College Algebra - Lecture 1 - Numbers

in this unit which I'm calling basics or subtitled remembrance of things past we'll look at some of the basic ideas that are needed for the rest of the college algebra course we'll look at numbers which you can think of as the raw materials for the course but they are not what the course is about the course is really about the operations that you perform on numbers we'll look at the language of mathematics a little bit we'll look at exponents and exponent notation we'll look at polynomial expressions and we'll finish up with a little bit of geometry so I hope you enjoy it and here we are the first part of basics we'll look at our numbers and under numbers the first thing we'll look at is an idea of mathematics that will help us describe numbers we'll talk about sets of objects first of all so let's look at that sets of objects now sets are what a mathematician calls a bunch of things now what will be in such a set or a bunch or a collection numbers expressions functions all these things that we'll be talking about later in the course but let me go ahead and give you an example for example for example let's look at the set of base 10 digits those are the digits we use to write all of our numbers in our number numeral system and I'll give you the notation as we go I'll use this curly brackets or braces to start the set and then I'll list all the digits inside with commas so we're four five six seven eight and nine and then end the brackets so these beginning and ending items are called set brackets sometimes they're called braces and they are what indicate the beginning in the end of the set all of the elements of the set there's so few of them I can simply list them all the elements are separated musi by commas this is an example of a finite set it's a set with only a finite number of elements in fact you can see that there are exactly ten elements and what we've done is we put it all together in one place it's a collection of objects that's all that a set is nothing nothing fancy just a simple idea let's look at a couple more examples here's an example this should be familiar to anyone that speaks English the set of 26 English letters I will write that starting off with the braces again and then list the letters a B C and D and you know there are so many that I don't want to list them all so what I will do is I will put dot dot dot in the middle and then a comma and then I will list the last few XY and Z these dots sometimes referred to as an ellipsis is a way of indicating that the set continues on and in this case I happen to know the ending so I cannot put on the the last few elements this again is another finite set now as you might expect in mathematics there are lots and lots of sets that are not finite we call those infinite sets so let me give you an example of one of those and yet again show you this notation example here is another set this is the set of positive even numbers now as we go on we'll talk about numbers in detail but I think you probably remember what these are numbers like 2 4 6 8 10 etc now you see I put the dots at the end and I put nothing afterwards I don't know how it ends because there's an infinite number of even numbers there's always more beyond wherever I stop so this is a perfect example of an infinite set and this is one way of writing down such a set we will learn other ways as we go but I wanted to show you both finite and an infinite set now there's one more idea that we want to touch on before we start talking about numbers and here I'll show you in an example what I mean here is a small set the set two four six there's a set with only three elements in it this is what's called this is what's called a subset of that larger set we saw on the other page 2 4 6 8 10 etc the infinite set of real even numbers you see that these three elements are contained in this set this is a smaller set this is a sub set of the larger set so if you were to look at this in a picture say if this shape describes the set then what we have here is a small part of that set called a sub set now I will use that terminology throughout the course and if we need to say anything more about sets and sets of objects we'll introduce that terminology as we go but now let's go back to the list and see what kind of numbers we will first look at will first look at natural numbers these are sometimes called the counting numbers what are they natural numbers these are numbers that we really take for granted these are numbers that we didn't have to invent as we have had to event other numbers these numbers can be thought of as just existing they are often denoted the set is often denoted by a boldface N and that's the way I'll write that and what is this set well its consists of these numbers 1 2 3 4 etc and then sometimes we do this also we'll put an N here to indicate the variable that I will use or the letter I will use to stand for such numbers and then a comma and continue on out to infinity so this is a commonly used variable for natural numbers and I'll use it from time to time these will be taken as given these are not invented there's nothing to describe these are simply the starting place for the number systems we will look at but you know there are lots of numbers within this set of numbers that are quite that are that are quite interesting so let's go ahead and talk about a few of them because they will have bearing later there's a set of numbers that are called prime numbers now why would I want to talk about those well as in chemistry it is helpful to look at what you might consider the atoms of the material that you are discussing prime numbers can be considered the atoms of the natural numbers for example let me go ahead and define what a prime number is P is prime call it a prime number if well first of all P is greater than 1 because 1 is not considered a prime number and what makes a prime number and atom P is divisible only by two things only by 1 which divides into every number and itself P what do these numbers look like well I'm sure you've seen these in your life the set of prime numbers looks like this the first one is 2 which happens to be the only even prime number then 3 5 7 not 9 because remember 9 would be 3 times 3 11 13 17 etc there are an infinite number of primes and this is what they look like numbers like this they're only divisible they can only be factored words that I will be describing soon into themselves times 1 so those are prime numbers now with prime numbers since we can think of them as the atoms of the natural numbers you might expect that we ought to be able to write all natural numbers in terms of prime numbers and so we can and this is such a famous result I just want to go ahead and introduce it to you it's called the fundamental theorem fundamental theorem of arithmetic arithmetic you may ask why are we studying arithmetic in an algebra course as you'll see it really is going to be the basis for much of what we do and it's probably a theorem that you haven't seen in this form but I think you know what it's going to say any natural number that is to say any of those counting numbers we saw before any natural number can be factored now I haven't officially defined the word factor but a factor is simply a thing that is multiplied so to factor a number like 12 would be to write it as a product of two other numbers it can be factored uniquely there's only one way you can factor it and that one way is except for the order of the factors except for the order of the factors because as you may remember from your knowledge of the numbers that numbers like two times three are the same thing as numbers three times two into a product of prime powers prime numbers prime powers so to read to say this one more time any natural number can be factored uniquely except for the order of the factors into a product of prime powers now let me give you some examples so you'll know what it is I'm saying here and once I do I think you'll see that this is familiar let me take the number I mentioned a moment ago twelve twelve as you know can be written as three times four that may be the first thing that comes to mind but four can be written as two squared so this can be written as 2 squared times 3 now 2 is a prime number to a power this is two times two that's what the two up there means and 3 is another prime number let me do a few more examples 70 can be written as 2 times 5 times 7 for example 111 can be written as 3 times 37 etc all the natural numbers can be written as products of prime numbers or possibly powers of prime numbers and as I said before and we'll define it again in a little bit factors are things that are multiplied now before I leave this topic I don't want to mislead you generally factoring numbers is hard if you start with a very large number and you try to factor it is very hard to do in fact that's being used in security systems these days for that very reason but the kind of factoring we will do is fairly simple and so I'll pause now and give you a chance to remind yourself of a few things now we'll continue our development of the numbers by looking at integers so what are the integers the integers is a larger set than the natural numbers the integers are often denoted by a capital Z you may say what does that have to do with integers there's no Z in the word integer well just for the historically minded among you this comes from the German words all N which stands for numbers so that's what the symbolism comes from and what are the integers how do they differ from the natural numbers well the way I'm going to define the integers is to give you a sense of why one would need the numbers so let's let imagine that we're moving ahead in the course here and we want to solve a certain very simple equation suppose we want to solve the equation X plus 1 equals 1 we want to find a number X that will make this equation true well you know from your own experience that there is only one number that will make this true we need what we need the number 0 but in the natural numbers there is no 0 so 0 has to be added to that and that will become one of the integers let me show you one other small equation to suggest why we need another set of numbers suppose we want to solve X plus 1 equals 0 using that new number zero now from what you know what would you have to add to one to get zero well you'd have to add we need a number like minus one but again the natural numbers don't have any negative numbers in them so we have to add these numbers in so when we add them all in we're going to get a set that is the integers here is that set the set Z and it will be infinite in both directions so I'll have to start off with dots and then I'll give you a few of the numbers minus 4 minus 3 minus 2 minus 1 then there's that special number 0 so I'll put it on its own line alone and then we'll continue 1 2 3 4 etc out to infinity now there's a more compact way to write this so I'll show you that if we put 0 at the front and then we put a plus and a minus sign here for 1 so we can imagine plus 1 and minus 1 being next to each other and then plus or minus 2 plus or minus 3 plus or minus 4 etc out to infinity now this is just more compact so this is a more compact way of writing this set but we have now expanded our set of numbers we no longer have simply the natural numbers notice that the natural numbers are right here they're included in the integers we have added 0 and we've added all these negative numbers and a reasonable way of describing the reason for that is to say that we needed to solve certain kinds of equations so let me show you a couple of things in an example about these numbers these are things that you probably have seen much of your life I'll just remind you of a couple of things for example if I take 0 times 5 and notice I've already started introducing a notation that you may not be familiar with I'm using a dot for multiplication again I will talk about that a little bit later on but you can see that I'm multiplying 0 times 5 well you know what happens when you multiply a number times 0 you get 0 so there's a fact from your past I hope and if we take 0 times 82 we're going to get zero it doesn't matter what number you choose zero times that number is zero what else can we say how about zero plus a number like zero plus seven well that is seven the zero contributes nothing so you know that adding zero to any number will result in the original number coming back how about negative numbers what are some of the properties you probably have seen before well suppose we take two numbers like minus 5 and minus 2 and multiply them together well you'll get 10 in fact to get plus 10 because a negative times a negative is a positive what about an addition problem take minus 5 plus 2 well as you know 2 minus 5 will give me a minus 3 etc so the kinds of facts that you you should be familiar with are the facts about the elementary integers and the properties they have so let's go ahead and look at another set of numbers that is built from integers so we'll go back to my list and after integers we will look at rational numbers this is where we take integers and we create a new set of numbers and I'll show you exactly how that's done rational numbers now rational numbers have their own bold-faced letter symbol Q and this time the Q stands for quotient so that seems to make good sense what does a rational number look like well then what you previously called in grammar school fractions P over Q that is a rational number what are the P and Q the P and the Q are integers so they come from that set Z that we saw a moment ago and we also say that Q is not 0 now we're forbidding the bottom to be 0 the bottom of the fraction is not 0 and I'll explain why in a little bit but we're going to say the division by 0 is not allowed it is undefined there's a good reason for that but this of course is what you call before as set a fraction you might also have used the word ratio which is where rational comes from in the description of these and now let's look at a few examples just to remind you what we're talking about here examples you know what these are 1/2 2/3 1113 574 over 201 etc those are all examples of rational numbers they're fractions some of them as in the case of the last one are improper fractions the top is bigger than the bottom but there's still rational numbers now let me note that once again we have increased our number system without losing anything all n is in queue in fact all Z the integers all of that set is in queue how do I know that well since if you take a number n from either Z or the natural numbers and you put it over 1 lo and behold you have a rational number but of course n over 1 is still N and so these numbers from here can be re-written as n over 1 and they appear as rational numbers and for example 8 over 1 of course is 8 and so on so that's what rational numbers look like now I said a moment ago that division by zero is not allowed in the natural in the in the rational numbers so let me go ahead and give you that warning and give you a reason why that is true so that we don't have any mysteries here warning division by zero is what's called undefined now you might say well it's undefined why doesn't somebody define it well as a reason it's undefined because you cannot define it as a number and end up without a contradiction so the answer is why the question rather is why why is it undefined here's why let us suppose let's take a simple case because if it fails here it's probably going to fail for the rest suppose 1 over 0 and I'll put it in quotes suppose 1 over 0 is taken to be a number well if one over 0 is taken to be a number it must share all the properties that the other numbers we've seen satisfy that is going to lead us to a problem watch then consider the following situation and this is just one of many possible situations I could describe consider the following 0 times this news purported number 1 over 0 now I can make two different arguments on the one hand I can say this must be 0 why here's my reason since as we saw a moment ago 0 times x equals 0 for any number that we've seen X you multiply a number by 0 it is 0 that's what I've done here multiplied this purported number by 0 so I claim it's 0 fine on the other hand I can also make the following argument that this is equal to 1 and what's my reason there since copying what I have here if you have x times 1 over X that's the same as x over X and of course that's 1 for any X well now we have the contradiction if I assume that you can divide by zero and that one over zero is a legitimate number then I end up with it being multiplied by zero I get either zero or one I can't have both so there is no way no matter what I define this to be I will end up with a problem like this that's a contradiction so one over zero can't be defined because if you define it you're going to have contradictions arise and you don't want that in your number system so in case anyone was wondering why division by zero is not allowed there's the reason alright let's look at some operations on rational number rational numbers and I'll say some operations on Q and these are fraction operations these are the kind of things you've had experience with over the course of your elementary school career but now we'll put them in a form that's more algebraic for example suppose you have the fraction a over B times and other fractions C over D what does that simplify to how do you multiply two fractions well this is the easiest of the operations this one's easy you multiply the top a C and over the bottom B D now there are a couple of things going on here with notation that we'll have to discuss later I'm using the dot here again for multiplication and also over here I'm using nothing I'm just putting the numbers next to each other for multiplication so I have AC over BD what about division what if I took the fraction or rational number a over B and I wanted to divide it by the fraction or rational numbers C over D how is that done well this is how it's done you just copy over the top one a over B and then we're going to multiply it times the bottom one but we're going to change the bottom one to what's called its reciprocal now that's a fancy older word it just means flip it over so if it's C over D here we'll multiply it by D over C and then of course this looks just like with the case above so this becomes ad over BC and that's how you divide two rational numbers the top one just stays the bottom one comes over multiplies and gets flipped over so that should be something that you've seen before and we've covered two operations now multiplication and division there are two more there's the addition and subtraction so let's just go ahead for the sake of completeness look at let's look at a over B plus or minus C over D now of all of the operations this is the hardest these are the hardest because but the reason is that a over B and C over D are divisions in themselves and pluses and minuses are different kind of operation so we have to decide how we're going to do this and the way that this is done is we get what's called a common denominator we make the fractions have the same number at the bottom and the easiest way to do this is to take a / b multiply it by whatever i need on the bottom and i'm going to take the product of these two numbers as my new denominator so I need a d down there and since I don't want to change anything I'll multiply a over B times D over D remember D over D is 1 so I don't change anything then plus or minus C over D I do the same thing except I multiply by B over B the advantage is now the bottom is BD and the bottom is BD here so I will get a new fraction with the denominator or a bottom BD and the top will be ad plus or minus BC and that is how you add or subtract two fractions this number at the bottom here by the way is often called a common denominator there's one of those older words denominator we'll just call it bottom for the most part and likewise we'll refer to the tops of the fractions instead of numerators we'll call them top most of the time alright let me go ahead and do a few numerical examples just to give you a chance to warm up a little bit example if I take two fifths and I divide it by 1/7 remembering the process I copy down two fifths and then I multiply it by this fraction flipped over that's seven over one and so I will have fourteen fifths and so I've divided two fractions and ended up with another fraction similarly I can do a simple addition maybe take 1/3 plus 1/5 I take the 1/3 I multiply the top and bottom by 5 over 5 plus one-fifth multiply top and bottom by 3 so my denominator will be 3 times 5 or 15 in both cases have a new fraction the bottom is 15 the top is 5 plus 3 so I end up with the fraction 8 15 now you'll have a chance in a little bit to practice some of this but these should be simple operations I hope another note I want to make before we have a chance to stop note in this example I will illustrate another idea suppose you're faced with the fraction 4 sixths well that is more complicated than it needs to be if you recognize that there's a factor common to the top and the bottom you can then remove it as follows if you realize the top is 2 times 2 and the bottom is 3 times 2 and of course here's where factoring comes in if you can do it you see that what you've got is a new fraction here 2 over 2 which is just 1 so this can simply be rewritten as 2 over 3 this is nice because now 2 & 3 have no factors in common they are in fact prime numbers and that's a much nicer fraction than the original and when you do this this whole process is called put into what's referred to as lowest terms put this into lowest terms and that just means you find out if they have common factors the top and the bottom and you rewrite them and those common factors it's called cancelling but of course you realize is just rewriting 2 over 2 as the number 1 all right one final note for rational numbers before we move on to the next set of numbers my final note here on rationals is do not here I'm going to give you a stricture do not use mixed numbers or mixed number notation this is a notation that's very common in the elementary schools like for example 3 and 1/4 and I'll tell you why because 3 and 1/4 means 3 plus 1/4 that's what you mean when you write down 3 and 1/4 you mean 3 plus 1/4 but now when we write numbers next to each other we're assuming for the most part multiplication so we don't want to use this notation and so I highly recommend that you do not do that it's an older notation and I would suggest that you simply ignore it and use another form all right let's go back to my list and look at one last set of numbers before we take a pause the last set of numbers are the irrational numbers so let me write that down the irrational numbers Irrational's Irrational's these are ones that are not rational and let me again give you a reason by presenting you with a simple equation that will require an irrational to be solved if you want to solve very simple equation like X to the power 2 equals 2 so I'm looking for a number here that when multiplied by itself that's what the 2 means I get 2 but there are no natural numbers lower than 2 that will have that property the only natural number two is one and one times one is one so if I'm going to find a number it's outside of my system what do we need well you remain may remember the notation that's used for this we need either plus or minus what's called the square root of two and you may notice the way I write this with a little tail I'll remind you of that later but this is a number that one squared gives me - this number is not an integer it's not a natural number it's not a rational number it is a new number we must add it to our set Q which was the last set of numbers we were looking at what other kinds of numbers might not exist in two well the set of irrational is rather large and it's a mixed bag it's really a motley bunch of numbers square root of 2 square root of 3 another famous number you may know PI another number from that rises in calculus II these are all examples of irrational numbers they are oddballs if you like they are not defined as quotients of other numbers like rational numbers in fact they're not defined in a direct manner at all they're simply all the numbers that are left over when you picked out the rational numbers there are a lot of them in fact let me just give you one little fact here in fact there are surprisingly there are more Irrational's then rationals these very odd strange numbers that don't seem to fit any pattern here there are in fact more like that than there are rational numbers now I won't go into that here but on that note we'll pause and you can explore the number systems we've already looked at now as you've had a little practice with rational numbers and the other numbers we looked at earlier will now put them together with the Irrational's and produce a set called the real numbers so let's examine that the real numbers also have a symbol that's very common they will be often denoted by a capital R bold-faced R oh and one thing while I'm here real numbers these numbers are no more or less real in the colloquial sense than any other number all numbers are equally real this is simply a technical term we put that up here this is a technical term that we use to describe the set I'm about to describe so there's nothing more real about these than any other numbers all right what is the set of real numbers well it is the set that contains Q and Q remember the rational numbers and all the Irrational's so you take all the rationals and all the Irrational's together and that forms what we refer to as the set of real numbers now this is the primary set that we are going to be looking at in college algebra this is the set we'll spend most of our time in now let's go back to my list because there's a way of talking about these numbers that is quite nice it takes advantage of geometry the real line from numbers to points so we're going to take numbers which are in the realm of arithmetic and algebra and connect them up with a geometric idea this will be called the real line so here's what we're doing give you the overall view we're taking a geometric object points alright points which lie on a line and we are going to put them in what's called one-to-one correspondence with the real numbers and real numbers of course are an arithmetic or algebraic object so we're connecting up algebra and geometry here and it'll be very fruitful when we have them connected to be easy to see one idea from the other so let's go ahead and write down what the real number line looks like it is of course a line so here's a line and I'll put little arrows at the end so I indicate that this goes on forever in both directions now as it stands now it's a geometric object with points what I'm going to do now is start labeling the points so that it gets algebraic notation connected with it so the real numbers become labels for the points on this line first of all I'm going to label what you might consider the center of the real number system the number 0 the numbers go to the right and to the left from there the rightward direction gives you positive numbers the leftward direction gives you negative numbers now 0 I will place anywhere so I'll even mark that down anywhere let me also point out that this is also referred to as the origin so you can think of it also as the origin the place where all the numbers sort of begin then arbitrarily I will pick a distance say to the right and I will mark that as my 1 unit distance so this is an arbitrary arbitrary 1 unit I just make a choice but once I have that I can now put all the other real numbers on here without any difficulty using that as my measuring device I can say then that 2 will be there 3 will be there four will be there and so on in the other direction - 1 - 2 - 3 and so on and you might say well where are all the other numbers where are say the rational numbers well we just need to label them between 0 & 1 at the halfway mark is probably a good place to put the number 1/2 if we went a little further B one we might find a place for oh the number ten ninth say and where would square root of two be well square root of two as it turns out would appear in about there square root of two the Irrational's also live on this line what about in the negative direction well down here somewhere maybe is the number minus thirteen 11s and so on all the numbers we've seen the rationals the natural numbers the integers and the Irrational's can be considered lying along this line and two just to emphasize what I said about direction let me write two large arrows here there is one going to the right here's another one going to the left the one to the right I'll write right in there and right means what it means increasing so as you move toward the right no matter where you start even if you start over here you move from minus 3 to minus 2 you've gained by 1 as you move to the right you gain as you move to the left that is decreasing the numbers go down as you go to the left so if you move from 2 to 1 it's gone down but if you've removed from minus 1 to minus 2 you've also gone down by 1 so you have left and right alright let's go back to my list and we'll talk about another aspect of the real numbers we looked at real numbers and the real line from numbers to points now we'll look at the fact that real numbers are ordered and you can see that from the line but we can say even more so our is ordered that means that some numbers are bigger than other numbers it's a very commonsensical idea and as I said before see the line I guess I could stop there and say that's all there is to it but I want to introduce a notation so let me show you something we can take the real numbers and we can write them a new way we can write them in this way we can say that they are the set of all the positive numbers now that includes all the numbers we've seen that happened to be positive zero that dividing number and then all the negative numbers now those are three sets that we've joined together to make the real numbers and the reason I brought this up is for the following definition suppose we have two numbers a and B and suppose they're in the real numbers so we have two numbers in the real numbers we are going to say the following we're going to talk about which one's bigger we will say that a is greater than B if a minus B is positive now think about what that means a minus B is positive only if a is bigger than B when you take away B you have something left over well that's what we want this to say we want this to mean bigger the way the notation works is you have this triangle shape angled shaped object the big side the open side is where the big number goes and the point the small side is where the small number goes let's go ahead and continue this a equals B if a minus B is well if they're equal and you subtract one from the other you should get zero and so you do and finally a is less than B if and this time B is the bigger one so where will a minus B be if B is the biggest one bigger one and you subtract it from a you're going to get a negative result so here is the way you can describe numbers one of these three properties will always hold when you have any two numbers out of the real numbers in other words we have order the real numbers have order any two numbers one is bigger than the other or they're equal all right well with that in mind let's extend that notation a bit likewise there's another notation we like to have we will define a greater than or equal to B and you heard me say it and that's what it is defined to be this means a is greater than B or possibly a is equal to B and we put it together in a single nice compact notation we also can define a is less than or equal to B in the same way this means that a is less than B or that a is equal to B so this allows us a little flexibility alright it's time to look at some examples here so in fact let's go to a new page so we have plenty of room here's an example X greater than 4 now this is an algebraic statement about members I want to write it using the real number line so I could turn it into a geometric statement well here's my real number line here say is 0 let me mark off 4 because that's the number I'm interested in and I want all to represent all the numbers X that are bigger than 4 now this doesn't include 4 so the way I will do that is I'll put an open dot at 4 that indicates there's no point involved and then I'll mark off all the points going to the right with a slightly thicker line and an arrow at the end so this inequality means this picture this is geometry folks this is algebra by connecting the two it becomes easier to see what this means it means all the numbers right above 4 and going off to the right forever let me give you another example it's very important that you visualize these things example here's another one that's more complicated minus 1 is less than X is less than or equal to 3 now we haven't seen this before let me go ahead and explain that this means what does this mean this means minus 1 is less than X that's the left-hand part and simultaneously at the same time X is less than or equal to 3 that's the second part so both of these things must hold at the same time now it's very easy to draw a picture that illustrates this in fact it's easier to draw the picture than it is to look at the double inequality here so the two numbers involved are minus 1 and 3 that's all I'm going to mark minus 1 and 3 I don't care about 0 it's not part of what I'm looking at this time now I want my X's to be greater than minus 1 and not include minus 1 so I'll put an open dot at minus 1 on the other direction I want X to be less than or equal to 3 but I do want 3 to be included so I'll put a closed dot there so I'll shade that in and what am I looking for all the numbers that are in between here so I'll try and shade this all in so it looks a little thicker and now this inequality up here is this geometric set so algebraic idea geometric idea that's what you want to try for when you're trying to understand this all right there is a warning I want to give you Oh morning and this is very common error so I want to make sure that you don't do this the statement that looks like this the statement that now put it in quotes say X is less than minus 2 or X is greater than or equal to 5 okay which means let me go ahead and show you the picture because that will help you understand what this means which means what well it means let's see there's the line there's minus 2 and there is 5 because -2 and 5 of the two numbers I'm interested in less than minus 2 means that we do not include minus 2 but we can go less than that and any number less than that which be all of these X greater than or equal to 5 we do include 5 and we go to the right any number that's greater than that is allowed so this set is a set in two pieces okay this this statement I'm sorry I forgot to put the end of my quotes there the statement X less than minus 2 or X greater than or equal to 5 which means this picture cannot cannot what cannot be collapsed or if you like the word re-written as and let me go ahead and bring this back so you see what I'm about to do I have X is less than minus 2 so you might want to think of X as being in the middle like it was before in the previous example X is greater than or equal to 5 it cannot be rewritten this way X less than minus 2 and greater than or equal to 5 because if you think about that for about a minute you'll see that that's impossible you're asking for numbers X which are simultaneously greater than or equal to 5 so numbers that are bigger than 5 or possibly equal to 5 and at the same time less than minus 2 so it would have to be a number that's both positive and negative which can't happen this is nonsense so when you see statements as in this earlier example X is greater than minus 1 less than or equal to 3 that has to be very carefully qualified you have to realize that there are numbers in there and then draw a picture if you draw the picture it's very difficult to make that error but I wanted to warn you against that all right before we leave this topic of order sometimes we like to talk about a group of real numbers that are all together on a real on the real line and they are referred to as intervals so intervals on the real line and let me go ahead and show you how we're going to notate this notation open ends you've seen from the previous examples of sometimes we don't want to include the number at one end of an interval open ends where say endpoints are not included as you saw in the previous examples we will use either an open parenthesis on the left or an open parenthesis on the right similarly for closed ends now these are endpoints that will include the point at the end this is where endpoints are included we will use another notation square brackets or square brackets going the other way so these are the common accepted standards for this kind of notation now let me describe the intervals what is an interval I said it was a set of real numbers there are in fact going to be nine types of intervals depending on what part of the real number line we choose nine types the first group there will be two groups the first group will be a bounded group and you'll see what I mean for example the first type of interval will be called an open interval it will look like this a comma B will write it exactly that way what will that mean look at the picture and you'll see why the notation was invented if this is the real number line and that those are the points a and B now I do not want to include a I do not want to include B but I do want all of the points that are in between the two so I'll go ahead and shade this region in here that is what this symbol in the middle represents the parentheses indicate that a is not included B is not included and a and B are simply the start and the end and all the numbers in between are indicated by this open interval let's go ahead and look at what are called half open or I guess if you're pessimistic half closed intervals so you can call them whatever you like they will have one end open and when in close so suppose a is the open end B is the closed end or the other way around a is the closed end B is the open end now I can draw pictures for both of these also mark off a and B a and B now in this first one I do not want to include a so I'll put an open hole at a I do want to include B see that's what the square brackets are includes B so I'll shade in B and then the interval is all the numbers in between and in this case including the point at the right endpoint if I look at this half open or half closed interval this time I want to include a so I'll close a shade in that point but I do not want to include B so I'll put an open hole there and then I'll just shade in everything in between more or less and now I have an interval where I include the left end point and finally as you might expect if we have open and half open or half closed we finally should have closed and that would be square brackets on both ends and you're probably ahead of me on this one if this is a and this is B we want to include both a and B and we shade in between and we now have an interval that includes both of the endpoints the left and the right endpoint and all of these intervals here all the ones you see here are bounded they don't go beyond a or B in any one of these cases that's what bounded means they have boundaries on both ends now let's look at the other type because I've only shown you four types here there are another five that I claimed exist these will come under the category of unbounded intervals so these are unbounded but before we talk about unbounded if we are going to have an interval that goes off in one direction forever or in the other direction forever I need a notation for that I don't have a notation yet so let's invent one we either going to go all the way off to the right or all the way off sorry all the way off to the left or all the way off to the right well here's the notation that's used for this to say go all the way off to the left is to say go to minus infinity to the right it's to go to infinity that can also be written plus infinity if you like now before we go any further in there any misconceptions these are not real numbers these are not real numbers what they are are just handy symbols they're very handy they will allow us to say go off to the left without stopping or go off to the right without stopping but they are not real numbers what that means is you can't add subtract multiply or divide them or do anything that you do with real numbers so now I'm ready to write down the last five intervals I promised you so we can have an interval that includes the left endpoint but goes off to the right forever you might notice that I've put in open parentheses on the right side that makes sense because I can't have a closed parenthesis there that would mean I include the last number but there is no last number so I have to put an open end whenever there's an infinity around what would the picture look like for that well if this is a I am including a this time and now I'm going off to the right without stopping unbounded similarly I can start a on the Left without including a so I'll have a picture where it's an open circle and then I go off to the right without stopping and then similarly I can go in the other direction minus infinity indicates go to the left without stopping B can be included or minus infinity B can be not included and these as you might expect will look something like this draw two lines there there's B in both cases the first one I include B and go to the left in this case I do not include B and go to the left now that's eight of the nine integrals in intervals rather what is the last one well the last one is everything nobody usually writes this as the last one but let me indicate it minus infinity to infinity what would this look like well that's it everything all of our now if you want to write all of our usually you're right simply are you don't write the interval but it's nice to know that it can be included in this notation that we've got so having this new notation we can rewrite what we did in previous examples so previous examples just to show you how this can be used can now be re-written as the following well this is a nice way to see that this notation is compact one of the examples I talked about X greater than four this means that X is in the set the interval from 4 to infinity greater than 4 means I do not include 4 that's why I have the open parenthesis on the left and greater than 4 doesn't have any boundary so I go off to infinity here's another one from before I had minus 1 less than X less than or equal to 3 what is that going to mean that means X is in the interval open minus 1 because I'm not including minus 1 and on the other end including 3 so this is a half open or half closed interval including 3 and my final example X less than minus 2 or remember X greater than or equal to 5 that means that X is in the first one X is less than minus 2 that's going to be in going from minus infinity to minus 2 open or because X can't be in both at the same time remember X is greater than or equal to 5 X in the interval starting and including 5 so it's closed bracket square brackets going off to infinity to the right so there I've rewritten the answers to those previous problems in this nice compact interval notation which you will see for the rest of the course so with that let's pause for a moment so you can practice some of these real number facts now let's continue our exploration to the real numbers by looking at the real line distance between points first of all we're going to make it very simple we're going to talk about the distance from zero one particular fixed point on the real number line so let me show you what I mean here is the real number line and here somewhere is 0 which is the origin if you recall now suppose we have a number on the line say I'll write it to the right the number a now it has a certain distance from zero we're going to give that distance a name but before I do give it that name I want to point out that there's another number that is the same distance from zero as a is if we just go to the left of zero to right about there the same distance a we come up to the number minus a now minus a also is the same distance away from zero so both of these numbers are the same distance away so with that in mind we're going to define what that distance is definition see before we talk about the distance between two arbitrary points we're going to take the distance between the number and arbitrary number and a fixed point the zero point and here it is the distance from a real number X okay we'll just call it X this time for the time being to 0 which is the origin is called now this is a name that probably could be better but will give you the name this is called the absolute value of x now I'll put underneath it what we probably should have called it it should probably be called the absolute distance I'll put that in quotes because you won't see that anywhere you'll see absolute value but we probably should call it absolute distance because it's simply the distance of this number from zero and it's called the absolute value of X and how do we denote it it is denoted as follows absolute value of X two parallel lines on either side of X so let me show you some examples so you'll see how this notation is used now remember what this is this is the distance of X from the origin from zero here's an example suppose we have this situation here's 0 here's 5 this distance here is the absolute value of 5 because it is the distance 5 is from 0 now we can actually calculate that because we know how far away 5 is it is five units away all right maybe you think that was too easy how about this suppose zero is here and we go to the left to say the number minus 4 now this is the distance of minus 4 from zero the absolute value of minus 4 what distance is that how many units do you walk to get to minus 4 well of course you walk 4 units so the absolute value of minus 4 is 4 we're developing a theory on how to use this notation aren't we we're seeing that if you're to the right then the absolute value of you your distance from 0 is indeed just yourself the positive number if you're to the left of 0 it looks like the left word indicator needs to be removed and you just write the l'absolu value of minus 4 is simply 4 now that's a bit colloquial let's go ahead and write down what we seem to have discovered another way of writing down what that symbol means so this symbol the absolute value of X which remember we might think of as the absolute distance of X that is to say the distance of this number from the origin zero our experience suggests the following it's the same thing as X whatever's inside there if X is a positive number if X is to the right of zero X is greater than zero if X is zero well of course what's the distance of zero from itself zero so if X is zero then the distance should be zero and then first I'll write the if part what if X is less than zero that means left of zero well as we found out with the absolute value of minus four let me bring that back and show you the absolute value of minus four turned out to be simply for the minus had no bearing on it well how would one get rid of a - well you know from your experience with negative numbers that if you take X X is less than zero so it has a minus or a negative internal to it by putting another minus out front that will make it positive and give us the absolute value or distance because distance after R all should be a positive number so just to write it out again this is the absolute value of X I've also called it the absolute distance we're by distance I mean distance from zero and you might just call it that distance of x from zero now we will explore throughout this course other situations where the absolute value is used but if you remember this fundamental idea that is a distance measure it will make all of those calculations easier now that we have distance between an arbitrary point and zero let's worry about what the distance would be between two points where one of them might not be zero so the distance between two points on the real line get that all down there all right and the two points may not include the origin so we might have something new what what I'll do here is we'll explore this again by example and I think we'll see how this is going to work so suppose we have maybe 0 is there and we have two numbers minus 3 and 5 and since I also want to refer to the points remember these are numbers so they are algebraic objects suppose I want to refer to the points where these numbers occur let me call this point P and this point Q it'll make things a little easier later then I ask myself what is the distance between those two the distance well it's very easy to see even though this is not to scale clearly that if I go five to the right there's five units involved and then I go three units to the left the total number of units is eight so the distance between these two points is eight now let's see if we can figure out a way to write that using the notation that we have notice based on that picture let me even redraw it so here is minus three and five all right here's here's zero in here notice the following if I take the absolute value of the difference of the two points - three - five or if you like the absolute value the other way 5 - the - 3 watch what happens - 3 - 5 as you know adds up to minus 8 and how far is minus 8 away from zero well 8 units on the other hand 5 minus a minus 3 that's 8 inside these absolute value lines and how far is 8 from zero also 8 so it looks like I might have an aha moment here and I might have something I can use as a definition so let me take that idea and write down what I think is a good definition definition if I want the distance between the points P and Q and for the sake of argument let me draw them again up here here's P say here's Q and maybe the number associated with P is a the number associated with Q is B I could say that that's the absolute value of B minus a or if I like it the other way the absolute value of a minus B remember in the previous example we saw that it didn't matter which way I did it I got the same distance so that would be the same as saying this is the distance from Q to P measuring the other way because it doesn't matter which way you measure it it should be the same number and so that's what we'll do we'll take this or like this either one of these as the definition of the distance between two points so for example just to toss in one final example if I want the distance between say minus 11 + 42 I simply take the absolute value of the difference and it doesn't matter which way I do it - 11 - 42 will work what is that that's the absolute value of minus 53 and how far is minus 53 from 0 53 units and there's the distance I was interested in so that's all that's involved in finding the distance between two points on the real number line real members as decimals reals as decimals now decimal numbers have been around for a very long time nowadays they're a lot more common because we use calculators a lot and so the numbers that we look at often appear in decimal form you should know some basic ideas about real numbers written as decimals so let me see if I can outline a few of those ideas for example let's look at first of all rational numbers rational x' alright we'll see what they look like in decimals and see if we can detect something a principle here also let me remind you what is decimal mean well the part that says deci refers to base 10 that refers to 10 so we're talking about a base 10 that's where we have the digits remember 0 1 2 up to 9 so I'm going to know about what rationals look like in decimal form so let me take a simple rational like 3/4 now how do I put this in decimal form you remain remember that the idea is to do a long division so let's go ahead and walk our way through this let's divide 4 into 3 and let's see if we can resurrect some of your experience I'll put a decimal point after 3 and a couple of zeros I don't know how many I'll need but that'll do for now I can add more if I need them and then I ask myself what would I multiply by 4 to get 3 well I can there is no number so I then look at 30 and I ask myself what could I multiply by 4 to get 30 or close to it well I will put a decimal above the decimal that is underneath multiplying 4 by 7 gives me 28 if I then subtract that number from 30 I have a remainder of 2 I bring this 0 down and I have a new number 20 this is the way the algorithm for long division works if you recall now I keep this up until I end up with a repeating set of digits up here or I stop now watch what happens next I have 20 here what could I multiply it by 4 to get 20 well 5 will do the trick 5 times 4 is 20 I then subtract that away and I'm left with nothing so I can stop there but I can also think of this is now a trivial division because all the other numbers will be 0 so what do I have I have that 3/4 is equal to 0.75 but what I really have is 0.75 0 0 dot dot dot the zeros repeat so I have what apparently is repeats okay now that's the first observation let's go ahead and look at a little more complicated number it doesn't look that bad when we write it down 1/7 but when we do the long division it takes a little effort so I wanted to work through one long one to give you a sense of how this works so I'm going to divide 7 into 1 well that can't be done so I'll put a dot there and then I'm going to need a few zeros I don't know how many they'll be just write a few of them there put a decimal point above now I realize that if I multiply 7 by 1 I get 7 if I subtract that away from the 10 above I end up with 3 then I carry the zero down I have 30 now I repeat the process four times seven will give me 28 if I subtract that from 30 I get to bring the zero down I have 20 multiply 7 by 2 I get 14 with a 6 remainder bring the 0 down I have 60 bear with me if I multiply by 8 I get 56 subtract that away I get 4 still going bring a 0 down 40 if I multiply by 5 I get 35 subtracted away I get a 5 left over looks like I'm going to run out of zeros on a minute 50 and then I multiply by 7 7 times 7 is 49 I subtract away 49 and I get one and now I recognize something notice that the first thing I worked with was a1 and now I've repeated so now with one and then bringing another zero down I'm going to have zero if I subtract away seven I'm going to repeat the process I just repeated here so it looks like although I have six different numbers that's not going to change forever this is now going to repeat so I'm running out of space here let me write that down a little more compactly so 1/7 is equal to point one four two eight five seven and then that repeats forever now you may remember a compact way of saying that that repeats forever is to put a bar over those six numbers that means that this just continues to repeat so once again we have that this repeats well that's only two examples but that certainly does suggest something in fact the following fact is true rationals have the following property rationals say Q using that notation is the set of repeating decimals you take all the decimals that repeat including the ones that seem to end remember we looked at 0.75 which was 3/4 and you might say well that's not a repeating decimal well sure it is if you consider the repeating number to be zero so repeating decimals include and are exactly equal to the set of all rational numbers well it makes it easier to guess what the set of Irrational's might look like the set of Irrational's which doesn't have any standard symbol would be the set of non repeating decimals so for example in that set of non repeating decimals is the number square root of two which is a famous irrational it starts off one point four one four two three or two one three excuse me etc there's another famous irrational pie which you probably know 3.14159 dot dot dot now I'm not going to kid you it is difficult to tell if a given number is an irrational number or not because you have to determine somehow that the decimals don't repeat and no matter how many you write out that's not enough to check that so determining whether a number is irrational or not is a difficult problem but this at least gives us a way of categorizing rationals and Irrational's one final thing before we leave the real number system I want to point out an ambiguity so let me go ahead and write that down this is another warning and this will be the last thing I say about the real numbers today warning and ambiguity of decimals this is a situation where the numbers that you think are always going to stand for the same thing don't and it's a little surprising if you've never seen this before I'll do this by example so you'll see what I'm going to deal with I am going to show you that three point nine nine nine where the 9s go out forever it is exactly the same number is four point zero zero zero out forever these are the same number they look different and you might suspect that they are different based on that but in fact they are the same number how will I do this well I'll do a verification meaning I will show you a little calculation that I hope will prove to you that these are the same all right starting at the top of this page let X be the three point nine nine nine out to infinity number in fact I'll box that in what I'd like to end up with at the bottom of my page is that X is also equal to four that will show you that those two numbers are the same here's the way the argument goes then multiply X by 10 you will get 10x on the left on the right what happens well you remember from decimals when you multiply a number that's in decimals by 10 it moves the decimal point over by 1 so this becomes 39.99 9 then I still go out forever now from this 10x let me subtract the X number which is 3 point 9 9 9 out forever on the left of this 10x minus X gives me 9x on the right 3 from 39 gives me 36 and the point 9 9 9s cancel each other out they subtract away so 9x equals 36 x equals 36 over 9 but 9 goes into 36 so x equals 4 and look at that X started out being 3 point 9 9 9 and now I've got X equal 4 that means those two numbers are the same surprisingly so that's an ambiguity of our decimal system where two numbers can appear to be different it only happens in this specific case where there's an infinite number of nines but it is enough to give you pause let me show you one more example and then we'll stop talking about real numbers another quick example looks like this just to show you what can occur inside a number point seven three two one five nine nine nine nine out to infinity this is going to be the same as point seven three two one six zero zero zero zero so the five nine becomes a six zero and so these two numbers are the same even though they look different and of course you'd like the second one better because you can call it a decimal that has zeros at the end so you can stop it after the six you don't have to worry about rewriting the nines forever all right well that's all I'm going to say now about real numbers it's time for you to do a little work and practice with some of the concepts about real numbers

Channel: UMKC

Views: 730,190

Rating: 4.8716631 out of 5

Keywords: sets of objects, finite set, Infinite set, subset, Natural numbers, prime numbers, Integers, Rational numbers, Irrational

Id: 1Amt_-uB9QQ

Length: 79min 34sec (4774 seconds)

Published: Mon May 04 2009