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  • College Algebra – Lecture 1 – Numbers

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    Watch at: 00:00 / 00:00:20Watch at: 00:20 / 00:40in this unit which I'm calling basics orsubtitled remembrance of things pastwe'll look at some of the basic ideasthat are needed for the rest of thecollege algebra course we'll look atnumbers which you can think of as theraw materials for the course but theyare not what the course is about thecourse is really about the operationsWatch at: 00:40 / 01:00that you perform on numbers we'll lookat the language of mathematics a littlebit we'll look at exponents and exponentnotation we'll look at polynomialexpressions and we'll finish up with alittle bit of geometry so I hope youenjoy it and here we areWatch at: 01:00 / 01:20the first part of basics we'll look atour numbers and under numbers the firstthing we'll look at is an idea ofmathematics that will help us describenumbers we'll talk about sets of objectsfirst of all so let's look at that setsWatch at: 01:20 / 01:40of objects now sets are what amathematician calls a bunch of thingsnow what will be in such a set or abunch or a collection numbersexpressions functions all these thingsthat we'll be talking about later in thecourse but let me go ahead and give youWatch at: 01:40 / 02:00an example for example for example let'slook at the set of base 10 digits thoseare the digits we use to write all ofour numbers in our number numeral systemand I'll give you the notation as we goI'll use this curly brackets or bracesto start the set and then I'll list allthe digits inside with commas so we'reWatch at: 02:00 / 02:20four five six seven eight and nine andthen end the brackets so these beginningand ending items are called set bracketssometimes they're called braces and theyare what indicate the beginning in theWatch at: 02:20 / 02:40end of the set all of the elements ofthe set there's so few of them I cansimply list them all the elements areseparated musi by commas this is anexample of a finite setit's a set with only a finite number ofelements in fact you can see that thereare exactly ten elements and what we'veWatch at: 02:40 / 03:00done is we put it all together in oneplace it's a collection of objectsthat's all that a set is nothing nothingfancy just a simple idea let's look at acouple more examples here's an examplethis should be familiar to anyone thatspeaks English the set of 26 EnglishWatch at: 03:00 / 03:20letters I will write that starting offwith the braces again and then list theletters a B C and D and you know thereare so many that I don't want to listthem all so what I will do is I will putdot dot dot in the middle and then acomma and then I will list the last fewWatch at: 03:20 / 03:40XY and Z these dots sometimes referredto as an ellipsis is a way of indicatingthat the set continues on and in thiscase I happen to know the ending so Icannot put on the the last few elementsthis again is another finite set now asWatch at: 03:40 / 04:00you might expect in mathematics thereare lots and lots of sets that are notfinite we call those infinite sets solet me give you an example of one ofthose and yet again show you thisnotation example here is another setthis is the set of positive even numbersWatch at: 04:00 / 04:20now as we go on we'll talk about numbersin detail but I think you probablyremember what these are numbers like 2 46 8 10 etc now you see I put the dots atthe end and I put nothing afterwards Idon't know how it ends because there'sWatch at: 04:20 / 04:40an infinite number of even numbersthere's always more beyond wherever Istop so this is a perfect example of aninfinite set and this is one way ofwriting down such a set we will learnother ways as we go but I wanted to showyou both finite and an infinite set nowthere's one more idea that we want toWatch at: 04:40 / 05:00touch on before we start talking aboutnumbersand here I'll show you in an examplewhat I mean here is a small set the settwo four sixthere's a set with only three elementsin it this is what's called this iswhat's called a subset of that largerWatch at: 05:00 / 05:20set we saw on the other page 2 4 6 8 10etc the infinite set of real evennumbers you see that these threeelements are contained in this set thisis a smaller set this is a sub set ofthe larger set so if you were to look atWatch at: 05:20 / 05:40this in a picture say if this shapedescribes the set then what we have hereis a small part of that set called a subset now I will use that terminologythroughout the course and if we need tosay anything more about sets and sets ofobjects we'll introduce that terminologyas we go but now let's go back to theWatch at: 05:40 / 06:00list and see what kind of numbers wewill first look at will first look atnatural numbers these are sometimescalled the counting numbers what arethey natural numbers these are numbersthat we really take for granted theseWatch at: 06:00 / 06:20are numbers that we didn't have toinvent as we have had to event othernumbers these numbers can be thought ofas just existing they are often denotedthe set is often denoted by a boldface Nand that's the way I'll write that andwhat is this set well its consists ofthese numbers 1 2 3 4 etc and thenWatch at: 06:20 / 06:40sometimes we do this also we'll put an Nhere to indicate the variable that Iwill use or the letter I will use tostand for such numbers and then a commaand continue on out to infinity so thisis a commonly used variable for naturalnumbers and I'll use it from time totime these will be taken as given theseWatch at: 06:40 / 07:00are not invented there's nothing todescribe these are simply the startingplace for the number systems we willlook at but you know there are lots ofnumbers within this set of numbers thatare quite that are that are quiteinteresting so let's go ahead and talkabout a few of thembecause they will have bearing laterWatch at: 07:00 / 07:20there's a set of numbers that are calledprime numbers now why would I want totalk about those well as in chemistry itis helpful to look at what you mightconsider the atoms of the material thatyou are discussing prime numbers can beWatch at: 07:20 / 07:40considered the atoms of the naturalnumbers for example let me go ahead anddefine what a prime number is P is primecall it a prime number if well first ofall P is greater than 1 because 1 is notWatch at: 07:40 / 08:00considered a prime number and what makesa prime number and atom P is divisibleonly by two things only by 1 whichdivides into every number and itself PWatch at: 08:00 / 08:20what do these numbers look like well I'msure you've seen these in your life theset of prime numbers looks like this thefirst one is 2 which happens to be theonly even prime number then 3 5 7 not 9because remember 9 would be 3 times 3 11Watch at: 08:20 / 08:4013 17 etc there are an infinite numberof primes and this is what they looklike numbers like thisthey're only divisible they can only befactored words that I will be describingsoon into themselves times 1 so thoseare prime numbers now with prime numbersWatch at: 08:40 / 09:00since we can think of them as the atomsof the natural numbers you might expectthat we ought to be able to write allnatural numbers in terms of primenumbers and so we can and this is such afamous result I just want to go aheadand introduce it to you it's called thefundamental theorem fundamental theoremWatch at: 09:00 / 09:20of arithmetic arithmetic you may ask whyare we studying arithmetic in an algebracourseas you'll see it really is going to bethe basis for much of what we do andit's probably a theorem that you haven'tseen in this form but I think you knowwhat it's going to say any naturalWatch at: 09:20 / 09:40number that is to say any of thosecounting numbers we saw before anynatural number can be factored now Ihaven't officially defined the wordfactor but a factor is simply a thingthat is multiplied so to factor a numberWatch at: 09:40 / 10:00like 12 would be to write it as aproduct of two other numbers it can befactored uniquely there's only one wayyou can factor it and that one way isexcept for the order of the factorsWatch at: 10:00 / 10:20except for the order of the factorsbecause as you may remember from yourknowledge of the numbers that numberslike two times three are the same thingas numbers three times two into aproduct of prime powers prime numbersWatch at: 10:20 / 10:40prime powers so to read to say this onemore time any natural number can befactored uniquely except for the orderof the factors into a product of primepowers now let me give you some examplesso you'll know what it is I'm sayinghere and once I do I think you'll seeWatch at: 10:40 / 11:00that this is familiar let me take thenumber I mentioned a moment ago twelvetwelve as you know can be written asthree times four that may be the firstthing that comes to mind but four can bewritten as two squared so this can bewritten as 2 squared times 3 now 2 is aprime number to a power this is twoWatch at: 11:00 / 11:20times two that's what the two up theremeans and 3 is another prime number letme do a few more examples 70 can bewritten as 2 times 5 times 7for example 111 can be written as 3times 37 etc all the natural numbers canbe written as products of prime numbersWatch at: 11:20 / 11:40or possibly powers of prime numbersand as I said before and we'll define itagain in a little bit factors are thingsthat are multiplied now before I leavethis topic I don't want to mislead yougenerally factoring numbers is hard ifWatch at: 11:40 / 12:00you start with a very large number andyou try to factor it is very hard to doin fact that's being used in securitysystems these days for that very reasonbut the kind of factoring we will do isfairly simple and so I'll pause now andWatch at: 12:00 / 12:20give you a chance to remind yourself ofa few things now we'll continue ourdevelopment of the numbers by looking atintegers so what are the integers theWatch at: 12:20 / 12:40integers is a larger set than thenatural numbers the integers are oftendenoted by a capital Z you may say whatdoes that have to do with integersthere's no Z in the word integer welljust for the historically minded amongyou this comes from the German words allWatch at: 12:40 / 13:00N which stands for numbers so that'swhat the symbolism comes from and whatare the integers how do they differ fromthe natural numbers well the way I'mgoing to define the integers is to giveyou a sense of why one would need thenumbers so let's let imagine that we'removing ahead in the course here and wewant to solve a certain very simpleWatch at: 13:00 / 13:20equation suppose we want to solve theequation X plus 1 equals 1 we want tofind a number X that will make thisequation true well you know from yourown experience that there is only onenumber that will make this true we needwhat we need the number 0 but in theWatch at: 13:20 / 13:40natural numbers there is no 0 so 0 hasto be added to that and that will becomeone of the integers let me show you oneother small equation to suggest why weneed another set of numbers suppose wewant to solve X plus 1 equals 0using that new number zero now from whatWatch at: 13:40 / 14:00you know what would you have to add toone to get zero well you'd have to addwe need a number like minus one butagain the natural numbers don't have anynegative numbers in them so we have toadd these numbers in so when we add themWatch at: 14:00 / 14:20all in we're going to get a set that isthe integers here is that set the set Zand it will be infinite in bothdirections so I'll have to start offwith dots and then I'll give you a fewof the numbers minus 4 minus 3 minus 2minus 1 then there's that special numberWatch at: 14:20 / 14:400 so I'll put it on its own line aloneand then we'll continue 1 2 3 4 etc outto infinity now there's a more compactway to write this so I'll show you thatif we put 0 at the front and then we puta plus and a minus sign here for 1 so weWatch at: 14:40 / 15:00can imagine plus 1 and minus 1 beingnext to each other and then plus orminus 2 plus or minus 3 plus or minus 4etc out to infinity now this is justmore compact so this is a more compactway of writing this set but we have nowexpanded our set of numbers we no longerWatch at: 15:00 / 15:20have simply the natural numbers noticethat the natural numbers are right herethey're included in the integers we haveadded 0 and we've added all thesenegative numbers and a reasonable way ofdescribing the reason for that is to saythat we needed to solve certain kinds ofequations so let me show you a couple ofthings in an example about these numbersWatch at: 15:20 / 15:40these are things that you probably haveseen much of your life I'll just remindyou of a couple of things for example ifI take 0 times 5 and notice I've alreadystarted introducing a notation that youmay not be familiar with I'm using a dotfor multiplication again I will talkWatch at: 15:40 / 16:00about that a little bit later on but youcan see that I'm multiplying 0 times 5well you know what happens when youmultiply a number times 0 you get 0 sothere's a fact from your past I hope andif we take 0 times82 we're going to get zero it doesn'tmatter what number you choose zero timesthat number is zero what else can we sayhow about zero plus a number like zeroWatch at: 16:00 / 16:19plus sevenwell that is seven the zero contributesnothing so you know that adding zero toany number will result in the originalnumber coming back how about negativenumbers what are some of the propertiesyou probably have seen before wellsuppose we take two numbers like minus 5and minus 2 and multiply them togetherWatch at: 16:19 / 16:40well you'll get 10 in fact to get plus10 because a negative times a negativeis a positivewhat about an addition problem takeminus 5 plus 2 well as you know 2 minus5 will give me a minus 3 etc so theWatch at: 16:40 / 17:00kinds of facts that you you should befamiliar with are the facts about theelementary integers and the propertiesthey have so let's go ahead and look atanother set of numbers that is builtfrom integers so we'll go back to myWatch at: 17:00 / 17:20list and after integers we will look atrational numbers this is where we takeintegers and we create a new set ofnumbersand I'll show you exactly how that'sdone rational numbers now rationalnumbers have their own bold-faced letterWatch at: 17:20 / 17:40symbol Q and this time the Q stands forquotient so that seems to make goodsense what does a rational number looklike well then what you previouslycalled in grammar school fractions Pover Q that is a rational number whatare the P and Q the P and the Q areWatch at: 17:40 / 18:00integers so they come from that set Zthat we saw a moment ago and we also saythat Q is not 0now we're forbidding the bottom to be 0the bottom of the fraction is not 0 andI'll explain why in a little bit butwe're going to say the division by 0 isnot allowed it is undefined there's aWatch at: 18:00 / 18:20good reason for that but this of courseis what you call before asset a fraction you might also have usedthe word ratio which is where rationalcomes from in the description of theseand now let's look at a few examplesjust to remind you what we're talkingabout here examples you know what theseWatch at: 18:20 / 18:40are 1/2 2/3 1113 574 over 201 etc thoseare all examples of rational numbersthey're fractionssome of them as in the case of the lastone are improper fractions the top isbigger than the bottom but there's stillrational numbers now let me note thatWatch at: 18:40 / 19:00once again we have increased our numbersystem without losing anything all n isin queue in fact all Z the integers allof that set is in queue how do I knowWatch at: 19:00 / 19:20that wellsince if you take a number n from eitherZ or the natural numbers and you put itover 1 lo and behold you have a rationalnumberbut of course n over 1 is still N and sothese numbers from here can bere-written as n over 1 and they appearas rational numbers and for example 8Watch at: 19:20 / 19:40over 1 of course is 8 and so on sothat's what rational numbers look likenow I said a moment ago that division byzero is not allowed in the natural inthe in the rational numbers so let me goahead and give you that warning and giveyou a reason why that is true so that weWatch at: 19:40 / 20:00don't have any mysteries here warningdivision by zero is what's calledundefinednow you might say well it's undefinedwhy doesn't somebody define it well as aWatch at: 20:00 / 20:20reason it's undefined because you cannotdefine it as a number and end up withouta contradiction so the answer is why thequestion rather is why why is itundefined here's why let us supposelet's take a simple case because if itfails here it's probably going to failfor the rest suppose 1 over 0 and I'llWatch at: 20:20 / 20:40put it in quotes suppose 1 over 0 istaken to be a number well if one over 0is taken to be a number it must shareall the properties that the othernumbers we've seen satisfy that is goingWatch at: 20:40 / 21:00to lead us to a problem watch thenconsider the following situation andthis is just one of many possiblesituations I could describe consider thefollowing 0 times this news purportednumber 1 over 0 now I can make twoWatch at: 21:00 / 21:20different arguments on the one hand Ican say this must be 0why here's my reason since as we saw amoment ago 0 times x equals 0 for anynumber that we've seen X you multiply aWatch at: 21:20 / 21:40number by 0 it is 0 that's what I'vedone here multiplied this purportednumber by 0 so I claim it's 0 fine onthe other hand I can also make thefollowing argument that this is equal to1 and what's my reason there sincecopying what I have here if you have xtimes 1 over X that's the same as x overWatch at: 21:40 / 22:00X and of course that's 1 for any X wellnow we have the contradictionif I assume that you can divide by zeroand that one over zero is a legitimatenumber then I end up with itbeing multiplied by zero I get eitherWatch at: 22:00 / 22:20zero or one I can't have both so thereis no way no matter what I define thisto be I will end up with a problem likethis that's a contradiction so one overzero can't be defined because if youdefine it you're going to havecontradictions arise and you don't wantWatch at: 22:20 / 22:40that in your number system so in caseanyone was wondering why division byzero is not allowed there's the reasonalright let's look at some operations onrational number rational numbers andI'll say some operations on Q and theseWatch at: 22:40 / 23:00are fraction operations these are thekind of things you've had experiencewith over the course of your elementaryschool career but now we'll put them ina form that's more algebraic for examplesuppose you have the fraction a over Btimes and other fractions C over D whatdoes that simplify to how do youmultiply two fractions well this is theWatch at: 23:00 / 23:20easiest of the operations this one'seasyyou multiply the top a C and over thebottom B D now there are a couple ofthings going on here with notation thatwe'll have to discuss later I'm usingthe dot here again for multiplicationand also over here I'm using nothing I'mWatch at: 23:20 / 23:40just putting the numbers next to eachother for multiplication so I have ACover BD what about divisionwhat if I took the fraction or rationalnumber a over B and I wanted to divideit by the fraction or rational numbers Cover D how is that done well this is howit's done you just copy over the top oneWatch at: 23:40 / 24:00a over B and then we're going tomultiply it times the bottom one butwe're going to change the bottom one towhat's called its reciprocal now that'sa fancy older word it just means flip itoverWatch at: 24:00 / 24:20so if it's C over D here we'll multiplyit by D over C and then of course thislooks just like with the case above sothis becomes ad over BC and that's howyou divide two rational numbers the topone just stays the bottom one comes overmultiplies and gets flipped over so thatWatch at: 24:20 / 24:40should be something that you've seenbefore and we've covered two operationsnow multiplication and division thereare two more there's the addition andsubtraction so let's just go ahead forthe sake of completeness look at let'slook at a over B plus or minus C over Dnow of all of the operations this is theWatch at: 24:40 / 25:00hardest these are the hardest becausebut the reason is that a over B and Cover D are divisions in themselves andpluses and minuses are different kind ofoperation so we have to decide how we'regoing to do this and the way that thisis done is we get what's called a commondenominator we make the fractions haveWatch at: 25:00 / 25:20the same number at the bottom and theeasiest way to do this is to take a / bmultiply it by whatever i need on thebottom and i'm going to take the productof these two numbers as my newdenominator so I need a d down there andsince I don't want to change anythingI'll multiply a over B times D over Dremember D over D is 1 so I don't changeWatch at: 25:20 / 25:40anything then plus or minus C over D Ido the same thing except I multiply by Bover B the advantage is now the bottomis BD and the bottom is BD here so Iwill get a new fraction with thedenominator or a bottom BD and the topWatch at: 25:40 / 26:00will be ad plus or minus BC and that ishow you add or subtract two fractionsthis number at the bottom here by theway is often called a common denominatorthere's one of those older wordsdenominator we'll just call it bottomWatch at: 26:00 / 26:20for the most part and likewise we'llrefer to the tops of the fractionsinstead of numerators we'll call themtop most of the timealright let me go ahead and do a fewnumerical examples just to give you achance to warm up a little bitexample if I take two fifths and IWatch at: 26:20 / 26:40divide it by 1/7 remembering the processI copy down two fifths and then Imultiply it by this fraction flippedover that's seven over one and so I willhave fourteen fifths and so I've dividedtwo fractions and ended up with anotherfraction similarly I can do a simpleWatch at: 26:40 / 27:00addition maybe take 1/3 plus 1/5 I takethe 1/3 I multiply the top and bottom by5 over 5 plus one-fifth multiply top andbottom by 3 so my denominator will be 3times 5 or 15 in both cases have a newfraction the bottom is 15 the top is 5Watch at: 27:00 / 27:20plus 3 so I end up with the fraction 815 now you'll have a chance in a littlebit to practice some of this but theseshould be simple operations I hopeanother note I want to make before wehave a chance to stop note in thisWatch at: 27:20 / 27:40example I will illustrate another ideasuppose you're faced with the fraction 4sixths well that is more complicatedthan it needs to be if you recognizethat there's a factor common to the topand the bottom you can then remove it asfollows if you realize the top is 2Watch at: 27:40 / 28:00times 2 and the bottom is 3 times 2 andof course here's where factoring comesin if you can do it you see that whatyou've got is a new fraction here 2 over2 which is just 1 so this can simply berewritten as 2 over 3 this is niceWatch at: 28:00 / 28:20because now 2 & 3 have no factors incommon they are in fact prime numbersand that's a much nicer fraction thanthe original and when you do thisthis whole process is called put intowhat's referred to as lowest termsWatch at: 28:20 / 28:40put this into lowest terms and that justmeans you find out if they have commonfactors the top and the bottom and yourewrite them and those common factorsit's called cancelling but of course yourealize is just rewriting 2 over 2 asthe number 1 all right one final noteWatch at: 28:40 / 29:00for rational numbers before we move onto the next set of numbers my final notehere on rationals is do not here I'mgoing to give you a stricturedo not use mixed numbers or mixed numberWatch at: 29:00 / 29:20notation this is a notation that's verycommon in the elementary schools likefor example 3 and 1/4 and I'll tell youwhy because 3 and 1/4 means 3 plus 1/4Watch at: 29:20 / 29:40that's what you mean when you write down3 and 1/4 you mean 3 plus 1/4 but nowwhen we write numbers next to each otherwe're assuming for the most partmultiplication so we don't want to usethis notation and so I highly recommendthat you do not do that it's an oldernotation and I would suggest that youWatch at: 29:40 / 30:00simply ignore it and use another formall right let's go back to my list andlook at one last set of numbers beforewe take a pause the last set of numbersare the irrational numbers so let mewrite that downthe irrational numbers Irrational'sWatch at: 30:00 / 30:20Irrational's these are ones that are notrational and let me again give you areason by presenting you with a simpleequation that will require an irrationalto be solved if you want to solve verysimple equation like X to the power 2equals 2 so I'm looking for a numberWatch at: 30:20 / 30:40here that when multiplied by itselfthat's what the 2 means I get 2 butthere are no natural numbers lower than2 that will have that property the onlynatural numbertwo is one and one times one is one soif I'm going to find a number it'soutside of my system what do we needWatch at: 30:40 / 31:00well you remain may remember thenotation that's used for this we needeither plus or minus what's called thesquare root of two and you may noticethe way I write this with a little tailI'll remind you of that later but thisis a number that one squared gives me -this number is not an integer it's not aWatch at: 31:00 / 31:20natural number it's not a rationalnumber it is a new number we must add itto our set Q which was the last set ofnumbers we were looking at what otherkinds of numbers might not exist in twowell the set of irrational is ratherWatch at: 31:20 / 31:40large and it's a mixed bag it's really amotley bunch of numbers square root of 2square root of 3 another famous numberyou may know PI another number from thatrises in calculus II these are allWatch at: 31:40 / 32:00examples of irrational numbers they areoddballs if you like they are notdefined as quotients of other numberslike rational numbers in fact they'renot defined in a direct manner at allthey're simply all the numbers that areleft over when you picked out therational numbers there are a lot of themWatch at: 32:00 / 32:20in fact let me just give you one littlefact here in fact there are surprisinglythere are more Irrational'sthen rationals these very odd strangeWatch at: 32:20 / 32:39numbers that don't seem to fit anypattern here there are in fact more likethat than there are rational numbers nowI won't go into that here but on thatnote we'll pause and you can explore thenumber systems we've already looked atWatch at: 32:39 / 33:00now as you've had a little practice withrational numbers and the other numberswe looked at earlier will now put themtogether with the Irrational's andproduce a set called the real numbers solet's examine that the real numbers alsoWatch at: 33:00 / 33:20have a symbol that's very common theywill be often denoted by a capital Rbold-faced R oh and one thing while I'mhere real numbers these numbers are nomore or less real in the colloquialsense than any other number all numbersare equally real this is simply atechnical term we put that up here thisis a technical term that we use toWatch at: 33:20 / 33:39describe the set I'm about to describeso there's nothing more real about thesethan any other numbers all right what isthe set of real numbers well it is theset that contains Q and Q remember therational numbers and all theIrrational'sWatch at: 33:39 / 34:00so you take all the rationals and allthe Irrational's together and that formswhat we refer to as the set of realnumbers now this is the primary set thatwe are going to be looking at in collegealgebra this is the set we'll spend mostof our time in now let's go back to mylist because there's a way of talkingWatch at: 34:00 / 34:20about these numbers that is quite niceit takes advantage of geometry the realline from numbers to points so we'regoing to take numbers which are in therealm of arithmetic and algebra andconnect them up with a geometric ideathis will be called the real lineWatch at: 34:20 / 34:40so here's what we're doing give you theoverall view we're taking a geometricobject points alrightpoints which lie on a line and we aregoing to put them in what's calledone-to-one correspondence with the realWatch at: 34:40 / 35:00numbers and real numbers of course arean arithmetic or algebraic object sowe're connecting up algebra and geometryhere and it'll be very fruitful when wehave them connected to be easy to seeone idea from the other so let's goahead and write down what the realnumber line looks like it is of course aline so here's a line and I'll putWatch at: 35:00 / 35:20little arrows at the end so I indicatethat this goes on forever in bothdirections now as it stands now it's ageometric object with points what I'mgoing to do now is start labeling thepoints so that it gets algebraicnotation connected with it so the realnumbers become labels for the points onthis line first of all I'm going tolabel what you might consider the centerWatch at: 35:20 / 35:40of the real number system the number 0the numbers go to the right and to theleft from there the rightward directiongives you positive numbers the leftwarddirection gives you negative numbers now0 I will place anywhere so I'll evenmark that down anywhere let me alsopoint out that this is also referred toWatch at: 35:40 / 36:00as the origin so you can think of italso as the origin the place where allthe numbers sort of begin thenarbitrarily I will pick a distance sayto the right and I will mark that as my1 unit distance so this is an arbitraryarbitrary 1 unit I just make a choiceWatch at: 36:00 / 36:20but once I have that I can now put allthe other real numbers on here withoutany difficulty using that as mymeasuring device I can say then that 2will be there 3 will be there four willbe there and so on in the otherdirection - 1 - 2 - 3 and so on and youWatch at: 36:20 / 36:40might say well where are all the othernumbers where are say the rationalnumbers well we just need to label thembetween 0 & 1 at the halfway mark isprobably a good place to put the number1/2 if we went a little further Bone we might find a place for oh thenumber ten ninth say and where wouldWatch at: 36:40 / 37:00square root of two be well square rootof two as it turns out would appear inabout theresquare root of two the Irrational's alsolive on this line what about in thenegative direction well down heresomewhere maybe is the number minusthirteen 11s and so on all the numbersWatch at: 37:00 / 37:20we've seen the rationals the naturalnumbers the integers and theIrrational's can be considered lyingalong this line and two just toemphasize what I said about directionlet me write two large arrows here thereis one going to the right here's anotherone going to the left the one to theWatch at: 37:20 / 37:40right I'll write right in there andright means what it means increasing soas you move toward the right no matterwhere you start even if you start overhere you move from minus 3 to minus 2you've gained by 1 as you move to theright you gain as you move to the leftthat is decreasing the numbers go downWatch at: 37:40 / 38:00as you go to the left so if you movefrom 2 to 1 it's gone down but if you'veremoved from minus 1 to minus 2 you'vealso gone down by 1 so you have left andright alright let's go back to my listand we'll talk about another aspect ofthe real numbers we looked at realWatch at: 38:00 / 38:20numbers and the real line from numbersto points now we'll look at the factthat real numbers are ordered and youcan see that from the line but we cansay even more so our is ordered thatmeans that some numbers are bigger thanother numbers it's a very commonsensicalWatch at: 38:20 / 38:40idea and as I said before see the line Iguess I could stop there and say that'sall there is to it but I want tointroduce a notation so let me show yousomething we can take the real numbersand we can write them a new way we canwrite them in this way we can say thatWatch at: 38:40 / 39:00they are the set of all the positivenumbersnow that includes all the numbers we'veseen that happened to be positive zerothat dividing number and then all thenegative numbers now those are threesets that we've joined together to makeWatch at: 39:00 / 39:20the real numbers and the reason Ibrought this up is for the followingdefinition suppose we have two numbers aand B and suppose they're in the realWatch at: 39:20 / 39:40numbers so we have two numbers in thereal numbers we are going to say thefollowing we're going to talk aboutwhich one's bigger we will say that a isgreater than B if a minus B is positiveWatch at: 39:40 / 40:00now think about what that means a minusB is positive only if a is bigger than Bwhen you take away B you have somethingleft over well that's what we want thisto say we want this to mean bigger theway the notation works is you have thistriangle shape angled shaped object theWatch at: 40:00 / 40:20big side the open side is where the bignumber goes and the point the small sideis where the small number goes let's goahead and continue this a equals B if aminus B is well if they're equal and yousubtract one from the otheryou should get zero and so you do andfinally a is less than B if and thisWatch at: 40:20 / 40:40time B is the bigger one so where will aminus B be if B is the biggest onebigger one and you subtract it from ayou're going to get a negative result sohere is the way you can describe numbersone of these three properties willWatch at: 40:40 / 41:00always hold when you have any twonumbers out of the real numbers in otherwords we have orderthe real numbers have order any twonumbers one is bigger than the other orthey're equalall right well with that in mind let'sWatch at: 41:00 / 41:20extend that notation a bit likewisethere's another notation we like to havewe will define a greater than or equalto B and you heard me say it and that'swhat it is defined to be this means a isWatch at: 41:20 / 41:40greater than B or possibly a is equal toB and we put it together in a singlenice compact notation we also can definea is less than or equal to B in the sameway this means that a is less than B orthat a is equal to B so this allows us aWatch at: 41:40 / 42:00little flexibility alright it's time tolook at some examples here so in factlet's go to a new page so we have plentyof room here's an example X greater than4 now this is an algebraic statementWatch at: 42:00 / 42:20about members I want to write it usingthe real number line so I could turn itinto a geometric statement well here'smy real number line here say is 0 let memark off 4 because that's the number I'minterested in and I want all torepresent all the numbers X that arebigger than 4 now this doesn't include 4Watch at: 42:20 / 42:40so the way I will do that is I'll put anopen dot at 4 that indicates there's nopoint involved and then I'll mark offall the points going to the right with aslightly thicker line and an arrow atthe end so this inequality means thispicture this is geometry folks this isWatch at: 42:40 / 43:00algebra by connecting the two it becomeseasier to see what this means it meansall the numbers right above 4 and goingoff to the right foreverlet me give you another example it'svery important that you visualize theseWatch at: 43:00 / 43:20things example here's another one that'smore complicated minus 1 is less than Xis less than or equal to 3 now wehaven't seen this before let me go aheadand explain that this means what doesthis mean this means minus 1 is lessWatch at: 43:20 / 43:40than X that's the left-hand part andsimultaneously at the same time X isless than or equal to 3 that's thesecond part so both of these things mustWatch at: 43:40 / 44:00hold at the same time now it's very easyto draw a picture that illustrates thisin fact it's easier to draw the picturethan it is to look at the doubleinequality here so the two numbersinvolved are minus 1 and 3 that's allI'm going to mark minus 1 and 3 I don'tcare about 0 it's not part of what I'mlooking at this time now I want my X'sWatch at: 44:00 / 44:20to be greater than minus 1 and notinclude minus 1 so I'll put an open dotat minus 1 on the other direction I wantX to be less than or equal to 3 but I dowant 3 to be included so I'll put aclosed dot there so I'll shade that inand what am I looking for all thenumbers that are in between here so I'lltry and shade this all in so it looks aWatch at: 44:20 / 44:40little thicker and now this inequalityup here is this geometric set soalgebraic idea geometric idea that'swhat you want to try for when you'retrying to understand this all rightthere is a warning I want to give youWatch at: 44:40 / 45:00Oh morning and this is very common errorso I want to make sure that you don't dothis the statement that looks like thisthe statement that now put it in quotessay X is less than minus 2 or X isWatch at: 45:00 / 45:20greater than or equal to 5 okay whichmeans let me go ahead and show you thepicture because that will help youunderstand what this means which meanswhat well it means let's see there's theline there's minus 2 and there is 5because -2 and 5 of the two numbers I'minterested in less than minus 2 meansWatch at: 45:20 / 45:40that we do not include minus 2 but wecan go less than that and any numberless than that which be all of these Xgreater than or equal to 5 we do include5 and we go to the right any numberthat's greater than that is allowed sothis set is a set in two pieces okayWatch at: 45:40 / 46:00this this statement I'm sorry I forgotto put the end of my quotes there thestatement X less than minus 2 or Xgreater than or equal to 5 which meansthis picture cannot cannot what cannotbe collapsed or if you like the wordWatch at: 46:00 / 46:20re-written as and let me go ahead andbring this back so you see what I'mabout to do I have X is less than minus2 so you might want to think of X asbeing in the middle like it was beforein the previous example X is greaterthan or equal to 5 it cannot beWatch at: 46:20 / 46:40rewritten this way X less than minus 2and greater than or equal to 5 becauseif you think about that for about aminute you'll see that that's impossibleyou're asking for numbers X which aresimultaneously greater than or equal toWatch at: 46:40 / 47:005 so numbers that are bigger than 5 orpossibly equal to 5 and at the same timeless than minus 2 so it would have to bea number that's both positive andnegative which can't happen this isnonsenseso when you see statements as in thisWatch at: 47:00 / 47:20earlier example X is greater than minus1 less than or equal to 3 that has to bevery carefully qualified you have torealize that there are numbers in thereand then draw a picture if you draw thepicture it's very difficult to make thaterror but I wanted to warn you againstthat all right before we leave thisWatch at: 47:20 / 47:40topic of order sometimes we like to talkabout a group of real numbers that areall together on a real on the real lineand they are referred to as intervals sointervals on the real line and let me goahead and show you how we're going toWatch at: 47:40 / 48:00notate this notation open ends you'veseen from the previous examples ofsometimes we don't want to include thenumber at one end of an interval openends where say endpoints are notWatch at: 48:00 / 48:20included as you saw in the previousexamples we will use either an openparenthesis on the left or an openparenthesis on the right similarly forclosed ends now these are endpoints thatwill include the point at the end thisWatch at: 48:20 / 48:40is where endpoints are included we willuse another notation square brackets orsquare brackets going the other way sothese are the common accepted standardsWatch at: 48:40 / 49:00for this kind of notation now let medescribe the intervals what is aninterval I said it was a set of realnumbers there are in fact going to benine types of intervalsdepending on what part of the realnumber line we choose nine types theWatch at: 49:00 / 49:20first group there will be two groups thefirst group will be a bounded group andyou'll see what I mean for example thefirst type of interval will be called anopen interval it will look like this acomma B will write it exactly that wayWatch at: 49:20 / 49:40what will that mean look at the pictureand you'll see why the notation wasinvented if this is the real number lineand that those are the points a and Bnow I do not want to include a I do notwant to include B but I do want all ofthe points that are in between the twoso I'll go ahead and shade this regionWatch at: 49:40 / 50:00in here that is what this symbol in themiddle represents the parenthesesindicate that a is not included B is notincluded and a and B are simply thestart and the end and all the numbers inbetween are indicated by this openinterval let's go ahead and look at whatWatch at: 50:00 / 50:20are called half open or I guess ifyou're pessimistic half closed intervalsso you can call them whatever you likethey will have one end open and when inclose so suppose a is the open end B isthe closed end or the other way arounda is the closed end B is the open endWatch at: 50:20 / 50:40now I can draw pictures for both ofthese also mark off a and B a and B nowin this first one I do not want toinclude a so I'll put an open hole at aI do want to include B see that's whatthe square brackets are includes B soI'll shade in B and then the interval isWatch at: 50:40 / 51:00all the numbers in between and in thiscase including the point at the rightendpoint if I look at this half open orhalf closed interval this time I want toinclude a so I'll close a shade in thatpoint but I do not want to include B soI'll put an open hole there and thenWatch at: 51:00 / 51:20I'll just shade in everything in betweenmore or less and now I have an intervalwhere I include the leftend point and finally as you mightexpect if we have open and half open orhalf closed we finally should haveclosed and that would be square bracketson both ends and you're probably aheadWatch at: 51:20 / 51:40of me on this one if this is a and thisis B we want to include both a and B andwe shade in between and we now have aninterval that includes both of theendpoints the left and the rightendpoint and all of these intervals hereWatch at: 51:40 / 52:00all the ones you see here are boundedthey don't go beyond a or B in any oneof these casesthat's what bounded means they haveboundaries on both ends now let's lookat the other type because I've onlyshown you four types here there areanother five that I claimed exist theseWatch at: 52:00 / 52:20will come under the category ofunbounded intervals so these areunbounded but before we talk aboutunbounded if we are going to have aninterval that goes off in one directionforever or in the other directionforeverI need a notation for that I don't havea notation yet so let's invent one weeither going to go all the way off toWatch at: 52:20 / 52:40the right or all the way off sorry allthe way off to the left or all the wayoff to the right well here's thenotation that's used for this to say goall the way off to the left is to say goto minus infinity to the right it's togo to infinity that can also be writtenplus infinity if you like now before wego any further in there anyWatch at: 52:40 / 53:00misconceptions these are not realnumbersthese are not real numbers what they areare just handy symbols they're veryWatch at: 53:00 / 53:20handy they will allow us to say go offto the left without stopping or go offto the right without stopping but theyare not real numbers what that means isyou can't add subtract multiply ordivide them or do anything that you dowith real numbers so now I'm ready towrite down the last five intervals Ipromised you so we can have an intervalWatch at: 53:20 / 53:40that includes the left endpoint but goesoff to the right forever you mightnotice that I've put in open parentheseson the right side that makes sensebecause I can't have a closedparenthesis there that would mean Iinclude the last number but there is noWatch at: 53:40 / 54:00last number so I have to put an open endwhenever there's an infinity aroundwhat would the picture look like forthat well if this is a I am including athis time and now I'm going off to theright without stopping unboundedsimilarly I can start a on the LeftWatch at: 54:00 / 54:20without including a so I'll have apicture where it's an open circle andthen I go off to the right withoutstopping and then similarly I can go inthe other direction minus infinityindicates go to the left withoutstopping B can be included or minusWatch at: 54:20 / 54:40infinity B can be not included and theseas you might expect will look somethinglike this draw two lines there there's Bin both cases the first one I include Band go to the left in this case I do notinclude B and go to the left now that'sWatch at: 54:40 / 55:00eight of the nine integrals in intervalsrather what is the last one well thelast one is everything nobody usuallywrites this as the last one but let meindicate it minus infinity to infinitywhat would this look like well that's iteverything all of ourWatch at: 55:00 / 55:20now if you want to write all of ourusually you're right simply are youdon't write the interval but it's niceto know that it can be included in thisnotation that we've got so having thisnew notation we can rewrite what we didin previous examples so previousexamples just to show you how this canWatch at: 55:20 / 55:40be used can now be re-written as thefollowing well this is a nice way to seethat this notation is compact one of theexamples I talked about X greater thanfour this means that X is in the set theWatch at: 55:40 / 56:00interval from 4 to infinity greater than4 means I do not include 4 that's why Ihave the open parenthesis on the leftand greater than 4 doesn't have anyboundary so I go off to infinity here'sanother one from before I had minus 1less than X less than or equal to 3 whatWatch at: 56:00 / 56:20is that going to mean that means X is inthe interval open minus 1 because I'mnot including minus 1 and on the otherend including 3 so this is a half openor half closed interval including 3 andmy final example X less than minus 2 orWatch at: 56:20 / 56:40remember X greater than or equal to 5that means that X is in the first one Xis less than minus 2 that's going to bein going from minus infinity to minus 2open or because X can't be in both atWatch at: 56:40 / 57:00the same time remember X is greater thanor equal to 5 X in the interval startingand including 5 so it's closed bracketsquare brackets going off to infinity tothe right so there I've rewritten theanswers to those previous problems inthis nice compact interval notationWatch at: 57:00 / 57:20which you will see for the rest of thecourse so with that let's pause for amoment so you can practice some of thesereal number factsWatch at: 57:20 / 57:40now let's continue our exploration tothe real numbers by looking at the realline distance between points first ofall we're going to make it very simplewe're going to talk about the distancefrom zero one particular fixed point onthe real number line so let me show youWatch at: 57:40 / 58:00what I meanhere is the real number line and heresomewhere is 0 which is the origin ifyou recall now suppose we have a numberon the line say I'll write it to theright the number a now it has a certaindistance from zero we're going to giveWatch at: 58:00 / 58:20that distance a name but before I dogive it that name I want to point outthat there's another number that is thesame distance from zero as a is if wejust go to the left of zero to rightabout there the same distance a we comeup to the number minus a now minus aWatch at: 58:20 / 58:40also is the same distance away from zeroso both of these numbers are the samedistance away so with that in mind we'regoing to define what that distance isdefinition see before we talk about thedistance between two arbitrary pointswe're going to take the distance betweenWatch at: 58:40 / 59:00the number and arbitrary number and afixed point the zero point and here itis the distance from a real number Xokay we'll just call it X this time forthe time being to 0 which is the originWatch at: 59:00 / 59:20is called now this is a name thatprobably could be better but will giveyou the name this is called the absolutevalue of xnow I'll put underneath it what weWatch at: 59:20 / 59:40probably should have called it it shouldprobably be called the absolute distanceI'll put that in quotes because youwon't see that anywhere you'll seeabsolute value but we probably shouldcall it absolute distance because it'ssimply the distance of this number fromzero and it's called the absolute valueWatch at: 59:40 / 01:00:00of X and how do we denote it it isdenoted as follows absolute value of Xtwo parallel lines on either side of Xso let me show you some examples soyou'll see how this notation is used nowremember what this is this is thedistance of X from the origin from zeroWatch at: 01:00:00 / 01:00:20here's an example suppose we have thissituation here's 0 here's 5 thisdistance here is the absolute value of 5because it is the distance 5 is from 0now we can actually calculate thatWatch at: 01:00:20 / 01:00:40because we know how far away 5 is it isfive units away all right maybe youthink that was too easyhow about this suppose zero is here andwe go to the left to say the numberminus 4 now this is the distance ofminus 4 from zero the absolute value ofWatch at: 01:00:40 / 01:01:00minus 4 what distance is that how manyunits do you walk to get to minus 4 wellof course you walk 4 units so theabsolute value of minus 4 is 4 we'redeveloping a theory on how to use thisnotation aren't we we're seeing that ifyou're to the right then the absoluteWatch at: 01:01:00 / 01:01:20value of you your distance from 0 isindeed just yourself the positive numberif you're to the left of 0 it looks likethe left word indicator needs to beremoved and you just write the l'absoluvalue of minus 4 is simply 4 now that'sa bit colloquial let's go ahead andwrite down what we seem to haveWatch at: 01:01:20 / 01:01:40discovered another way of writing downwhat that symbol means so this symbolthe absolute value of Xwhich remember we might think of as theabsolute distance of X that is to saythe distance of this number from theorigin zero our experience suggests theWatch at: 01:01:40 / 01:02:00following it's the same thing as Xwhatever's inside there if X is apositive number if X is to the right ofzero X is greater than zero if X is zerowell of course what's the distance ofzero from itself zero so if X is zeroWatch at: 01:02:00 / 01:02:20then the distance should be zero andthen first I'll write the if part whatif X is less than zero that meansleft of zero well as we found out withthe absolute value of minus four let mebring that back and show you theabsolute value of minus four turned outWatch at: 01:02:20 / 01:02:40to be simply for the minus had nobearing on it well how would one get ridof a - well you know from yourexperience with negative numbers that ifyou take X X is less than zero so it hasa minus or a negative internal to it byputting another minus out front thatWatch at: 01:02:40 / 01:03:00will make it positive and give us theabsolute value or distance becausedistance after R all should be apositive number so just to write it outagain this is the absolute value of XI've also called it the absolutedistanceWatch at: 01:03:00 / 01:03:20we're by distance I mean distance fromzero and you might just call it thatdistance of x from zero now we willexplore throughout this course othersituations where the absolute value isused but if you remember thisWatch at: 01:03:20 / 01:03:40fundamental idea that is a distancemeasure it will make all of thosecalculations easier now that we havedistance between an arbitrary point andzero let's worry about what the distancewould be between two points where one ofthem might not be zero so the distanceWatch at: 01:03:40 / 01:04:00between two points on the real line getthat all down there all right and thetwo points may not include the origin sowe might have something new what whatI'll do here is we'll explore this againWatch at: 01:04:00 / 01:04:19by example and I think we'll see howthis is going to work so suppose we havemaybe 0 is there and we have two numbersminus 3 and 5 and since I also want torefer to the points remember these arenumbers so they are algebraic objectssuppose I want to refer to the pointsWatch at: 01:04:19 / 01:04:40where these numbers occur let me callthis point P and this point Q it'll makethings a little easier later then I askmyself what is the distance betweenthose two the distance well it's veryeasy to see even though this is not toscale clearly that if I go five to theWatch at: 01:04:40 / 01:05:00right there's five units involved andthen I go three units to the left thetotal number of units is eight so thedistance between these two points iseight now let's see if we can figure outa way to write that using the notationthat we have notice based on thatWatch at: 01:05:00 / 01:05:19picture let me even redraw it so here isminus three and five all right here'shere's zero in here notice the followingif I take the absolute value of thedifference of the two points- three - five or if you like theabsolute value the other way 5 - the - 3Watch at: 01:05:19 / 01:05:40watch what happens - 3 - 5 as you knowadds up to minus 8 and how far is minus8 away from zerowell 8 units on the other hand 5 minus aminus 3 that's 8 inside these absoluteWatch at: 01:05:40 / 01:06:00value lines and how far is 8 from zeroalso 8 so it looks like I might have anaha moment here and I might havesomething I can use as a definition solet me take that idea and write downwhat I think is a good definitiondefinition if I want the distanceWatch at: 01:06:00 / 01:06:19between the points P and Q and for thesake of argument let me draw them againup here here's P say here's Q and maybethe number associated with P is a thenumber associated with Q is B I couldsay that that's the absolute value of BWatch at: 01:06:19 / 01:06:40minus a or if I like it the other waythe absolute value of a minus B rememberin the previous example we saw that itdidn't matter which way I did it I gotthe same distance so that would be thesame as saying this is the distance fromQ to P measuring the other way becauseWatch at: 01:06:40 / 01:07:00it doesn't matter which way you measureit it should be the same number and sothat's what we'll do we'll take this orlike this either one of these as thedefinition of the distance between twopoints so for example just to toss inone final example if I want the distancebetween say minus 11 + 42 I simply takeWatch at: 01:07:00 / 01:07:19the absolute value of the difference andit doesn't matter which way I do it - 11- 42 will work what is that that's theabsolute value of minus 53 and how faris minus 53 from 0 53 units and there'sthe distance I was interested in sothat's all that's involved in findingWatch at: 01:07:19 / 01:07:40the distance between two points on thereal number linereal members as decimals reals asWatch at: 01:07:40 / 01:08:00decimalsnow decimal numbers have been around fora very long time nowadays they're a lotmore common because we use calculators alot and so the numbers that we look atoften appear in decimal form you shouldknow some basic ideas about real numbersWatch at: 01:08:00 / 01:08:20written as decimals so let me see if Ican outline a few of those ideas forexample let's look at first of allrational numbers rational x' alrightwe'll see what they look like indecimals and see if we can detectsomething a principle here also let meWatch at: 01:08:20 / 01:08:40remind you what is decimal mean well thepart that says deci refers to base 10that refers to 10 so we're talking abouta base 10 that's where we have thedigits remember 0 1 2 up to 9 so I'mgoing to know about what rationals looklike in decimal form so let me take aWatch at: 01:08:40 / 01:09:00simple rational like 3/4 now how do Iput this in decimal form you remainremember that the idea is to do a longdivision so let's go ahead and walk ourway through this let's divide 4 into 3and let's see if we can resurrect someof your experience I'll put a decimalpoint after 3 and a couple of zeros IWatch at: 01:09:00 / 01:09:20don't know how many I'll need butthat'll do for now I can add more if Ineed them and then I ask myself whatwould I multiply by 4 to get 3 well Ican there is no number so I then look at30 and I ask myself what could Imultiply by 4 to get 30 or close to itwell I will put a decimal above theWatch at: 01:09:20 / 01:09:40decimal that is underneath multiplying 4by 7 gives me 28 if I then subtract thatnumber from 30 I have a remainder of 2 Ibring this 0 down and I have a newnumber 20 this is the way the algorithmfor long division works if you recallnow I keep this up untilWatch at: 01:09:40 / 01:10:00I end up with a repeating set of digitsup here or I stop now watch what happensnext I have 20 here what could Imultiply it by 4 to get 20 well 5 willdo the trick5 times 4 is 20 I then subtract thataway and I'm left with nothing so I canstop there but I can also think of thisWatch at: 01:10:00 / 01:10:20is now a trivial division because allthe other numbers will be 0 so what do Ihave I have that 3/4 is equal to 0.75but what I really have is 0.75 0 0 dotdot dot the zeros repeat so I have whatWatch at: 01:10:20 / 01:10:40apparently is repeats okay now that'sthe first observation let's go ahead andlook at a little more complicated numberit doesn't look that bad when we writeit down 1/7 but when we do the longdivision it takes a little effort so IWatch at: 01:10:40 / 01:11:00wanted to work through one long one togive you a sense of how this works soI'm going to divide 7 into 1 well thatcan't be done so I'll put a dot thereand then I'm going to need a few zeros Idon't know how many they'll be justwrite a few of them there put a decimalpoint above now I realize that if IWatch at: 01:11:00 / 01:11:20multiply 7 by 1 I get 7 if I subtractthat away from the 10 above I end upwith 3 then I carry the zero down I have30 now I repeat the process four timesseven will give me 28 if I subtract thatfrom 30 I get to bring the zero down Ihave 20 multiply 7 by 2 I get 14 with aWatch at: 01:11:20 / 01:11:406 remainder bring the 0 down I have 60bear with me if I multiply by 8 I get 56subtract that away I get 4 still goingbring a 0 down 40 if I multiply by 5 IWatch at: 01:11:40 / 01:12:00get 35 subtracted away I get a 5 leftover looks like I'm going to run out ofzeros on a minute 50 and then I multiplyby 7 7 times 7 is 49 I subtract away49 and I get one and now I recognizesomething notice that the first thing IWatch at: 01:12:00 / 01:12:20worked with was a1 and now I've repeatedso now with one and then bringinganother zero down I'm going to have zeroif I subtract away seven I'm going torepeat the process I just repeated hereso it looks like although I have sixWatch at: 01:12:20 / 01:12:40different numbers that's not going tochange forever this is now going torepeat so I'm running out of space herelet me write that down a little morecompactly so 1/7 is equal to point onefour two eight five seven and then thatWatch at: 01:12:40 / 01:13:00repeats forever now you may remember acompact way of saying that that repeatsforever is to put a bar over those sixnumbers that means that this justcontinues to repeat so once again wehave that this repeats well that's onlytwo examples but that certainly doessuggest somethingin fact the following fact is trueWatch at: 01:13:00 / 01:13:20rationals have the following propertyrationals say Q using that notation isthe set of repeating decimals you takeWatch at: 01:13:20 / 01:13:40all the decimals that repeat includingthe ones that seem to end remember welooked at 0.75 which was 3/4 and youmight say well that's not a repeatingdecimal well sure it is if you considerthe repeating number to be zero sorepeating decimals include and areexactly equal to the set of all rationalnumbers well it makes it easier to guessWatch at: 01:13:40 / 01:14:00what the set of Irrational's might looklike the set of Irrational's whichdoesn't have any standard symbol wouldbe the set of non repeating decimalsWatch at: 01:14:00 / 01:14:20so for example in that set of nonrepeating decimals is the number squareroot of two which is a famous irrationalit starts off one point four one fourtwo three or two one three excuse meetc there's another famous irrationalpie which you probably know 3.14159 dotWatch at: 01:14:20 / 01:14:40dot dot now I'm not going to kid you itis difficult to tell if a given numberis an irrational number or not becauseyou have to determine somehow that thedecimals don't repeat and no matter howmany you write out that's not enough tocheck that so determining whether aWatch at: 01:14:40 / 01:15:00number is irrational or not is adifficult problem but this at leastgives us a way of categorizing rationalsand Irrational's one final thing beforewe leave the real number system I wantto point out an ambiguity so let me goahead and write that down this isanother warning and this will be thelast thing I say about the real numbersWatch at: 01:15:00 / 01:15:20today warning and ambiguity of decimalsthis is a situation where the numbersthat you think are always going to standfor the same thing don't and it's aWatch at: 01:15:20 / 01:15:40little surprising if you've never seenthis before I'll do this by exampleso you'll see what I'm going to dealwith I am going to show you that threepoint nine nine nine where the 9s go outforever it is exactly the same number isfour point zero zero zero out foreverthese are the same number they lookdifferent and you might suspect thatWatch at: 01:15:40 / 01:16:00they are different based on that but infact they are the same number how will Ido this well I'll do a verificationmeaning I will show you a littlecalculation that I hope will prove toyou that these are the same all rightstarting at the top of this page let Xbe the three point nine nine nine out toWatch at: 01:16:00 / 01:16:20infinity number in fact I'll box that inwhat I'd like to end up with at thebottom of my page is that X is alsoequal to four that will show you thatthose two numbers are the same here'sthe way the argument goesthen multiply X by 10 you will get 10xon the left on the right what happensWatch at: 01:16:20 / 01:16:40well you remember from decimals when youmultiply a number that's in decimals by10 it moves the decimal point over by 1so this becomes 39.99 9 then I still goout forevernow from this 10x let me subtract the Xnumber which is 3 point 9 9 9 outWatch at: 01:16:40 / 01:17:00forever on the left of this 10x minus Xgives me 9x on the right 3 from 39 givesme 36 and the point 9 9 9s cancel eachother out they subtract away so 9xequals 36 x equals 36 over 9 but 9 goesWatch at: 01:17:00 / 01:17:20into 36 so x equals 4 and look at that Xstarted out being 3 point 9 9 9 and nowI've got X equal 4 that means those twonumbers are the same surprisingly sothat's an ambiguity of our decimalWatch at: 01:17:20 / 01:17:40system where two numbers can appear tobe different it only happens in thisspecific case where there's an infinitenumber of nines but it is enough to giveyou pause let me show you one moreexample and then we'll stop talkingabout real numbers another quick examplelooks like this just to show you whatWatch at: 01:17:40 / 01:18:00can occur inside a number point seventhree two one five nine nine nine nineout to infinity this is going to be thesame as point seven three two one sixzero zero zero zero so the five ninebecomes a six zero and so these twoWatch at: 01:18:00 / 01:18:20numbers are the same even though theylook different and of course you'd likethe second one better because you cancall it a decimal that has zeros at theend so you can stop it after the six youdon't have to worry about rewriting thenines forever all right well that's allI'm going to say now about real numbersit's time for you to do a little workand practice with some of the conceptsWatch at: 01:18:20 / 01:18:40about real numbers