Lesson 1 - Real Numbers (Pre-Algebra Tutor)

hi and welcome to the pre-algebra tutor now I know everyone watching this video is coming from a background where you know how to add you know how to subtract you know how to multiply and divide you've had some exposure to fractions you know you kind of have some idea about how to manipulate fractions and decimals and things like that and your basic math and then you get into this topic of pre-algebra and then after that of course you're gonna go into algebra 1 and algebra 2 and all this other math that follows and the tendency when you start a course titled pre-algebra is to maybe get a little frightened because the title of it sounds difficult I mean the word algebra it just kind of has a natural ring to it of something that maybe is difficult well I'm here to tell you that algebra and of course the pre-algebra that we're gonna do here in this course it's not difficult at all it really isn't it's just a bunch of rules that you have to fully understand and once you know those rules doing the stuff is not really difficult at all it's just like learning a language there's lots of rules with language how to make a sentence how to speak how to make verbs and nouns and all that stuff and you know all this because you were you grew up and you learn how to speak by listening to other people well in this course the pre-algebra course and all the other courses that follow we're gonna learn in exactly the same way we're not gonna learn with a bunch of boring lectures we're going to learn by doing problems so you're gonna see a lot of problems and I'm going to explain the topics of course but we're gonna have a lot of problems up here that are going to show you how this works by examples so you're gonna see how it's done by basically doing it by following along and things like that now what I'm going to encourage you to do for every section of this course is to watch the whole lesson through without pausing the video a whole lot of times I don't want you to have a note sheet out trying to take notes I don't want you to try to pause and work the problems in between in between me and working the problems I want you to watch and absorb just like you do when you were two years old or one year old and just learning how to talk you didn't have any note paper out then you were just listening to everybody talk now after you think you understand the topic of course you can watch the video again you can pause and work out your problems then of course but I don't want you to do that the first way through so what is this topic of pre-algebra anyway I want to give you a little bit of motivation you are about to embark on a really awesome journey because your basic math your fractions and all that stuff very very important absolutely essential to everything that we're doing in this course and all the other future courses but really algebra is when you first are able to get the tools to be able to solve more complicated problems you know how do how did people you know invent the first airplane or maybe invent the first spacecraft or invent you know anything you know that could go into the real robotics that can travel and go inside of a burning building and retrieve things and and anything like that in the modern world has all really been designed by scientists or engineers or mathematicians work on these things and they all have to use math and because because of that algebra is the very first stepping stone to begin being able to have those tools to do those things and the reason it's so powerful really is because algebra and pre-algebra they're all really about solving for things that you don't know you you always have a problem where you're trying to find the answer to some question and so you set the problem up you may not know what the answer to that question is in the beginning but by solving the equation or setting up the equation that you you're trying to use to solve this the situation that you have you'll find the answer it's a really powerful tool so we're gonna start with baby steps I'm not going to assume you have any idea what algebra is or what pre-algebra is and we're gonna take it one step at a time now before we get into this section this section is titled real numbers and we're going to talk about what a real number is and all that stuff it's just basically learning some definitions of numbers unfortunately you have to cover this in the beginning because every pre-algebra book is gonna cover real numbers off in the very beginning so I'm going to put this at the beginning as well but before I kind of dive into the topic of this section I want to just go and say algebra pre-algebra has this reputation as being difficult in my opinion for two reasons students get put off for two reasons one it's because algebra or pre-algebra it's all really the same thing so if you hear me say the word algebra I mean we're learning algebra in this class it's just the building foundation for for what we're gonna learn in algebra but two things that trip students up and make them think this is difficult is the concept of a negative number you really have never used a negative number before really learning anything about algebra too much anyway but negative numbers and I'm just going to give you a quick little definition a negative number is not a difficult thing to understand basically if I tell you that you have positive three pencils you have positive three pencils how many pencils do you have well you have three pencils I mean that's what it is you have three pencils in your hand we always use those numbers but they're always positive we always knew they were positive we just didn't say the word so if I have a positive three pencils that's how many pencils I have three of them now if I say that instead of three pencils I have negative five pencils negative five pencils the negative is basically telling me that I don't really have any pencils at all in fact I actually owe you five pencils that's what that really means so when you see negative numbers I want you to think about it in terms of what it really is it's not a magical thing that's this mystical number that that suddenly has a negative sign in front and it's so difficult it's just telling you that I don't really have that many objects I actually owe you that many objects so and you think about it because that's exactly what it kind of makes sense if I have a negative of something it means I don't really actually have it at all I think I owe you that many of whatever it is we're talking about so if I have negative two sticks of gum I don't actually have any gum I owe you two sticks of gum that's what that means so we're gonna talk about negative numbers off in this course you know a little bit later actually during the whole entire course but I want you to calm down a little bit and realize that it's not that difficult it just means I owe you something that's basically what a negative number is the second thing that trips people up in algebra is the concept of a variable you see X a lot or a or B you see that running around the textbooks a lot and people open a book and they see a bunch of negative numbers and they see a bunch of letters and it gets scary because you've never dealt with that stuff when you had well anytime you see a letter in algebra try not to get upset try not to get worried about it what it really is is it's telling you that you don't know what the value of that number is so we may be trying to to calculate something but we don't know the answer yet so we assign a letter to represent what we're trying to find like maybe we're trying to calculate the surface area of this sheet of paper you know the surface area how many square inches or how many square centimeters you know would fill up the sheet of paper if we were to calculate the length and the width we'd multiply them together and we would get the answer but maybe maybe we want to write it down you know and represent the area as the letter a because let's say we don't know what the area is so we might put the letter A and we might use that letter in an equation somewhere because we need to know the area of the object to calculate something else we might represent it with the letter A and so that letter A just simply means I don't know if it's five square inches of area or if it's you know 16 square millimeters of area I don't know what it is I just know that I have the area of something so I'm going to use that variable is what we call it the variable so those are the two things that trip you up and trip people up in algebra and it they're not difficult negative numbers just means I owe you something I don't have that many I owe you that many and a variable is just a letter in algebra that we use because we don't know the value of whatever it is we're talking about but we're going to calculate that in the end and that's what solving equations is all about trying to calculate the value for you know the unknown that's what it is so I wanted to get that out of the way because a lot of people get upset about that when they start to learn algebra and get worried about it so we're gonna cover all this stuff in great detail I just wanted to give you a little bit of a advance preview now getting to the topic here we're going to learn about numbers and so we're going to talk about real numbers so let's just get right to it a real number these are all definitions that you'll see in your book a real number whatever textbook you happen to be using you'll always see a definition for a real number my definitions are not exactly what you're going to see in the book my definitions are more you know layman's terms definitions to kind of get the point across but these are the definitions that make sense to me it's any number that can be located on a number line and we haven't even really talked about what a number line is yet we're gonna get into that very very quickly but basically I'm gonna give you one more sentence I'm gonna put it in parentheses here and I'm gonna put its every number or everything except imaginary numbers okay so basically there are these things called real numbers every number that you've ever used in your whole life has always been a real number every fraction every decimal here we're gonna be talking about negative numbers every negative number every positive number every repeating number pi any number you've ever used has always been a real number because a real number is any number that can be located on a number line and we're gonna draw a lot of number lines in a minute to talk about that the basic idea of a number line is you know you have something like this this is a number line it's basically a line and you have zero here in the middle and here you have one two three four here so these are the positive numbers going this way and this arrow means these numbers go on forever because you know numbers go on and on forever and then here you have a negative one negative two negative three negative four and of course negative five negative six negative seven as you go on this way so these are the numbers you've been dealing with all in your basic math and of course we have these negative numbers because I have a less than zero pencils or whatever I owe you some pencils over here that's what the negatives mean so any number that you can plot here is a real number and so when I have here everything except imaginary numbers basically an imaginary number if you've been exposed to it before great you already know what it is if you have it then don't worry too much about it because it's not the purpose of this course to teach you what an imaginary number is but a real number is every number you've ever used you know everything except for an imaginary number so some examples of real numbers the number four I mean that's the number that you've used all your life the number six the number zero the number negative three the number two point three the number two point six I'm just making numbers up here the number pi which as you know is 3.14159 and it goes on and on forever the number negative one point six six and this little bar over the the digit means that these digits repeat so this is negative one point 666 666 like that on and on forever etc okay so any number at all that you've ever used in your life you know anything except the imaginary numbers the so-called imagined numbers anything you can plot on this line so I could find the number for put a dot there I could find the number 6 over here I could find the number 0 pi is 3.14 so here's 3 and here's 4 so 3.14 would fall right in between there negative 1 point 6 6 6 would be in between negative 1 and negative 2 etc and we're gonna do a lot of plotting with the number line here in a second but I just want to tell you a real number is basically any number that you could plot decimal fraction because 1/2 is 0.5 you could plot that right there so that is a real number all right now the next one is one that you probably haven't heard of too much until now it's called an irrational number irrational number now what do you think of when you hear the word irrational irrational number what do you think of when I think somebody's irrational I think that they're a little bit crazy they're not really with us they're they're really doing their own thing they're kind of unpredictable that's what irrational means something that doesn't quite behave right that's what irrational really is so irrational numbers are numbers that really don't quite behave right that's exactly what they what they are so let's write the definition there are numbers that cannot it cannot be written as a fraction they cannot be written as a fraction and let me go and write one thing of parentheses here we'll change color to make it clear and these are gonna basically be non repeating infinite decimals all right so some examples of an irrational number it's better to sort of just show you by example with what they are some examples of irrational numbers they're very simple they're things like pi because pi is equal to 3.14159 bla bla bla bla bla bla bla bla and it goes on and on forever and these digits that come after pi if you got a computer and you actually looked at the digits of pi out to a thousand decimal places or a million decimal places you would find that these digits never repeat they never follow any pattern it's not like you know it'll say one five four or five nine one five four five nine one five four five nine no no it pi doesn't behave like that it's a very special number if you look at all the digits they never actually ever repeat ever and believe me people have been trying to calculate pi out to a billion decimal places to see if it ever does repeat and it never does so we say pi is irrational because it kind of goes on and on like this there's no pattern to it it's not repeatable and you cannot write pi as a fraction some of you guys may think that or may have heard that you can write pi as a fraction I think it's 22 over 7 that's not really true that's that's an approximation the number pi cannot be written as a fraction and that's why it is irrational another another number that you just an example number that you cannot write as a fraction is the square root of two if you take the square root of two and we're going to talk a lot about these things a little bit later too if you take the square root of two that's going to be something like 1.4142 if you don't quite know exactly what square roots are yeah we're going to get to this a little bit later but this is a number you can put it in your calculator and you press the square root button and your gonna see a bunch of digits that go here and if you calculate those digits on and on and on forever you'll see there's no pattern there's no repetition and you definitely cannot write this as a fraction the only way to get this number is by is by this so you can see that my list is pretty short there's lots of other irrational numbers but they're not things that really we deal with every day we see pile all the time and that's why I put it first because that's the one you actually know about though that's the one irrational number that most people have ever actually seen before that's why they're called irrational because you cannot write them as a fraction they're non repeating infinite decimals and that's what we have here so if we have irrational numbers that are little weird wacky numbers that don't quite obey what we you know think is normal then the next type of number you might guess would be the rational numbers so let's go and take a look at at that so the rational numbers all right now what do you think of when someone is rational or something is rational the word rational means they're well-behaved they obey the rules they are things that have a nice nature to them and they're very calm cool and collected and they're rational so these numbers should be numbers that are quote unquote well behaved type of numbers and so that's exactly what they want rational numbers are numbers that can be written as a fraction all right so they can be written as a fraction if they're rational and in fact almost all numbers can be written as a fraction in one form or another so most numbers are actually rational most numbers are actually rational so let's look at a couple of them what about the number 0.5 is this number rational well it can be written as 1/2 so it is rational you can write that as a fraction the number is 0.25 is that rational well you can write that as 1/4 so it can be written as a fraction so it's rational what about the whole number 2 well you can write this as 2 over 1 so that is a rational number because it can be written as a fraction what about the number negative 5 well you can write that as negative 5 divided by 1 5 5 over 1 so it is it is rational what about something a little bit weird like zero point six six six six and then you put a little bar so the sixes go on forever well it looks like that is irrational but in fact this decimal can be written as two-thirds if you put 2/3 in your calculator that's what's gonna pop out so that is rational this is actually rational what about the fraction one-third well of course it's already fraction so it's rational and what about the fraction negative 7/8 well it's already a fraction it's a negative negative has nothing to do with it it's if it can be written as a fraction negative fraction counts just as much as a positive fraction so of course it's already in fractional form so you see pretty much any number that you could possibly think can be written as a fraction even the weird decimals as long even if they're infinite decimals if there's any kind of pattern to it like this one is a pattern because it's got six is going on forever but you have other decimals out there like you know zero point one five four one five four one five four one five four one five four on and on forever that has a pattern to it anything like that that has a pattern you will be able to write as a fraction now I may not be able to tell you what that fraction is but it is going to be able to be written as a fraction it's only when the numbers are kind of going off squirrely with no pattern at all and there's very few of those that we really deal with every day that's we call irrational most members are actually rational which is what we have here on the board okay now the next thing I want to talk about are what we call integers now we are going to be talking about the word integers a lot I mean in fact all of these numbers are important but we're going to be using the word integer a lot don't let it scare you integer is a positive negative or zero number with no decimal now like I said in your book you're probably not gonna see that definition this is my definition but your book will have something similar so for instance you might have the number or 0 1 2 3 4 5 dot dot dot dot in other words 6 7 8 9 10 11 12 on and on and on to infinity and then of course on the negative side we've got negative 1 negative 2 negative 3 negative 4 negative 5 dot dot dot so integers are really all of these numbers that don't have any decimal point I mean in other words 4.5 is not an integer 1.2 is not an integer 0.3 is not an integer integers have to be these numbers that are these nice whole numbers it can be negative it can be positive it can be 0 those are called integers and you'll see we'll just use this term a lot now algebra you'll see it coming up but that is my definition of what an integer is I think it's a good one just to show you what they really are there's nothing more to it than that it's just basically the numbers positive or negative without any decimals a whole number now the next thing here the next couple of things we're going to talk about here a whole number you know it's useful to know these only because they're in your book and because you're gonna see them on a test but really you're not really talking too much about whole numbers I mean you'll just see why in a second a whole number are the positive integers and you also include 0 so 1 2 3 4 5 . so when you say something is a whole number it means the positive integers and also you kind of include 0 in there so it's just the definition I mean there's nothing special about whole numbers it's just a label integers are the positive and the negative numbers without any decimals and you also include zero whole numbers are the positive ones if you also include zero and there's also another one that your problem that you will see in your book let me switch colors here and those are the natural numbers all right the natural numbers and those actually let me take this off like that the natural numbers just to show you by example those are 1 2 3 4 5 6 dots so basically these are the positive integers and you don't include 0 that's the only difference between a whole number and a natural number the reason they're called natural is because when you're a kid and you're counting jellybeans if you have a physical object here in front of you like this pencil or this marker so I have one marker and then if I have a pile of markers I'll start counting them 1 2 3 4 5 so those were considered natural because those are the the numbers that use to count things typically 0 even though it's a it's a definitely an important number you're not counting objects with the number 0 because usually if you're counting something you have more than 0 so that's why they call them natural I guess what I'm trying to say is these 2 things down here I mean they're useful to know for your test but you're really not gonna use them too much in algebra you know you're not going to be asked too much in the future you know hey go ahead lists the whole numbers here you know in this problem it's just not something you're really gonna see but it's a definition integers you will see a lot and so that's super important the main thing to take away from these boards here is that the world is divided into the real the real numbers of the general umbrella of all numbers that you could possibly plot on a number line and those are divided up into the irrational which are numbers that you cannot write as a fraction and then also the rational numbers that you can write as a fraction and then getting even more specific the integers are the numbers here like you see here with positive or negative numbers without any decimals in 0 whole numbers are the positive numbers including 0 and natural numbers are the positive numbers when you don't include zero I'm gonna draw a little picture for you here in a second that's going to hopefully make that a little bit clearer now the next topic or the next the next type of number is something that's useful again you'll you'll see it on your test you'll see it in your pre-algebra course but I'm not gonna sit here and lie to you it's not something that you're going to use in every section coming up here and the rest of the algebra sections but it's something that you probably have heard about and it's called a prime number in the only reason that books really even talk about prime numbers is because they're kind of special numbers a prime number is a whole number other than zero a little line through it means zero so you don't confuse it with the letter O zero and one other than the number zero and one that is divisible by only the number one and itself and every time you write the definition of a prime number most people their eyes cross and they just don't quite get it because it's kind of a complicating sounding thing it's whole numbers other than zero and one that is divisible by only one and itself it just doesn't seem to make a lot of sense when you write it down but but you'll see why it's written this way let me give you an example of some prime numbers and the number two is a prime number the reason it's prime is because the number two is only divisible by the number by the number one you know when I say it's only divisible I mean only divisible with no remainder so you can only divide this number by the number one because two divided by one is gonna give you two and you can divide it by the number two which is itself two divided by two is one you can divide it by anything else two divided by three gives you a decimal two divided by 10 gives you a decimal 2 divided by you know 4 gives you something else you can only divide it by the number one and the number two which is itself that makes it a prime number the number three you can only divide this number by let me go across here the number three you can only divide it by the number one and the number three you can't divide this number by two because that would be three halves that's a decimal you can't divide it by the number five because if you do you're gonna get a decimal answer when we say it's divisible by we mean divisible where you don't have any decimal left over so three can only be divided by one and itself five can only be divided by five and 1/7 can only be divided by seven and 111 can only be divided by eleven and 113 can only be divided by thirteen and 117 can only be divided by 17 and 119 and can only be divided by 19 and 123 can only be divided by 23 and 1 and you could go on and on and you can get a computer to calculate all the prime numbers out as far as you want to go and that's fine people do that but just know that it's sort of an infinite series of numbers that have these special properties now let's look at something that's not prime just to kind of give you an example 8 the number 8 is not prime and the reason it's not prime is because I'll just put because be EC you can divide it by 2 and you can divide it by 4 you know 8 divided by 2 gives you 4 and 8 divided by 4 gives you 2 so those are two additional things besides the number 1 and the number 8 you can divide into there and so it's not prime the number 10 is not prime because you can divide a number 10 by 2 and you can also divide it by the number 5 evenly one more example the number 12 is not prime because you can divided by a whole lot of numbers you can divide it by 2 you can divide it by 3 you could divide it by 4 and you can divide it by 6 so you see the higher up you go in the numbers you're usually going to be able to divide by lots of different numbers that will divide evenly but every number in this the only numbers you can divide into there is the number one and itself which itself is the number itself okay now also notice that prime numbers exclude zero and one from being prime by definition that's just a definition of the prime number so that's what it is ladies and gentlemen those are the types of numbers that we're going to learn in this section let me draw a quick graphic because I think it's going to make things a little bit clearer and then after that we will work I guess a couple of practice problems to kind of show us where we're going so first we have what we call the real numbers right which is basically everything everything except for what the so called imaginary numbers which you typically don't learn in algebra but just keep in the back of mind that there's something called an imaginary number you've never used it till now it's something that's kind of a little more of an advanced concept but everything else leftover is there is what we call real that's why they're real we can touch them it can we can use them to count marbles things like that and a real number is basically everything so that's the top level of the hierarchy and coming out of the real numbers you can kind of split this into the rational numbers right the rational numbers and of course the irrational numbers right now just to refresh your memory something like an irrational number it would be like the number you know hi 3.1415 no I know that you cannot write this as a fraction all of the rational numbers can be written as fraction all of the irrational numbers are just these random decimals that go on and on forever and you can't actually write them as a fraction let's change colors now this path kind of ends right there now the rational numbers can be broken down further so the rational numbers you have what we call the fractions right so that would be you know just waiting like you would think 1/2 2/3 negative 1/4 God got thought just any kind of fraction that you would think of is irrational because remember rational numbers can't be written as a fraction so of course the fractions are rational right and then you have decimals now when I say decimals these are decimals that that have some kind of pattern to them even if they go off to infinity they have some kind of pattern or they just terminate so something like 0.25 0.33 3 bar on top so 0.25 can be written as 1/4 is 0.33 3 3 3 3 3 off to infinity that can be written as 1/3 any of these decimals that have any kind of infinite pattern or just a decimal that stops can be written as a fraction so it's rational and you also have what we call the integers and those are things like you know negative 2 1 0 1 2 done that dot you know in both directions right so all the negative numbers there all the positive numbers there that are don't have any decimals and also the number 0 those are integers right and any of these integers can be written as a fraction also because the number 2 can be written as 2 over 1 the number negative 2 can be written as negative 2 over 1 so you see that the world really consists of these real numbers that are broken into two camps the rational numbers can all be written as fractions and they would consist of you know the fractions because they're already fractions the decimals that can be written as fractions and the integers that can all be written as fractions - so pretty much every number you've ever used is over here somewhere right the only number is a little weird are these things called irrational numbers and there's not too many examples of them that you really are already familiar with but just know that they're out there something like pi you could probably go and calculate different different irrational numbers without a problem and those are numbers that just go on and on forever with no real no real patter to them and you cannot write them as a as a fraction that's really the definition let's erase the board and let's get some practice with identifying these different types of numbers okay now what we're going to do now is write some numbers on the board we're gonna write them over here and then we're going to go ahead and look and try to identify what type of numbers that they are and put a little check mark in the different you know in the difference in the different columns and you'll find out that most numbers have check marks in different columns more than one column because you can see from our little hierarchy tree that we had over there a number can be you know can have different labels associated with it so let's go ahead and do that let's start with the first number the first number was saying the number 30 and let's just go right across the top and see if that that is uh there so it's a real number because as we said before pretty much any number that we have that doesn't have an I in there for imaginary is always going to be real rational is it rational can it be written as a fraction well of course it can because 30 divided by 130 over 1 is is possible is it irrational no it's not irrational because it's not an infinite decimal it goes on and on forever is it a whole number yes it is because a whole number remember are the numbers that are 0 1 2 3 4 5 6 and then go on like that including 0 okay is it an integer well yes it is an integer because remember integers are all the negative numbers all the positive numbers without any decimal and including zero so that is an integer is it a prime number well it's not a prime number because I can divide 30 by the number 10 I / the number 15 ICANN / lots of different numbers and still have a even division so this number is real rational whole integer and we can even have more columns with natural number and all this other stuff they're just labels I'm just trying to give you a little bit of practice here alright what about the number 0 well the zero is a real number just like we said pretty much any numbers real unless it has an imaginary thing attached to it it's rational because you can write it as a fraction the number 0 / 0 / 5 is going to be equal to 0 0 / 10 it's gonna be equal to 0 so you can write it as a fraction so it's rational it's not irrational basically a number cannot be rational and irrational at the same time either you can write it as a fraction or you can't that's what that boils down to it is a whole number because whole numbers are the numbers 0 1 2 3 4 5 and on up it is an integer because integers include 0 and prime number is excluded from being 0 or 1 and we talked about that when we talk about prime numbers all the prime numbers there they don't count 0 & 1 that go up from there the next one I want to talk about 0.5 0 is it real of course it's real pretty much all these are gonna be real is it rational can it be written as a fraction well yes because we can write it as 1/2 it's not irrational because it's gonna be one of the other you can't you you know you can write it as fraction so that's that's not gonna be there is it a whole number well no it's not a whole number because whole numbers are 0 1 2 3 4 5 etc integers are all the positive and negative guys but they have to have no decimals so it's not an integer is it a prime number well it's not a prime number either because those are all of those are whole numbers as well that that are divisible by itself and the number 1 so the number 0.5 is only real and rational ok the next let's talk about something a little different negative 3/8 so we have a fraction here negative 3/8 it's a real number is it rational well yes it is because it can be written as a fraction it's already as a fraction it's not irrational for the same reason it's not a whole number because those are you know 0 1 2 3 4 5 whatever it's not an integer and to not prime because these are all of these are basically numbers like 0 1 2 3 or 4 5 they can't have fractions or decimals involved so this is just real and rational ok let's look at the number 1 it's real it's not imaginary so it has to be real it's rational because you can write the number 1 as 1 over 1 it's not irrational for the same reason it is a whole number because whole numbers are 0 1 2 3 4 5 6 7 on up it is an integer because those are the positive and the negative numbers with no decimals so that fits into that category it is not prime though because prime numbers from our definition we just had on the other board cannot include 0 & 1 so it's not prime all right now let's look at the number pi is it a real number yes it is real because is there's nothing imaginary about the number pi there's no I there is it rational no it's not because you cannot write the number pi as a fraction is it irrational yes it is our first irrational number is it a whole number well no it's not because it's got a decimal involved is it an integer no because it's got a decimal involved is it a prime number no because it's got a decimal involved those last three columns are only going to apply to numbers they don't have any decimal so that's basically it pi is real and irrational let's look at the number 2 point 1 2 just the decimal 2 point 1 2 is it real oh yeah it's real doesn't have any imaginary parts to it is it rational and I'm gonna tell you yes it is rational you can write it as a as a fraction and you may not know right away if these decimals are written as fractions but the way the way that you really look at it is if you have a number that has a decimal that just stops eventually like after 2 or 3 decimal places or whatever you can always write it as a fraction I mean in this case for instance this guy 2 point 1 2 you can write it as 212 divided by a hundred or two hundred twelve over a hundred because if you put this in your calculator it's gonna move the decimal point two points back and it's gonna give you what you're looking for if you if you dump this in your calculator you find that it equals this so you can write it as a as a fraction but you're not always going to be able to think of the fraction that that you can write it as but you know that if it if it stops or if it has a repeating pattern of any kind like two-point 105 105 105 105 any kind of repeating pattern you will be able to write it as a fraction all right so it's real and it's rational it's not irrational because we can write it as fraction and it none of these apply because you know we have we don't have it in kind of we have a decimal point involved what about the number one point one one and we'll put a bar over the one one so what this means is it's one point 1 1 1 1 1 1 11 out to infinity it's real because there's no imaginary number involved and I'm telling you based on our definition that it is rational because there's a pattern here it does go on forever but it it the pattern is that all is all once so there is a fraction that would equal that no I don't know what it is but it's just sort of a mathematical truth that it that it's going to be that way it's not irrational because you can read it as fraction it's not any of these whole number integer or anything else because there's a decimal involved what about negative 2 point 5 well yeah it's real doesn't matter if it's negative or not it's a real number it doesn't have any eye or imaginary part to it it is rational just like I said the decimal point stops now in this particular case I know how to write that as a fraction you can write that as negative 5 halves if you put 5 and divided by 2 in your calculator negative sign out front you'll get negative two point five okay but as I said you don't need to know the fraction that that's equal equal to you just need to know sort of the rule and none of these other things apply because you know there's a decimal involved now what about negative square root of 2 if you get the square root button on your calculator and hit put 2 in there hit the square root stick a negative out front it's real I mean it doesn't have any imaginary parts to it it is not rational though because if you look at it the numbers that you'll see in your calculator even if get a computer you'll find that there's never any pattern so you there's no way to write that as a fraction and so it's irrational and it's definitely not a whole number or an integer or a prime number because there's decimals involved there and what about the number five well it's real it is rational because we can write it as 5 over 1 it's not irrational for the same reason it is a whole number because we can you know whole numbers are 0 1 2 3 4 5 and on and on it is an integer for the same reason because those are positive and negative numbers with no decimals it is a prime number also because the number 5 can only be divided by the number one and the number 5 itself I can't be divided by anything else and if you try to divided by anything else you're going to get a decimal so that's basically it you see these numbers can have multiple labels they can have multiple labels and that's because there's overlap the integers have overlapped with the whole numbers I mean the integer is more broad there and of course every prime number is also going to be an integer and etc you'll just sort of see that the only thing that is sort of going to be one or the other is going to be rational versus irrational either you can write it as a fraction or you can't so that concludes this section I know it probably wasn't the the most thrilling section just to learn about numbers and how to classify them but you know every book starts with this material just because everybody has to know the definitions you know math is just like a foreign language you have to know the definitions if you do then when someone says irrational number you won't you won't worry about it you'll understand what they're talking about and so you'll do well if you don't know what an irrational number is or a prime number then when you see that on your test you're gonna get you know stressed out upset worried and then you're not gonna do well because you're worried about it so make sure you understand this material it's important for the sections that follow we're gonna take you baby steps one step at a time through this class so that by the end of it you know coming in here you have no idea what algebra is about coming out the other end you're gonna have a fair amount of really good pre-algebra skills under your belt ready to tackle algebra 1 and do really well in those courses I'm Jason I hope you've enjoyed the lesson let's go on into the next one and continue on through the course

Channel: Math and Science

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Keywords: prealgebra, pre-algebra, real numbers, pre-algebra review, math, algebra, help, solve, education, teaching, learning algebra, algebra 1, equations, pre algebra videos, prealgebra and adding integers, integers, rational numbers, irrational numbers, natural numbers, whole numbers, mathematics, numbers, properties of numbers

Id: VFyfQmmsgec

Length: 44min 0sec (2640 seconds)

Published: Wed Feb 03 2016