud. This gets at a objectio that a practical perso could have for the way we ve defied our operatios thus far. Certaily, the least upper bouds exist. But they are the output of a algorithm that ever termiates. To actually use real umbers as a stad i for approximatios with error itervals, we eed to be able at each step of a ever termiatig algorithm to have cotrol o the error. Notice we did have that kid of cotrol i the example with 2. Whe we had decimal places, we kew we were withi of the aswer. I the ext lecture, we will get at both the practical ad theoretical versios of this problem by itroducig the defiitio of the limit. We will see that uderstadig that a limit exists is more tha kowig what the limit is. It also ivolves estimatig how fast the limit coverges. I practical terms, this meas calculatig a error iterval aroud the limitad. 5